The Eleven Dimensional Supergravity Equations, Resolutions And

The eleven dimensional supergravity equations, resolutions and Lefschetz fiber metrics by Xuwen Zhu B.S., Peking University (2010) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2015 c Massachusetts Institute of Technology 2015. All rights reserved. ○

Author...... Department of Mathematics Apr 5, 2015

Certified by...... Richard B. Melrose Simons Professor of Mathematics Thesis Supervisor

Accepted by ...... William Minicozzi Chairman, Department Committee on Graduate Studies 2 The eleven dimensional supergravity equations, resolutions and Lefschetz fiber metrics by Xuwen Zhu

Submitted to the Department of Mathematics on Apr 5, 2015, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics

Abstract This thesis consists of three parts. In the first part, we study the eleven dimensional supergravity equations on B7 S4 considered as an edge manifold. We compute the indicial roots of the linearized× system using the Hodge decomposition, and using the edge calculus and scattering theory we prove that the moduli space of solutions, near the Freund–Rubin states, is parameterized by three pairs of data on the bounding 6-sphere. In the second part, we consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result ofObitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous with integral powers but possible multiplicities, at the preimage of the singular fibers in terms of parameters of size comparable to the length of the shrinking geodesic. This is joint work with Richard Melrose. In the third part, the resolution of a compact group action in the sense described by Albin and Melrose is applied to the conjugation action by the unitary group on self-adjoint matrices. It is shown that the eigenvalues are smooth on the resolved space and that the trivial tautological bundle smoothly decomposes into the direct sum of global one-dimensional eigenspaces.

Thesis Supervisor: Richard B. Melrose Title: Simons Professor of Mathematics

3 4 Acknowledgments

First and foremost, I would like to express my sincere gratitude to my advisor, Richard Melrose, for introducing me into this field, suggesting interesting problems and pro- viding tremendous support for my Ph.D study and research. His deep insight, patient guidance and encouragement over the years has been invaluable to me. As noted below, this thesis is in part joint work with him. I am deeply thankful to Rafe Mazzeo for inspiring discussions, András Vasy for conversations on related problems for future works, Michael Singer for our mathematical discussions and his help along the way, Robin Graham for illuminating comments on the supergravity project and Colin Guillarmou for his interest and comments on this thesis. I would like to thank Victor Guillemin and Tomasz Mrowka for agreeing to serve on my thesis committee and their help during my Ph.D study. I would also like to thank Gigliola Staffilani and David Jerison for their constant support and guidance during my five years at MIT. I am grateful to Fang Wang, Lu Wang, Linan Chen, Michael Eichmair, Chris Kottke, Ailsa Keating, Ting Zhou, Kiril Datchev, Semyon Dyatlov, Shelby Kimmel, Lenore Cowen, Emma Zbarsky, Boris Hanin, Zuoqin Wang and Dean Baskin for being great friends and mentors. To all my other friends here, I wish to express my appreciation for their support. Finally, I would like to thank my family, especially my parents, for their love, understanding and constant encouragement.

5 6 Contents

1 Introduction 11 1.1 Eleven dimensional supergravity theory on edge manifolds ...... 12 1.2 Resolution of the canonical fiber metrics for a Lefschetz fibration . 18 1.3 Resolution of eigenvalues ...... 23

2 The eleven dimensional supergravity equations on edge manifolds 27 2.1 Introduction ...... 27 2.1.1 Equations derived from the Lagrangian ...... 29 2.1.2 Edge metric and edge Sobolev space ...... 30

2.1.3 Poincaré–Einstein metric on B7 ...... 31 2.1.4 Main theorem ...... 32 2.2 Gauged operator construction ...... 35 2.2.1 Equations derived from Lagrangian ...... 35 2.2.2 Change to a square system ...... 37 2.2.3 Gauge condition ...... 37 2.3 Fredholm property of the linearized operator ...... 41 2.3.1 Linearization of the operator 푄 ...... 42 2.3.2 Indicial roots computation ...... 46 2.3.3 Fredholm property ...... 52 2.3.4 Large eigenvalues ...... 54 2.3.5 Individual eigenvalues with 휆 = 0, 16, 40 ...... 58 ̸ 2.3.6 Individual eigenvalues with 휆 = 0, 16, 40 ...... 61 2.3.7 Boundary data for the linear operator ...... 62

7 2.4 Nonlinear equations: application of the implicit function theorem . . 67 2.4.1 Domain defined with parametrization ...... 68 2.4.2 Regularity of the solution ...... 72 2.5 Edge operators ...... 75 2.5.1 Edge vector fields and edge differential operators . . . 75 2.5.2 Hybrid Sobolev space ...... 76 2.6 Computation of the indicial roots ...... 77 2.6.1 Hodge decomposition ...... 77 2.6.2 Indicial roots ...... 78

3 Resolution of the canonical fiber metrics for a Lefschetz fibration 87 3.1 The plumbing model ...... 92 3.2 Global resolution and outline ...... 98 3.3 Bounds on (훥 + 2)−1 ...... 100 3.4 Formal solution of (훥 + 2)푢 = 푓 ...... 106 3.5 Polyhomogeneity for the curvature equation ...... 111

4 Group resolution 115 4.1 Introduction ...... 115 4.2 Proof of the theorem ...... 117

8 List of Figures

1-1 Degenerating surfaces with a geodesic cycle shrinking to a point . . . 19 1-2 Lefschetz fibration 휓 : 푀 푍 ...... 20 → 1-3 The metric resolution ...... 22

2-1 Indicial roots of the linearized supergravity operator on C ...... 48

9 10 Chapter 1

Introduction

Microlocal analysis has a lot of applications in partial differential equations and analysis of problems with geometry and physics background. In this thesis, we apply microlocal techniques and the theory of pseudo-differential operators to two problems on noncompact and singular manifolds. The first one is the eleven dimensional supergravity equations on edge manifolds for which I give a characterization of all the solutions near the Freund–Rubin solution [47]. The second project, in collaboration with Richard Melrose, we work on the complete expansion of the constant scalar curvature fiber metric in the case of a Lefschetz fibration [31], which arises naturally as the singular behavior across the divisors introduced in the Deligne–Mumford compactification of the moduli space of Riemann surfaces.

To get information of operators in a singular geometry setting, people study the Schwarz kernel and the behavior of the model operators on the double space. To study the behavior of the kernel and construct parametrices on this double space, blow up action is introduced [34, 32] and this approach has been utilized to solve many geometric problems [35, 26, 27, 24, 44]. The third part of this thesis contains an example of resolutions, which is an application of the resolution of a compact group action on a compact manifold described by Albin and Melrose [1], to the case of 푈(푛) action on self-adjoint matrices [48].

11 1.1 Eleven dimensional supergravity theory on edge manifolds

Supergravity theories arise as the representations of super Lie algebras in various dimensions. They can be viewed as low energy approximations to string theory with classical equations of motion. In particular, the case of dimension eleven has been studied by physicists since the 1970s [43, 42, 6, 2]. It was shown that there is a unique system in eleven dimensions and the theories in lower dimensions can be obtained from it through dimension reduction [37]. Since it is related to the AdS/CFT correspondence and brane dynamics, this subject has recently attracted more atten- tion [3, 45].

We are interested in a particular case, namely, the bosonic sector of eleven dimensional supergravity theory. The nonlinear system couples a metric 푔 (gravity) and a 4-form 퐹 (extra field), and is derived from a Lagrangian constructed on the eleven dimensional manifold 푋 = B7 S4: × ∫︁ 1 ∫︁ ∫︁ 1 퐿(푔, 퐹 ) = 푅푑푉푔 ( 퐹 퐹 + 퐴 퐹 퐹 ) (1.1) 푋 − 2 푋 ∧ * 푋 3 ∧ ∧

where 퐴 satisfies 푑퐴 = 퐹 . The first term is the classical Einstein–Hilbert action term, while the second and the third terms are, respectively, of Yang–Mills and Chern- Simons type for a field. The supergravity system, the variational equation for (1.1) is 1 훾1훾2훾3 1 훾1훾2훾3훾4 푅훼훽 = (퐹훼훾 훾 훾 퐹 퐹훾 훾 훾 훾 퐹 푔훼훽) 12 1 2 3 훽 − 12 1 2 3 4 푑 퐹 = 1 퐹 퐹 (1.2) * − 2 ∧ 푑퐹 = 0.

As a special case, there is a family of solutions to the full system given by the product of a scaled spherical metric on S4 and an Einstein metric ℎ on B7 with Ricci curvature satisfying Ric(ℎ) = 푐2ℎ. The product solutions are given by − 9 푔 = ℎ 푔 4 , 퐹 = 푐푑푉 4 , 푐 R. (1.3) × 푐2 S S ∀ ∈ 12 In particular for 푐 = 6 these are known as Freund–Rubin solutions [11] with 1/4 of the standard S4 metric and an Einstein metric on H7. When we restrict the search of solutions to a product metric, the following theorem by Graham–Lee [15] showed that the existence of Poincaré–Einstein solutions near the hyperbolic metric prescribed by data at conformal infinity:

Theorem 1.1 ([6]). Let 푀 = B푛+1 be the unit ball and ℎ^ the standard metric on S푛. For any smooth Riemannian metric 푔^ on S푛 which is sufficiently close to ℎ^ in 퐶2,훼 norm if 푛 > 4, or 퐶3,훼 norm if 푛 = 3, for some 0 < 훼 < 1, there exists a smooth metric 푔 on the interior of 푀, with a 퐶0 conformal compactification with conformal infinity [^푔] and Ric(푔) = 푛푔. −

Combined with (1.3), the 4-form 퐹 being a multiple of the 4-sphere volume form gives a family of solutions parametrized by conformal metrics on the bounding 6- sphere. This product solution corresponds to the edge structure in the sense of Mazzeo [25, 23]. An edge structure is defined on a manifold 푀 where the boundary has a fibration over a compact manifold as follows,

휋 : 퐹 / 휕푀 (1.4)

퐵

which, is our case, is the product fibration

휋 : S4 / 휕푀

The space of edge vector fields 푒(푀) is a Lie algebra consisting of those smooth vector 풱 fields on M which are tangent to the boundary and such that the induced vectorfields on the boundary are tangent to the fibres of 휋. Let (푥, 푦1, 푦2, ...푦6) be coordinates 7 of the upper half space model for hyperbolic space H , and 푧푗 be coordinates on the

13 4 sphere S . Then locally 푒 is spanned by 풱

푥휕푥, 푥휕푦, 휕푧.

The edge forms are the dual to the edge vector fields 푒, with a basis: 풱 푑푥 푑푦 , , 푑푧. 푥 푥

The edge tensors and co-tensors are the products of those basis forms, and the solutions we look for are in the sections of the edge bundles.

In Mazzeo’s paper [25] the Fredholm property of certain elliptic edge operators has been discussed. It is related to the invertibility of the corresponding normal operator 푁(퐿), which is the lift of the operator to the front face of the double stretched space 2 푋푒 . If we write the edge operator in local coordinates as:

∑︁ 푗 훼 훽 퐿 = 푎푗,훼,훽(푥, 푦, 푧)(푥휕푥) (푥휕푦) (휕푧) , (1.5) 푗+|훼|+|훽|≤푚 then the normal operator is

∑︁ 푗 훼 훽 푁(퐿) = 푎(0, 푦,˜ 푧)(푠휕푠) (푠휕푢) 휕푧 , (1.6) 푗+|훼|+|훽|≤푚

2 where (푠, 푢, 푦,˜ 푧, 푧˜) is a coordinate on the front face of 푋푒 . The invertibility of the normal operator is in turn related to its action on functions polyhomogeneous at the

2 left boundary of 푋푒 , of which the expansion is determined by the indicial operator, which by definition is

−푠 푠 퐼푝[푃 ](푠)푣 = 푥 푃 (푥 푣) −1 , (1.7) |휋 (푝) where 푝 is a point in the base and 푠 is a complex number. and in local coordinate

퐼(퐿) is written out (using a conjugation of Mellin transform 푀푖푠) as

(︂ )︂ ∑︁ 푗 훽 −1 퐼[퐿](푠) = 푀푖푠 푎푗,0,훽(0, 푦, 푧)(푠휕푠) 휕푧 푀푖푠 (1.8) 푗+|훽|≤푚

14 The inverse of the indicial operator 퐼(퐿)(휃)−1 exists and is meromorphic on the com-

plement of a discrete set spec푏 퐿, which is the indicial roots of L. Those indicia roots provide information of the operator, and more precisely, the parametrix of an edge operator is constructed on the stretched double product where the Schwartz kernel is lifted with polyhomogeneous expansions.

Theorem 1.2 ([25]). If an elliptic edge operator 퐿 Diff푚(푀) has constant indicial ∈ 푒 푡 roots over the boundary and its normal operator 퐿0 and its adjoint 퐿0 has the unique continuation property, then L is essentially injective (resp. surjective) for a weight parameter 훿 / Λ = Re 휃 + 1/2 : 휃 spec 퐿 and 훿 0 (resp. 훿 0), and in either ∈ { ∈ 푏 } ≫ ≪ case has closed range.

With the edge vector fields one can define the edge Sobolev spaces

퐻푠(푀) = 푢 퐿2(푀) 푘푢 퐿2(푀), 0 푘 푠 . (1.9) 푒 { ∈ | 풱푒 ∈ ≤ ≤ }

For purpose of regularity we are also interested in hybrid spaces with additional tangential regularity. The existence of solutions with infinite smooth b-regularity gives the solution with polyhomogeneous expansions. Therefore we set the Sobolev space with boundary and edge regularity as:

퐻푠,푘(푀) = 푢 퐻푠(푀) 푖푢 퐻푠(푀), 0 푖 푘 푒,푏 { ∈ 푒 | 풱푏 ∈ 푒 ≤ ≤ }

푠,푘 By the commuting relation [ 푏, 푒] 푏, 퐻 (푀) is well defined, that is, independent 풱 풱 ⊂ 풱 푒,푏 to the order of applying edge and b-vector fields. These Sobolev spaces are defined so that edge operators maps between suitable spaces, i.e., for any m-th order edge operator 푃 Diff푚 푀, ∈ 푒

푃 : 퐻푠,푘(푀) 퐻푠−푚,푘(푀), 푚 푠. (1.10) 푒,푏 → 푒,푏 ≤

Following the idea of Graham–Lee [15] of constructing solutions that are close to the hyperbolic metric, we are interested in those solutions to (1.2) that are quasi- isometric to the Freund–Rubin solution in the edge class, that is, as sections of edge

15 bundles Sym2(푒푇 푀) 푒 ⋀︀4 푇 *푀. Kantor in his thesis [20] first considered this problem ⊕ and constructed a family of solutions to the linearized equations, which correspond to change of the 4-form along one particular direction. Our result is a generalization of the results of Graham–Lee and Kantor, in that we considered the variation of the metric and the 4-form together.

The structure of the proof is as follows. In section 2.2 we fix the gauge of this system, using the DeTurck gauge-breaking term 휑(푡, 푔) introduced in [15] and show that by adding this gauge term we get an operator Q, which is a map on the space of symmetric 2-tensors and closed 4-forms:

푄 : 푆2(푇 *푀) ⋀︀4 (푇 *푀) 푆2(푇 *푀) ⋀︀4 (푇 *푀) ⊕ 푐푙 → ⊕ 푐푙 ⎛ ⎞ ⎛ ⎞ 푔 Ric(푔) 휑(푡, 푔) 퐹 퐹 ⎝ ⎠ ⎝ − − ∘ ⎠ (1.11) 퐹 ↦→ 푑 (푑 퐹 + 1 퐹 퐹 ). * * 2 ∧ The solution to the gauged equation uniquely determines a solution to the original equations.

In section 2.3 we compute the indicial roots of the linearized gauged equations. The system splits according to the degree of forms on the product manifold, and further breaks down into blocks under the Hodge decomposition on the 4-sphere. The indicial roots apppear in pairs, symmetric around the line Re 푧 = 3, and are parametrized by the eigenvalues of the 4-sphere (see Figure 2-1 for the indicial roots distribution). Then according to different behaviors of the indicial roots weuse different strategies. For large eigenvalues, for which the indicial roots separate away from the 퐿2 line, a parametrix is constucted using the small edge calculus introduced by Mazzeo [25], showing that the operator is an isomorpism. The Fredholm property of the smaller eigenvalues that are still separated away from 퐿2 line is individually discussed in a similar manner using normal operator construction; then we use the fact there are no finite-dimensional 푆푂(7) invariant sections in 퐿2 on hyperbolic space for symmetric tensors and forms [7], to show that the operator is injective on any space that is contained in 퐿2. For the three pairs with real part of indicial roots equal

16 to 3, the projection of the operator becomes a 0-problem and we follow the method in Mazzeo–Melrose [29] and Guillarmou [18] to construct two generalized inverses

± −1 푅 = lim휖 0(푑푄 푖휖) . Then we use scattering matrix construction on hyperbolic → ± space given by Graham–Zworski [16] and Guillarmou–Naud [19] to show explicitly that real-valued kernel of the linearized operator is prescribed by three pairs of data on the boundary 6-sphere.

In section 2.4 we apply the implicit function theorem, using the fact that the nonlinear terms are all quadratic, so the nonlinear solutions are also parametrized by these three terms.

Now we state the main theorem. Let

∞ 6 ⋀︀3 * 6 푉1 := 푣1 퐶 (S ; 푇 S ): 6 푣1 = 푖푣1 . { ∈ *S }

± ± Let 푉2 , 푉3 be the smooth functions on the 6-sphere tensored with a finite dimensional 1-form space on S4:

∞ 6 푐푙 4 푉2 := 푣2 휉16 : 푣2 퐶 (S ; R), 휉16 퐸16(S ) { ⊗ ∈ ∈ }

∞ 6 푐푙 4 푉3 := 푣3 휉40 : 푣3 퐶 (S ; R), 휉40 퐸40(S ) { ⊗ ∈ ∈ }

푐푙 4 푐푙 where 퐸16(S ) and 퐸40 are closed 1-forms with eigenvalue 16 and 40 on the 4-sphere.

We also require three numbers that define the leading term in the expansion of the solution, which come from indicial roots:

± ± 휃 = 3 6푖, 휃 = 3 푖√21116145/1655, 휃3 = 3 푖3√582842/20098. (1.12) 1 ± 2 ± ±

If we fix an element [ℎ^] in the conformal boundary data to the leading order, the solution is parametrized by a small perturbation from the data on the bundle

∞ 6 3 퐶 (S ; 푖=1푉푖). The metric part of the solution to the leading order is given by the ⊕ conformal infinity [ℎ^], whereas the form part to the leading order is given by the

+ − + 휃푖 휃 oscillatory data 푣푖 푥 + 푆푖(푣푖)푥 .

17 To state the theorem, we also give the following notations. Denote

1 푢0 := (푔 7 푔 4 , 6 Vol 4 ) H × 4 S S

^ 6 and [ℎ] is close to 푔^0 as metric on S . A small neighborhood in the bundle is given by

∞ 6 3 푈 퐶 (S ; 푖=1푉푖) ⊂ ⊕

Theorem 1.3. For 훿 (0, 1), 푠 2 and 푘 0, in the space of solutions to 푄(푢) = 0 ∈ ≥ ≫ 3−훿 ∞ ^ in 푥 퐻푏 (푀; 푊 ), a neighborhood of 푢0 is smoothly parametrized by [ℎ] and 푈. For a smooth section 푣 푈 with a sufficiently small 퐻푘 norm and a [ℎ^], there is a unique ∈ 푔 푥−훿퐻∞(푀; Sym2(푒푇 *푀)) and a 4-form 퐹 푥−훿퐻∞(푀; 푒 ⋀︀4(푇 *푀)) prescribed ∈ 푏 ∈ 푏 −훿 푠,푘 by those data, such that (푔 ℎ, 퐹 푉0) 푥 퐻 (푀; 푊 ) and 푄(푢) = 0. − − ∈ 푒,푏

1.2 Resolution of the canonical fiber metrics for a Lefschetz fibration

In this project joint with Richard Melrose, we give a complete description of the behavior of the constant scalar curvature fiber metric on a Lefschetz fibration with fiber genus 2. For a Riemann surface with 푔 2, the classical uniformization ≥ ≥ theorem guarantees the existence of a metric with constant scalar curvature 1. One − may ask the question about the existence and behavior of a constant scalar curvature metric if the geometry becomes singular, namely, we take a nontrivial geodesic cycle and let its length go to zero, which is illustrated in Figure 1-1. This fits naturally with the setting of a Lefschetz fibration. The class ofLefschetz fibrations we consider is for a compact connected almost-complex 4-manifold 푊 and a smooth map, with complex fibers 퐹 , to a Riemann surface 푍

휓 : 퐹 / 푊 (1.13)

푍

18 Figure 1-1: Degenerating surfaces with a geodesic cycle shrinking to a point

which is pseudo-holomorphic, has surjective differential outside a finite subset of 푊 and near each of these singular points is reducible to the following normal crossing model:

{︁ 3 1}︁ 푃푡 = (푧, 푤, 푡) C ; 푧푤 = 푡, 푧 1, 푤 1, 푡 (푧, 푤, 푡) ∈ | | ≤ | | ≤ | | ≤ 2 ∋ {︁ 1}︁ D 1 = 푡 C; 푡 . (1.14) −→ 2 ∈ | | ≤ 2

Lefschetz fibrations play an important role in 4-manifold theory. Donaldson [8] showed that a 4-dimensional simply-connected compact symplectic manifold admits a Lefschetz fibration over a sphere up to a stabilization, and Gompf showed the converse [14].

Obitsu and Wolpert [39] [46] studied a degenerating family of Riemann surfaces 푅푡 where near each singularity the model is 푃푡 and the metric is given by the plumbing metric (︂휋 log 푧 휋 log 푧 )︂2 푑푠2 = | | csc | | 푑푠2, 푃푡 log 푡 log 푡 0 | | | | (1.15) (︂ 푑푧 )︂2 푑푠2 = | | . 0 푧 log 푧 | | | | Fiberwise it has constant scalar curvature 1, and approaches the singular metric − 2 푑푠0 on the cylinder as 푡 tends to 0. By grafting with the regular parts, they constructed the expansion of metrics on the global manifold.

19 q

ψ Z2 p t D ∈ 1/2 Figure 1-2: Lefschetz fibration 휓 : 푀 푍 →

20 2 Theorem 1.4 ([39], [46]). Let 푑푠푐푐 be the hyperbolic metric on the degenerated family

푅푡 with 푚 vanishing cycles, 훥 the associated Laplacian, and 푑푠푝푙 the plumbing metric 2 that comes from gluing 푑푠푃푡 with the regular part, then the metric has the following expansion

(︃ 푚 )︃ 2 2 (︂ 4)︂ 2 2 휋 ∑︁ (︁ 1 )︁ −1 ∑︁ (︁ 1 )︁ 푑푠푐푐 = 푑푠푝푙 1 (훥 2) (Λ(푧푗) + Λ(푤푗)) + 푂 − 3 log 푡푗 − log 푡푗 푗=1 | | | | (1.16)

4 where the function Λ is given by Λ(푧푗) = (푠 휒 −1 )푠 , 푠푧 = log 푧푗 . 푧 휋 D1/2 푧 | |

Our result is a generalization of Theorem 1.4 by showing that, in the resolved

plumbing space, 푔푐푐 is conformal to 푔푝푙, where the conformal factor has a complete expansion in the variable log 푡 . | | In order for the metric to be smooth, we introduce the resolved plumbing space which involves three steps of construction. The first step in the resolution is the blow up, in the real sense, of the singular fibers; this is well-defined in view ofthe transversality of the self-instersection but results in a tied manifold since the boundary faces are not globally embedded. The second step is to replace the ∞ structure by 풞 its logarithmic weakening, i.e. replacing each (local) boundary defining function 푥 by

ilog 푥 = (log 푥−1)−1.

This gives a new tied manifold mapping smoothly to the previous one by a homeo- morphism. These two steps can be thought of in combination as the ‘logarithmic blow up’ of the singular fibers. The final step is to blow up the corners, of codimension two, in the preimages of the singular fibers. This results in a manifold with corners,

푀mr, with the two boundary hypersurfaces denoted 퐵I, resolving the singular fiber,

and 퐵II arising at the final stage of the resolution. The parameter space 푍 is similarly resolved to a manifold with corners by the logarithmic blow up of each of the singular points.

21 Figure 1-3: The metric resolution

It is shown below that the Lefschetz fibration lifts to a smooth map

휓mr 푀mr / 푍mr (1.17) which is a b-fibration. In particular it follows from this that smooth vector fieldson

푀mr which are tangent to all boundaries and to the fibers of 휓mr form the sections b of a smooth vector subbundle of 푇 푀mr of rank two. The boundary hypersurface 퐵II has a preferred class of boundary defining functions, an element of which is denoted

휌II, arising from the logarithmic nature of the resolution, and this allows a Lie algebra of vector fields to be defined by

∞ b * ∞ 2 ∞ 푉 (푀mr; 푇 푀mr), 푉 휓 (푍mr) = 0, 푉 휌II 휌 (푀mr). (1.18) ∈ 풞 풞 ∈ II풞

−1 The possibly singular vector fields of the form 휌II 푉, with 푉 as in (3.4), also form all 퐿 the sections of a smooth vector bundle, denoted 푇 푀mr. This vector bundle inherits a complex structure and hence has a smooth Hermitian metric, which is unique up to a positive smooth conformal factor on 푀mr. The main result of this paper is:

Theorem 1.5. The fiber metrics of fixed constant curvature on a Lefschetz fibration,

퐿 in the sense discussed above, extend to a continuous Hermitian metric on 푇 푀mr which is related to a smooth Hermitian metric on this complex line bundle by a log-

22 smooth conformal factor.

The plumbing metric can be extended (‘grafted’ as in [39]) to give an Hermitian

퐿 metric on 푇 푀mr which has curvature 푅 equal to 1 near 퐵II and to second order − 2푓 at 퐵I. We prove the theorem above by constructing the conformal factor 푒 for this metric which satisfies the curvature equation, ensuring that the new metric has curvature 1 : − (훥 + 2)푓 + (푅 + 1) = 푒2푓 + 1 + 2푓 = 푂(푓 2). (1.19) − This equation is first solved in the sense of formal power series (with logarithms) at both boundaries, 퐵I and 퐵II, which gives us an approximate solution 푓0 with

2푓0 ∞ ∞ 훥푓0 = 푅 + 푒 + 푔, 푔 푠 (푀mr). − ∈ 푡 풞

Then a solution 푓 = 푓0 + 푓˜ to (3.9) amounts to solving

(︁ ˜ )︁ 푓˜ = (훥 + 2)−1 2푓˜(푒2푓0 1) + 푒2푓0 (푒2푓 1 2푓˜) 푔 = 퐾(푓˜). − − − − −

Here the non-linear operator 퐾 is at least quadratic in 푓˜ and the boundedness of 1 −1 − 2 푀 (훥 + 2) on 휌II 퐻b (푀mr) for all 푀 allow the Inverse Function Theorem to be ∞ ∞ applied to show that 푓˜ 푠 (푀mr) and hence that 푓 itself is log-smooth. ∈ 푡 풞 In §3.1 the model space and metric are analysed and in §3.2 the global resolution is described and the proof of the theorem above is outlined. The linearized model involves the inverse of 훥+2 for the Laplacian on the fibers and the uniform behavior, at the singular fibers, of this operator is explained in §3.3. The solution of thecurva- ture problem in formal power series is discussed in §3.4 and using this the regularity of the fiber metric is shown in §3.5.

1.3 Resolution of eigenvalues

This project is an explicit construction for an example of the Albin–Melrose resolution of compact group action on a compact manifold [1]. Such resolution of group actions

23 is interesting because on the resolved space, the group-equivariant objects are well- defined and smooth. Albin–Melrose gave a general scheme of how to resolve thegroup actions according to the index of isotropy types, namely, an iterative scheme where the smallest isotropy type is blown up and then the next level of stratum could be uniform.

Theorem 1.6 ([1]). A compact manifold with corners, M, with a smooth boundary intersection free action by a compact Lie group G, has a canonical full resolution, 푌 (푀), obtained by iterative blow-up of minimal isotropy types.

Here we consider the action of the unitary group 푈(푛) on the space of 푛-dimensional self-adjoint matrices 푆(푛) and construct the resolved space 푆[(푛) with a fixed isotropy type, that is, 푆[(푛)/푈(푛) is a smooth manifold. We introduce the following two definitions of resolutions:

Definition 1.1 (eigenresolution). By an eigenresolution of 푆, we mean a manifold with corners 푆^, with a surjective smooth map 훽 : 푆^ 푆 such that the self-adjoint → matrices have a smooth (local) diagonalization when lifted to 푆^, with eigenvalues lifted to smooth functions on 푆^.

Definition 1.2 (full eigenresolution). A full eigenresolution is an eigenresolution

with global eigenbundles. The eigenvalues are lifted to 푛 smooth functions 푓푖 on 푆^, and the trivial n-dimensional complex vector bundle on 푆^ is decomposed into 푛 smooth ^ 푛؀ 푛 ^ line bundles 푆 C = 퐸푖 such that 훽(푥)푣푖 = 푓푖(푥)푣푖, 푣푖 퐸푖(푥), 푥 푆. × 푖=1 ∀ ∈ ∀ ∈ The matrices belong to different isotropy types, in this case, are indexed bythe clustering of eigenvalues. First the two dimensional matrix case is explicitly computed, where the only singularities are the multiples of the identity matrix. Then for higher dimensional matrices, a local product structure using the Grassmannian is described, so we show that when there is a uniform spectral gap in the neighborhood, there is a local product decomposition into two lower rank matrices and a neighborhood in the Grassmannian. Then using this decomposition, we show that the blow up action can be done iteratively, each time blowing up the smallest isotropy type, which in our case is the matrices with the smallest number of different eigenvalues.

24 Theorem 1.7 ([48]). The iterative blow up of the isotropy types in 푆 (in an order compatible with inclusion of the conjugation class of the isotropy group) yields an eigenresolution. In particular, radial blow up gives a full eigenresolution.

We also discuss the difference between radial blow up and projective ones. We show in the example of two dimensional matrices that only after radial blow up the trivial C2 bundle splits into two global line bundles, while in the projective case there is no global splitting, which gives an example of the discussions in [1] that projective blow up does not induce a global resolution.

25 26 Chapter 2

The eleven dimensional supergravity equations on edge manifolds

2.1 Introduction

Supergravity is a theory of local supersymmetry, which arises in the representations of super Lie algebras. Nahm [37] showed that the dimension of the system is at most eleven in order for the system to be physical, and in this dimension if the system exists then it would be unique. The existence of such systems was shown later by Cremmer–Scherk [6] by constructing a specific system. Recently, Witten [45] showed that under AdS/CFT correspondence the M-theory is related to the 11-dimensional supergravity system, and as a result people start to work on this subject again [3]. Systems of lower dimensions can be obtained by dimensional reduction, which breaks into many smaller subfields, and in general there are many such systems. The full eleven dimensional case, with only two fields, is in many ways the simplest to consider.

A supergravity system is a low energy approximation to string theories, and can be viewed as a generalization of Einstein’s equation:푅훼훽 = 푛푔훼훽. We are specifically interested in the bosonic sections in the supergravity theory, which is a system of equations on the 11-dimensional product manifold 푀 = B7 S4 that solves for a × metric, 푔, and a 4-form, 퐹 . Derived as the variational equations from a Lagrangian,

27 the supergravity equations are as follows:

1 훾1훾2훾3 1 훾1훾2훾3훾4 푅훼훽 = (퐹훼훾 훾 훾 퐹 퐹훾 훾 훾 훾 퐹 푔훼훽) 12 1 2 3 훽 − 12 1 2 3 4 푑 퐹 = 1 퐹 퐹 (2.1) * − 2 ∧ 푑퐹 = 0

The nonlinear supergravity operator has an edge structure in the sense of Mazzeo [25], which is a natural generalization in the context of the product of a conformally compact manifold and a compact manifold. We consider those solutions that are sections of the edge bundles, which are rescalings of the usual form bundles. The Fredholm property of certain elliptic edge operators is related to the invertibility of the corresponding normal operator 푁(퐿), which is the lift of the operator to the front 2 face of the double stretched space 푋푒 . The invertibility of the normal operator is in 2 turn related to its action on functions polyhomogeneous at the left boundary of 푋푒 , which is determined by the indicial operator. The inverse of the indicial operator

−1 퐼휃(퐿) exists and is meromorphic on the complement of a discrete set spec푏 퐿, which is the indicial roots of 퐿. In this way the indicial operator as a model on the boundary determines the leading order expansion of the solution. One solution for this system is given by a product of the round sphere with a

Poincaré–Einstein metric on B7 with a volume form on the 4-sphere, in particular the Freund–Rubin solution [11] is contained in this class. Recall that a Poincaré–Einstein manifold is one that satisfies the vacuum Einstein equation and has a conformal boundary. In the paper by Graham and Lee [15], they constructed the solutions which are 퐶푛−1,훾 close to the hyperbolic metric on the ball B푛 near the boundary, and showed that every such perturbation is prescribed by the conformal data on the boundary sphere. We will follow a similar idea here for the equation (2.1), replacing the nonlinear Ricci curvature operator by the supergravity operator, considering its linearization around one of the product solutions, and using a perturbation argument to show that all the solutions nearby are determined by the metric and form data on the boundary. Kantor studied this problem in his thesis [20], where he computed the indicial

28 roots of the system and produced one family of solutions by varying along a specific direction of the form. Here we use a different decomposition that gives the same indicial roots and show that all the solutions nearby are prescribed by boundary data for the linearized operator, more specifically, the indicial kernels corresponding to three pairs of special indicial roots.

2.1.1 Equations derived from the Lagrangian

The 11-dimension supergravity theory contains the following information on an 11- dimensional manifold 푀: gravity metric 푔 Sym2(푀) and a 4-form 퐹 ⋀︀4(푀). In ∈ ∈ this theory, the Lagrangian 퐿 is defined as

∫︁ 1(︂ ∫︁ ∫︁ 1 )︂ 퐿(푔, 퐴) = 푅푑푉푔 퐹 퐹 + 퐴 퐹 퐹 (2.2) 푀 − 2 푀 ∧ * 푀 3 ∧ ∧

Here R is the scalar curvature of the metric g, A is a 3-form such that F is the field strength 퐹 = 푑퐴. The first term is the calssical Einstein-Hilbert action term, where the second and the third one are respectively Yang-Mills type and Maxwell type term for a field. Note here we are only interested in the equations derived from the variation of Lagrangian, therefore 퐴 needs not to be globally defined since we only need 푑퐴 because the variation only depends on 푑퐴:

(︂ ∫︁ )︂ ∫︁

훿퐴푖 퐴푖 퐹 퐹 = 3 훿퐴푖 퐹 퐹 (2.3) 푈푖 ∧ ∧ 푈푖 ∧ ∧ which shows that the variation is 퐹 퐹 which does not depend on 퐴푖. ∧ The supergravity equations, derived from the Lagrangian above, are (2.1). To deal with the fact the Ricci operator is not elliptic, we follow [15] and add a gauge

* breaking term 휑(푔, 푡) = 훿 푔훥푔푡퐼푑 to the first equation. Then we apply 푑 to the 2nd 푔 * equation, and combine this with the third equation to obtain the gauged supergravity system: 푄 : 푆2(푇 *푀) ⋀︀4(푀) 푆2(푇 *푀) ⋀︀4(푀) ⊕ → ⊕ 29 ⎛ ⎞ ⎛ ⎞ 푔 Ric(푔) 휑(푔,푡) 퐹 퐹 ⎝ ⎠ ⎝ − − ∘ ⎠ (2.4) 퐹 ↦→ 푑 (푑 퐹 + 1 퐹 퐹 ) * * 2 ∧ which is the nonlinear system we will be studying.

2.1.2 Edge metric and edge Sobolev space

Edge differential and pseudodifferential operators were formally introduced byMazzeo [25]. The general setting is a compact manifold with boundary, M, where the boundary has in addition a fibration 휋 : 휕푀 퐵, → with typical fiber F. In the setting considered here, 푀 = B7 S4 is the product of × a seven dimensional closed ball identified as hyperbolic space and a four-dimensional sphere. The fibration here identifies the four-sphere as fibre:

7 4 6 4 6 휋 : 휕(H S ) = S S S . × × →

The space of edge vector fields 푒(푀) is a Lie algebra consisting of those smooth 풱 vector fields on M which are tangent to the boundary and such that theinduced vector field on the boundary is tangent to the fibreof 휋. Another vector field Lie algebra we will be using is 푏 which is the space of all smooth vector field tangent to 풱 the boundary. As a consequence,

푒 푏, [ 푒, 푏] 푏. (2.5) 풱 ⊂ 풱 풱 풱 ⊂ 풱

Let (푥, 푦1, 푦2, ...푦6) be coordinates of the upper half space model for hyperbolic 7 4 space H , and 푧푗 be coordinates on the sphere S . Then locally 푏 is spanned by 풱 푥휕푥, 휕푦, 휕푧, while 푒 is spanned by 풱

푥휕푥, 푥휕푦, 휕푧.

30 The edge forms are the dual to the edge vector fields 푒, with a basis: 풱 (︁푑푥 푑푦 )︁ , , 푑푧 . 푥 푥

The 2-tensor bundle is formed by the tensor product of the basis forms. Edge differential operators form the linear span of products of edge vector fields over smooth 푚 functions. Denote the set of 푚-th order edge operator as Diff푒 (푀). We will see that the supergravity operator 푄 is a nonlinear edge differental operator. The edge-Sobolev spaces are given by

퐻푠(푀) = 푢 퐿2(푀) 푉 푘푢 퐿2(푀), 0 푘 푠 . 푒 { ∈ | 푒 ∈ ≤ ≤ }

However, for purpose of regularity we are also interested in hybrid spaces with additional tangential regularity. The exsitence of solutions with infinite smooth b- regularity gives the solution with polyhomogeneous expansions. Therefore we set the Sobolev space with boundary and edge regularity as:

퐻푠,푘(푀) = 푢 퐻푠(푀) 푉 푖푢 퐻푠(푀), 0 푖 푘 푒,푏 { ∈ 푒 | 푏 ∈ 푒 ≤ ≤ }

푠,푘 By the commuting relation (2.5), 퐻푒,푏 (푀) is well defined, that is, independent to the order of applying edge and b-vector fields. These Sobolev spaces are defined so that edge operators maps between suitable spaces, i.e., for any m-th order edge operator 푃 Diff푚 푀, ∈ 푒

푃 : 퐻푠,푘(푀) 퐻푠−푚,푘(푀), 푚 푠. (2.6) 푒,푏 → 푒,푏 ≤ for which the proof is contained in section 2.5.

2.1.3 Poincaré–Einstein metric on B7

Now let us go back to the Poincaré–Einstein metric. As mentioned above, the product of a Poincaré–Einstein metric with a sphere metric provides a large family of solutions

31 to this system, which is known as Freund–Rubin solutions. More specifically, for any Poincaré–Einstein metric ℎ with curvature 6푐2, the following metric and 4-form − gives a solution to equations 푄(푢) = 0:

(︁ 9 )︁ 푢 = ℎ 푔 4 , 푐푑푉 4 (2.7) × 푐2 S S

According to [15], there is a large class of Poincaré–Einstein metrics which can be obtained by perturbing the hyperbolic metric on the boundary. More specifically, there is the following result:

Theorem 2.1 ([6]). Let 푀 = B푛+1 be the unit ball and ℎ^ the standard metric on S푛. For any smooth Riemannian metric 푔^ on S푛 which is sufficiently close to ℎ^ in 퐶2,훼norm if 푛 > 4 or 퐶3,훼 norm if 푛 = 3, for some 0 < 훼 < 1, there exists a smooth metric 푔 on the interior of 푀, with a 퐶0 conformal compactification satisfying

Ric(푔) = 푛푔, 푔 has conformal infinity [^푔]. −

We are mainly interested in the solutions that are perturbations of such a family of solutions, in particular, we will focus on the solutions with 푐 = 6 above with hyperbolic metric on the ball and a scaled metric on the sphere, i.e. on 푋 = H7 S4: × (︁ 1 )︁ 푔퐻 푔푆, 6푑푉 4 , (2.8) × 4 S

which is also known as the Freund–Rubin solution.

2.1.4 Main theorem

In our theorem, we will fix a Poincaré–Einstein metric ℎ on B7 which is sufficiently 1 close to the hyperbolic metric. From the discussion above we know (ℎ 푔 4 , 6푑푉 4 ) × 4 S S satisfy the gauged supergravity equation (2.4). As in Graham–Lee’s paper, the

Poincaré–Einstein metrics are paramtrized by the boundary data on S6, i.e. near any fixed Poincaré–Einstein metric, there exists a unique solution to the Einstein

32 equation for a small (smooth) perturbation of the boundary conformal data.

For supergravity equations, we have additional parametrization data. Let

∞ 6 ⋀︀3 * 6 푉1 := 푣1 퐶 (S ; 푇 S ) 6 푣1 = 푖푣1 . { ∈ | *S }

± ± Let 푉2 , 푉3 be the smooth functions on the 6-sphere tensored with a finite dimensional 1-form space on S4:

∞ 6 푐푙 4 푉2 := 푣2 휉16 : 푣2 퐶 (S ; R), 휉16 퐸16(S ) { ⊗ ∈ ∈ }

∞ 6 푐푙 4 푉3 := 푣3 휉40 : 푣3 퐶 (S ; R), 휉40 퐸40(S ) { ⊗ ∈ ∈ }

푐푙 4 푐푙 where 퐸16(S ) and 퐸40 are closed 1-forms with eigenvalue 16 and 40 on the 4-sphere. We also require three numbers that define the leading term in the expansion of the solution, which come from indicial roots:

± ± 휃 = 3 6푖, 휃 = 3 푖√21116145/1655, 휃3 = 3 푖3√582842/20098. (2.9) 1 ± 2 ± ±

If we fix an element [ℎ^] in the conformal boundary data to the leading order, the solution is parametrized by a small perturbation from the data on the bundle

∞ 6 3 퐶 (S ; 푖=1푉푖). The metric part of the solution to the leading order is given by the ⊕ conformal infinity [ℎ^], whereas the form part to the leading order is given by the

+ − + 휃푖 휃 oscilatory data 푣푖 푥 + 푆푖(푣푖)푥 . To state the theorem, we give the following notations. Denote

(︁ 1 )︁ 푢0 := 푔 7 푔 4 , 6 Vol 4 H × 4 S S

^ 6 and [ℎ] is close to 푔^0 as metric on S . A small neighborhood in the bundle is given by

∞ 6 3 푈 퐶 (S ; 푖=1푉푖) ⊂ ⊕

Theorem 2.2. For 훿 (0, 1), 푠 2 and 푘 0, in the space of solutions to 푄(푢) = 0 ∈ ≥ ≫ 33 3−훿 ∞ ^ in 푥 퐻푏 (푀; 푊 ), a neighborhood of 푢0 is smoothly parametrized by [ℎ] and 푈. For a smooth section 푣 푈 with a sufficiently small 퐻푘 norm and a [ℎ^], there is a unique ∈ 푔 푥−훿퐻∞(푀; Sym2(푒푇 *푀)) and a 4-form 퐹 푥−훿퐻∞(푀; 푒 ⋀︀4(푇 *푀)) prescribed ∈ 푏 ∈ 푏 −훿 푠,푘 by those data, such that (푔 ℎ, 퐹 푉0) 푥 퐻 (푀; 푊 ) and 푄(푢) = 0. − − ∈ 푒,푏

Our approach is based on the implicit function theorem. We consider the operator

−1 푄푣( ) = 푄( +푣). A right inverse of the linearization, denoted (푑푄푣) , is constructed, · · −1 푠,푘 and we show that 푄푣 (푑푄푣) is an isomorphism on the Sobolev space 퐻 (푀; 푊 ) ∘ 푒,푏 corresponding to the range of 푑푄푣.

To get the isomorphism result, we note that the model operator on the boundary is 푆푂(5)-invariant, and therefore utilize the Hodge decomposition of functions and forms on 푆4. We decompose the equations into blocks and compute the indicial roots of each block. The indicial roots are defined by the indical operator on each fiber

휋−1(푝): −푠 푠 퐼푝[푄](푠)푣 = 푥 푄(푥 푣) −1 . |휋 (푝) Indicial roots are those 푠 that the indicial operator has a nontrivial kernel. These roots are related to the leading order of the solution expansions near the boundary.

−1 Once the indicial roots are computed, we construct the operator (푑푄푣) . The operator exihibits different properties for large and small spherical eigenvalues. For large ones, the operator is already invertible by constructing a parametrix in the

−1 small edge calculus. For small eigenvalues, two resolvents 푅± = lim휖→0(푑푄 푖휖) ± are constructed. We show that those elements corresponding to indicial roots with real part equal to 3 are the boundary perturbations needed in the theorem.

In section 2.2 we show the derivation of the equations from the Lagrangian and discuss the gauge breaking condition. In section 2.3 we study the linearized operator and show it is Fredholm on suitable edge Sobolev spaces. In section 2.4 we construct the solutions for the nonlinear equations using the implicit function theorem.

34 2.2 Gauged operator construction

2.2.1 Equations derived from Lagrangian

The supergravity system is derived as the variational equations for the following Lagrangian:

∫︁ 1(︂ ∫︁ ∫︁ 1 )︂ 퐿(푔, 퐴) = 푅푑푉푔 퐹 퐹 + 퐴 퐹 퐹 . 푋 − 2 푋 ∧ * 푋 3 ∧ ∧

Now we compute its variation along two directions, namely, the metric and the form direction. The first term is the Einstein-Hilbert action, for which the variation in 푔 is

(︂ ∫︁ )︂ ∫︁ (︂ )︂ 푅 훼훽 훿푔 푑푉푔푅 = 푅훼훽 푔훼훽훿푔 푑푉푔 . (2.10) − 2

Now we compute the variation of the second term 퐹 퐹 in the metric direction, ∧ * which is

(︂ ∫︁ )︂ ∫︁ ∫︁ 1 2 휂2휉2 휂3휉3 휂4휉4 휂1휉1 1 훼훽 훿푔 퐹 퐹 = 퐹휂 ...휂 퐹휉 ...휉 푔 푔 푔 훿푔 푑푉푔 퐹 퐹 푔훼훽훿푔 . 2 ∧ * 4! 1 4 1 4 − 4 ∧ * (2.11)

Combining the two variations and setting them equal to zero, we get

1 1 휂1휂2휂3 1 푅훼훽 푅푔훼훽 = 퐹훼휂 휂 휂 퐹 퐹, 퐹 푔훼훽. (2.12) − 2 12 1 2 3 훽 − 4⟨ ⟩

Here , is the inner product on forms: ⟨∙ ∙⟩

1 휂1...휂4 퐹, 퐹 = 퐹휂 ...휂 퐹 . (2.13) ⟨ ⟩ 4! 1 4

Taking the trace of the equation, we get

1 푅 = 퐹, 퐹 . (2.14) 6⟨ ⟩ 35 Finally, substituting 푅 in the equation, we get

1 훾1훾2훾3 1 훾1훾2훾3훾4 푅훼훽 = (퐹훼훾 훾 훾 퐹 퐹훾 훾 훾 훾 퐹 푔훼훽), (2.15) 12 1 2 3 훽 − 12 1 2 3 4

which gives the first equation in supergravity system2.1.

The variation with respect to the 3-form 퐴 is

∫︁ 1 1 ∫︁ ∫︁ 1 훿퐴푆 = 훿퐹 퐹 훿퐴 퐹 퐹 퐴 훿퐹 퐹 = 훿퐴 (푑 퐹 + 퐹 퐹 ), (2.16) ∧* − 6 ∧ ∧ − 3 ∧ ∧ − ∧ * 2 ∧

which gives the second supergravity equation:

1 푑 퐹 + 퐹 퐹 = 0. (2.17) * 2 ∧

Since 퐹 is the differential of 퐴, so we have the third equation

푑퐹 = 0. (2.18)

Product solutions are obtained as follows: let 푋7 be an Einstein manifold with negative scalar curvature 훼 < 0 and 퐾4 be an Einstein manifold with positive scalar curvature 훽 > 0. Consider 푋 퐾 with the product metric; then we have × ⎛ ⎞ 푀 6훼푔퐴퐵 0 푅푖푐 = ⎝ ⎠ (2.19) 퐾 0 3훽푔푎푏

Let 퐹 = 푐푉퐾 . A straightforward computation shows

⎛ ⎞ 2 푀 푐 2푔퐴퐵 0 (퐹 퐹 )훼훽 = ⎝ − ⎠ . (2.20) ∘ 12 퐾 0 4푔푎푏

Therefore any set (푐, 훼, 훽) satisfying

푐2/6 = 6훼, 푐2/3 = 3훽 (2.21) − 36 corresponds to a solution to the supergravity equation.

2.2.2 Change to a square system

In order to write the equations as a square system, we change the second equation to second order by applying 푑 . Combining with the closed condition 푑퐹 = 0, 푑 푑 퐹 * * * is the same as 훥퐹 . This leads to the following square system

Ric 푔 퐹 퐹 = 0 (2.22) − ∘ 1 훥퐹 + 푑 (퐹 퐹 ) = 0 (2.23) 2 * ∧

Proposition 2.1. After changing to the square system, the kernel is the same as the kernel of the original system.

Proof. The only change here is that we introduced 푑 = 푀 (푑 4 + 푑 7 ) 푀 , in which * * S H * 푀 and 푑 4 are both isomorphisms. Therefore we only need to consider the possible * S kernel introduced by 푑H7 in the solution space. On the hyperpolic space, there are no 2 퐿 kernel for 푑H7 because of the representation, and the nonlinear terms after taking off the kernel of the linearized operatoris ofdecay 푥3+훿.

2.2.3 Gauge condition

Following [15] in the setting of Poincaré–Einstein metric, we add a gauge operator to the curvature term where 푔 is the background metric:

* −1 휑(푡, 푔) = 훿푡 (푡푔) 훿푡퐺푡푔.

Here

1 푘 푗 [퐺푡푔]푖푗 = 푔푖푗 푔 푡푖푗, [훿푡푔]푖 = 푔 − 2 푘 − 푖푗,

* 훿푔 is the formal adjoint of 훿푔, which can be written as

1 [훿*푤] = (푤 + 푤 ), 푔 푖푗 2 푖,푗 푗,푖 37 and

−1 −1 푗푘 [(푡푔) 푤]푖 = 푡푖푗(푔 ) 푤푘.

By adding the gauge term we get an operator 푄, which is a map from the space of symmetric 2-tensors and closed 4-forms to the space of symmetric 2-tensors and closed 4-forms:

푄 : 푆2(푇 *푀) ⋀︀4 (푀) 푆2(푇 *푀) ⋀︀4 (푀) ⊕ 푐푙 → ⊕ 푐푙 ⎛ ⎞ ⎛ ⎞ 푔 Ric(푔) 휑(푡, 푔) 퐹 퐹 ⎝ ⎠ ⎝ − − ∘ ⎠ (2.24) 퐹 ↦→ 푑 (푑 퐹 + 1 퐹 퐹 ) * * 2 ∧ which will be the main object to study.

As discussed in [15], 푅푖푐(푔) + 푛푔 휑(푡, 푔) = 0 holds if and only if 푖푑 :(푀, 푔) − → (푀, 푡) is harmonic and 푅푖푐(푔)+푛푔 = 0. We will show that the gauged equations here yield the solution to the supergravity equations in a similar manner. We first prove a gauge elimination lemma for the linearized operator. Ascan be seen from (2.24), only the first part (the map on 2-tensors) involves the gauge term, therefore we restrict the discussion to the first part of 푑푄. We use 푑푄푔(푘, 퐻) to denote the linearization of the tensor part of 푄 along the metric direction at the point (푔, 퐹 ), which acts on (푘, 퐻). First we give the following gauge-breaking lemma for the linearized operator, which is adapted from Theorem 4.2 in [20].

Lemma 2.1. If (푘, 퐻) satisfies the linearized equation 푑푄푔(푘, 퐻) = 0, then there ˜ ˜ exists a 1-form 푣 and 푘 = 푘 + 퐿푣♯ 푔 such that 푑푆푔(푘, 퐻) = 0.

To prove the proposition, we first determine the equation to solve for sucha 1-form 푣.

Lemma 2.2. If a 1-form 푣 satisfies

푟표푢푔ℎ 1 훼 ((훥 푅푖푐)푣)휆 = (2 푘훼휆 휆푇 푟푔(푘)) (2.25) − 2 ∇ − ∇ 38 ˜ then 푘 = 푘 + 퐿푣♯ 푔 satisfies the gauge condition

푑휑(푡, 푔)푔(푘˜) = 0.

Proof. Let Ψ(푣, 푔) be the map

푘 * 훼훽 푘 * 푘 Ψ(푣, 푔) = (휑 ♯ 푔) (Γ (휑 ♯ 푔) Γ (푡)). 푣 훼훽 푣 − 훼훽

This satisfies

* * 훿푔 푔퐷푣Ψ(0, 푡)(푣) = 퐷푡휑(퐿푣♯푡푔), 훿푔 푔퐷푔Ψ(0, 푡)(푘) = 퐷푡휑(푘).

Therefore in order to get 퐷푡휑(푘˜) = 0, we only need

퐷푣Ψ(0, 푡)(푣) = 퐷푔Ψ(0, 푡)(푘). −

The left hand side can be reduced to

훼훽 푘 푘 휇 푟표푢푔ℎ 푘 푔 훼 훽푣 푅 푣 = (훥 푅푖푐)푣 − ∇ ∇ − 휇 −

and right hand side is

1 훼훽 푘휆 푔 푔 ( 훼푘훽휆 + 훽푘훼휆 휆푘훼훽). 2 ∇ ∇ − ∇

Lowering the index on both side, we get

푟표푢푔ℎ 1 훼 ((훥 푅푖푐)푣)휆 = (2 푘훼휆 휆푇 푟푔(푘)). − 2 ∇ − ∇

Next we discuss the solvability of the operator defined above in (2.25).

39 1 Lemma 2.3. If 훿 < 1, then at the point 푔0 = 푔퐻 푔푆 the operator | | × 4

훥푟표푢푔ℎ 푅푖푐 : 푥훿퐻2(푒푇 *푀) 푥훿퐿2(푒푇 *푀) − →

is an isomorphism.

Proof. Using the splitting

푒 * * 푒 * 7 *푒 * 4 푇 푀 = 휋 푇 H 휋 푇 S ∼ H ⊕ S

and the product structure of the metric, we write the operator as

훥푟표푢푔ℎ 푅푖푐 = 훥푟표푢푔ℎ + 훥푟표푢푔ℎ diag( 6, 12). − 퐻 푆 − −

It decomposes into two parts: trace and trace-free 2-tensors. Decomposing into eigenfunctions on the 4-sphere, consider the following two operators:

∞ 7 ∞ 7 퐿푡푟 = 훥 + 휆 24 : 퐶 (H ) 퐶 (H ), (2.26) H − →

푟표푢푔ℎ ′ 푒 ⋀︀* 7 푒 ⋀︀* 7 퐿푡푓 = 훥 + 휆 + 6 : H H (2.27) H →

Consider the smallest eigenvalue in each case: 휆푡푟 = 16, 휆푡푓 = 0. The indicial radius

for 퐿푡푟 is 1 and for 퐿푡푓 is 4. Then using theorem 6.1 in [25], for 훿 < 1 the operator 퐿푡푟 | | as a map from 푥훿퐻2(H7) 푥훿퐿2(H7) has closed range and is essentially injective. → Moreover, since the kernel of this operator lies in the 퐿2 eigen space of H7 which vanishes, therefore the operator is actually injective. By self-adjointness, it is also

surjective on the same range for 훿. The same argument holds for 퐿푡푓 , which is also 훿 2 푒 ⋀︀* 7 훿 2 푒 ⋀︀* 7 Fredholm and an isomorphism on 푥 퐻 ( H ) 푥 퐿 ( H ) for 훿 < 4. → | | 푟표푢푔ℎ Combining the statements for 퐿푡푟 and 퐿푡푓 , we conclude that 훥 푅푖푐 is an − isomorphism between 푥훿퐻2(푒푇 *푀) 푥훿퐿2(푒푇 *푀) for 훿 < 1. → | |

The isomorphism holds true for metrics nearby, by a simple perturbation argu-

40 ment.

푟표푢푔ℎ Corollary 1. At any metric 푔 which is close to 푔0, for 훿 < 1, 훥 푅푖푐 is an | | − isomorphism as a map

훥푟표푢푔ℎ 푅푖푐 : 푥훿퐻2(푒푇 *푀) 푥훿퐿2(푒푇 *푀). − →

With the lemmas above, we can prove the proposition.

Proof of Proposition 2.1. From Lemma 1 we know 훥푟표푢푔ℎ 푅푖푐 : 푥훿퐻2(푒푇 *푀) − → 푥훿퐿2(푒푇 *푀) is an isomorphism, therefore there exists a one-form 푣 satisfying 2.2. ˜ ˜ Then from Lemma 2.2, 푘 = 푘 + 퐿푣♯ 푔 satisfies 퐷푔휑(푘) = 0. Putting it back to the

linearized equation, we get 푑푆푔(푘,˜ 퐻) = 0.

Next we prove the nonlinear version of gauge elimination by using integral curves.

Proposition 2.2. If a metric and a closed 4-form (푔, 푉 ) satisfies the gauged equations 푄(푔, 푉 ) = 0, then there is an diffeomorphism 푔 푔˜ such that 휑(˜푔, 푡) = 0 and (˜푔, 푉 ) ↦→ is a solution to equation( 2.1) i.e. 푆(˜푔, 푉 ) = 0.

Proof. Consider the integral curve on the manifold of metrics close to the product ˜ metric, defined by the vector field at each point 푔 with value 푘푔 = 푘푔 + 퐿푣♯ 푔. Then

along this curve, it satisfies 퐷푔푄(푘˜푔, 퐻) = 0. Therefore 푄(˜푔, 푉 ) = 0. We also need to show that 푔˜ has the same regularity as 푔. It suffices to show that the vector field is smooth enough. Since 푣 solved above in Lemma 2.2 is polyhomogeneous, then the Lie derivative 퐿 ♯ 푔 = 훼푣훽 + 훽푣훼 is one order less smooth than 푣 ∇ ∇ 푣. However by integration we gain one order of regularity back. Therefore 푔˜ has the same regularity as 푔.

2.3 Fredholm property of the linearized operator

We now consider the linearization of the gauged supergravity operator near the base

1 metric and 4-form (푔0, 퐹 ) = (푔 7 푔 4 , 6 Vol 4 ). The first step is to compute the indi- H × 4 S S cial roots and indicial kernels of this linearizaed operator, which is done with respect

41 of the harmonic decomposition of the 4-sphere. As the eigenvalues becomes larger, the indicial roots become further apart. Specifically, all of them are separated by real part 3, with only three pairs of exceptions, where the indicial roots corresponding to the lowest three eigenvalues lie on the 퐿2 line which has real part equal to 3. Once we identify these indicial roots, we proceed differently according to whether the indicial roots land on the 퐿2 line or not. We show that for most of the indicial roots, the decomposed linearized operator is Fredholm on suitable edge Sobolev spaces. This is done by using small edge calculus and 푆푂(5) invariance of the structure. For the three exceptional pairs we use scattering theory to construct two generalized inverses, which encode the boundary data that parametrize the kernel of the linear operator. We then describe the kernel of this linearized operator in terms of the two generalized inverses, and a scattering matrix construction that gives the Poisson operator. Nearby the Poincaré–Einstein metric product, a perturbation argument shows that the space given by the difference of the two generalized inverses is transversal tothe range space of the linearized operator and therefore this space gives the kernel of the linearized operator, which later will provide the kernel parametrization for the nonlinear operator.

2.3.1 Linearization of the operator 푄

The nonlinear supergravity operator contains two parts: the gauged curvature op-

erator Ric 휑푔,푡 with its nonlinear part 퐹 퐹 , and the first order differential of the − ∘ 4-form 푑 퐹 with its nonlinear part 퐹 퐹 . Note that since the Hodge operator * ∧ * depends on the metric, the linearized operator couples the metric and the 4-form in both of the equations.

Though we only consider the linearization about the base poduct metric 푔 7 H × 1 4 푔S4 , the computation below applies to other Poincaré–Einstein metrics satisfying the relation 2.7 mentioned in the introduction. Since near the boundary the metric is the

same as H7 S4, the discussion about edge operators in later sections would remain × the same.

42 Proposition 2.3. The operator 푄 : 푊 푊 has the following linearization at the → point (푔0, 퐹 ):

2 푒 * 푒⋀︀4 2 푒 * 푒⋀︀4 푑푄푔 ,퐹 : Γ(Sym ( 푇 푀) (푀)) Γ(Sym ( 푇 푀) (푀)) 0 ⊕ → ⊕ ⎛ ⎞ ⎛ ⎞ 푘 훥푘 + LOT ⎝ ⎠ ⎝ ⎠ ↦→ 퐻 푑 (푑 퐻 + 6 Vol 퐻 + 6푑 푘1,1 + 3푑(tr 7 (푘) tr 4 (푘)) Vol ) * * S ∧ *H H − S ∧ H (2.28) where the lower order term matrix LOT is as follows:

⎛ ⎞ 푘퐼퐽 6 trS(푘)푡퐼퐽 + trH(푘)푡퐼퐽 + 2 S 퐻0,4푡퐼퐽 6푘1,1 3 S 퐻1,3 LOT = ⎝ − − * − * ⎠ 6푘1,1 3 퐻1,3 4푘푖푗 + 8 tr (푘)푡푖푗 퐻0,4푡푖푗 − *S S − *S

We break the computation into a curvature part and a form part as follows.

Lemma 2.4. For 푘 Sym2(푒푇 *푀), the linearization of the gauge broken Ricci op- ∈ erator at the base hyperbolic metric is

1 푟표푢푔ℎ 푑푔 (Ric 휑푔,푡)(푘) = 훥 푘 + 푅(푘), (2.29) 0 − 2 푔0

where ⎛ ⎞ 7푘퐼퐽 + TrH7 (푘)푔퐼퐽 0 푅(푘) = ⎝ − ⎠ . 0 16푘푖푗 Tr 4 (푘)푔푖푗 − S Proof. Following the result in [15], the linearization of the gauged operator at the base metric 푡 is

1 푟표푢푔ℎ 훼훽 1 훽 훽 푑푡(Ric 휑푔,푡)(푘) = 훥 푘 + 푘 푅훽훾훿훼 + (푅 푘훽훿 + 푅 푘훽훾). (2.30) − 2 푡 2 훾 훿

Specifically, if the metric is constant sectional curvature near the boundary of 푀 which is the case for the conformal compact metric here (with sectional curvature 1), the curvature term is diagonalized and can be written as −

푅훼훽훿훽 = (푔훼훿푔훾훽 푔훼훽푔훾훿), − − 43 so the linearization of this total operator is as above.

Lemma 2.5. The linearization of 퐹 퐹 along the 2-tensor direction and 4-form ∘ direction are: for 푘 Sym2(푒푇 *푀) ∈

푑푔 ,6 Vol (퐹 퐹 )(푘) 0 S ∘ ⎛ ⎞ 1 1 1 36 푇 푟푆(푘)푡퐼퐽 36 푘 + 2 푊, 퐻 푡 144 푊, 푊 푘퐼푗 ⎜ − ⟨ ⟩ ⟨ ⟩ ⎟ ⎜ ⎟ = ⎜ 푗 푗 ⎟ , ⎜ 1 (︀ 3푊 푊 2 3 푡푖1푙1 푘 푡푙2푗1 ⎟ ⎜ 1 12 푎푖1푖2푖3 푏푗1 푙1푙2 ⎟ ⎝ 144 푊, 푊 푘퐼푗 − ⎠ ⟨ ⟩ 4 푖1푖2푖3 푗1푙1 푙2푖1 1 )︀ + 퐹푖 푖 푖 푖 퐹 푡 푘푙 푙 푡 푡푎푏 푊, 푊 푘푎푏 12 1 2 3 4 푗1 1 2 − 12 ⟨ ⟩ (2.31)

and for 퐻 푒 ⋀︀4(푇 *푀) ∈ ⎛ ⎞ 1 1 72 푐2 푆 퐻(0,4)푡퐴퐵 3 ( 푆퐻(1,3))퐴푏 ⎜ * * ⎟ 1 푖 푖 푖 푑푔 ,6 Vol (퐹 퐹 )(퐻) = ⎜ 1 2 3 ⎟ . (2.32) 0 S ⎜ 6 퐻푎 푊푏푖1푖2푖3 ⎟ ∘ ⎝ 3( 푆퐻(1,3))퐴푏 ⎠ * 1 푖1푖2푖3푖4 푊푖 푖 푖 푖 퐻 푡푎푏 − 72 1 2 3 4

Proof. The proof is by direct computation. Note that

⎛ ⎞ 퐻퐻 퐻푆 퐷푡,푊 (퐹 퐹 )(퐻) = ⎝ ⎠ , ∘ 퐻푆 푆푆 where

1 푖1푖2푖3푖4 1 1 퐻퐻퐴퐵 = 푊푖 푖 푖 푖 퐻 푡퐴퐵 = 푐2ℎ푡퐴퐵 = 푐2 퐻(0,4)푡퐴퐵; (2.33) −72 1 2 3 4 −72 72 *S

푖1푖2푖3푖4 ((Vol )푖 푖 푖 푖 (푉푆) = 1, Vol = 1, 퐻(0,4) = ℎ Vol .) S 1 2 3 4 *S S S

1 푖1푖2푖3 퐻푆퐴푏 = 퐻퐴푖 푖 푖 푊 = 3( 퐻(1,3))퐴푏, (2.34) 12 1 2 3 푏 *S

1 푖1푖2푖3 1 푖1푖2푖3푖4 푆푆 = 퐻 푊푏푖 푖 푖 푊푖 푖 푖 푖 퐻 푡푎푏. (2.35) 6 푎 1 2 3 − 72 1 2 3 4 44 Then for metric variation 푘 Sym2(푇 *푀) ∈ ⎛ ⎞ 퐻퐻 퐻푆 퐷푡,푊 (퐹 퐹 )(푘) = ⎝ ⎠ , ∘ 퐻푆 푆푆

1 퐻푆 = 푊 푊 푖1푖2푖3푖4 푘 , (2.36) 144 푖1푖2푖3푖4 퐼푗 (︁ )︁ 1 4 푖1푖2푖3 푗1푙1 푙2푖1 1 푖1푖2푖3푖4 퐻퐻퐴퐵 = 퐹푖 푖 푖 푖 퐹 푡 푘푙 푙 푡 푡퐴퐵 푊1 푖 푖 푖 푊 푘퐴퐵 , (2.37) 12 12 1 2 3 4 푗1 1 2 − 12 1 2 3 4

(︁ 1 푗2푗3 푖1푙1 푙2푗1 4 푖1푖2푖3 푗1푙1 푙2푖1 푆푆푎푏 = 3푊푎푖 푖 푖 푊 푡 푘푙 푙 푡 + 퐹푖 푖 푖 푖 퐹 푡 푘푙 푙 푡 푡푎푏 12 − 1 2 3 푏푗1 1 2 12 1 2 3 4 푗1 1 2 )︁ 1 푖1푖2푖3푖4 푊푖 푖 푖 푖 푊 푘푎푏 , (2.38) − 12 1 2 3 4

푖1푖2푖3푖4 which using inner product 푊푖 푖 푖 푖 푊 = 푊, 푊 will give the expressions above. 1 2 3 4 ⟨ ⟩

Next we compute the linearization of the 2nd equation:

Lemma 2.6. The linearization of the equation

1 푑 퐹 + 퐹 퐹 = 0, * 2 ∧

along the form direction and tensor direction are respectively:

1 푑푔 ,퐹 (푑 퐹 + 퐹 퐹 )(퐻) = 푑 퐻 + 퐻 퐹, (2.39) 0 0 * 2 ∧ * ∧

1 푑푔 ,퐹 (푑 퐹 + 퐹 퐹 )(푘) = 6푑 푘(1,1) + 3푑(tr (푘) tr (푘)) Vol (2.40) 0 0 * 2 ∧ *H H − S H

Proof. The linearization along the form direction is straight-forward, as the terms are linear and quadratic on 퐹 . Along the metric direction, the linearization comes from

45 the Hodge star:

퐷( 퐹 )훽 훽 ..훽 (푘) * 1 2 7 1 훼1..훼4 1 훼1..훼4 4 훼2..훼4 훾1훼1 = 퐷( 4! 푉훽1..훽7 푊훼1..훼4 )(푘) = 4! (훿푉 )훽1..훽7 푊훼1..훼4 + 4! 푉훾1훽1..훽7 (훿푔) 푊훼1..훼4

1 1 훼훽 훼1..훼4 1 훼2..훼4 훾1휉 휓훼1 = 2 4! 푡 푘훼훽푉훽1..훽7 푊훼1..훼4 + 6 푉훾1훽1..훽7 푡 푘휉휓푡 푊훼1..훼4

= 6푑 푘(1,1) + 3푑(tr (푘) tr (푘)) Vol *H H − S H (2.41) which gives the expressions above.

Proof of Propsition 2.3. Combining everything together, the linearized equations are

푑 (6푑 푘(1,1) + 3푑(tr (푘) tr (푘)) Vol +푑 퐻 + 퐻 Vol푏 푏퐻) = 0 * *H H − S H * ∧ (2.42) 1 푟표푢푔ℎ 훼훽 1 훽 훽 2 훥푔0 푘 + 푘 푅훽훾훿훼 + 2 (푅훾 푘훽훿 + 푅훿 푘훽훾) + 퐿푂푇 = 0

which after arrangement gives the linearization in (2.28).

2.3.2 Indicial roots computation

Having obtained the linearized operator 푑푄, we next compute its indicial roots on the boundary of H7, which together with its indicial kernels parametrize the boundary values of this linear operator. Utilizing the Hodge decomposition on the 4-sphere, the operator acts on a space of sections on H7 tensored with finite dimensional subspaces of Γ(푇 *S4).

Definition 2.1 (Hodge decomposition projection). Let 휆 be one of the eigenvalues 4 ⋀︀푖 * 4 for the Hodge laplacian on Γ( 푖=0 (푇 S )) and define the eigenvalue projection ⊕ 2 푒 * 4 ⋀︀ 푒 * operator 휋휆 on the the sections of the bundle 푊 = Sym ( 푇 푀) 푇 푀 to be the ⊕ projection that maps to the corresponding part on sphere.

Note here we have a collection of eigenvalues on both functions and forms.

Lemma 2.7. Sections of the bundle 푊 decompose according to eigenvalues 휆.

2 Proof. We identify the symmetric edge 2-tensor bundle with (Sym (푒푇 *H7)) (푒푇 *H7 ⊕ 2 푇 *S4) Sym (푇 *S4), and decompose the 4-form bundle according to its degree on ⊗ ⊕ 46 7 4 푒 ⋀︀4 * 푒 ⋀︀푖 * 7 ⋀︀푗 * 4 H and S , i.e. 푇 푀 = 푖+푗=4 푇 H 푇 S . And for each element of the ⊕ ⊗ 푒 ⋀︀* * 7 ⋀︀* * 4 form 푢 푣 with 푢 Γ( 푇 H ), 푣 Γ( 푇 S ) the projection operator 휋휆 maps ⊗ ∈ ∈ it to 푢 휋휆푣, which by linearity extends to the whole bundle 푊 . ⊗ It follows that the operator decomposes to an infinite collection of operators, each acting on a subbundle.

Lemma 2.8. The operator 푑푄 preserves the eigenspaces of S4, and we have the following decomposition of the operator

∑︁ 휆 ∑︁ 푑푄 = 푑푄 := 휋휆 푑푄 휋휆 ∘ ∘ 휆≥0 휆

Proof. We only need to show that that Hodge laplacian 훥 commutes with the linearized operator. Since the linear operator is composed from 훥ℎ표푑푔푒, 훥푟표푢푔ℎ (which are related by Bochner formula), Hodge operator, differential, and scalar operator, all of * which commute with 훥, 푑푄 therefore commutes with the eigenvalue projections.

Now we define indicial roots and indicial kernels below for the edge operator.

Recall that 휕푀 is the total space of fibration over 푌 = 휕B7.

Definition 2.2 (Indicial operator). Let 퐿 : Γ(퐸1) Γ(퐸2) be an edge operator → between two vector bundles over 푀. For any boundary point 푝 푌 , and 푠 C, the ∈ ∈ indicial operator of 퐿 at point 푝 is defined as

퐼푝[퐿](푠) : Γ(퐸1 −1 ) Γ(퐸2 −1 ) |휋 (푝) → |휋 (푝)

−푠 푠 (퐼푝[퐿](푠))푣 = 푥 퐿(푥 푣˜) −1 |휋 (푝) where 푣˜ is an extension of 푣 to a neighborhood of 휋−1(푝). The indicial roots of

퐿 at point 푝 are those 푠 C such that 퐼푝[퐿](푠) has a nontrivial kernel, and the ∈ corresponding kernels are called indicial kernels.

In the conformally compact case, the indicial operator is a bundle map from 퐸1 푝 | to 퐸2 푝 (which is simpler than a partial differential operator as in the general edge | 47 IIIIII

(z) = 3 ℜ Figure 2-1: Indicial roots of the linearized supergravity operator on C case). Moreover, since we have an 푆푂(7) symmetry for the operator, the indicial roots will be constants for any boundary point 푝 S6. ∈ Proposition 2.4. The indicial roots of operator 푑푄 are symmetric around Re 푧 = 0, with three special pairs of roots

± ± 휃 = 3 6푖, 휃 = 3 푖√21116145/1655, 휃3 = 3 푖3√582842/20098. 1 ± 2 ± ± and all other roots lying in Re 푧 3 1 . {‖ − ‖ ≥ } Proof. With the harmonic decomposition on sphere S4, the linearized operator 푑푄 is block-diagonalized and we compute the indicial roots for the linear system 푑푄 in Section 2.6. We summarize the results below and Figure 2-1 is an illustration of the indicial roots distribution. The indicial roots fall into the following three categories:

1. The roots corresponding to harmonic forms:

(a) The equation for trace-free 2-tensors on 퐻7 arising from the first component of (2.28) is

(훥푆 + 훥퐻 2)푘^퐼퐽 = 0, − and the corresponding indicial equation is

2 ( 푠 + 6푠)푘퐼퐽 = 0. − 48 We have indicial roots

+ − 푆1 = 0, 푆1 = 6.

This corresponds to the perturbation of the hyperbolic metric to a Poincaré– Einstein metric.

(b) The equation for trace-free 2-tensors on 푆4 is

푟표푢푔ℎ^ ^ ^ 훥푆 푘푖푗 + 훥퐻 푘푖푗 + 8푘푖푗 = 0

where indicial equation is

2 ( 푠 + 6푠 + 8)푘^푖푗 = 0, −

and the indicial roots are

푆± = 3 √17. 2 ±

푑퐻 퐻(4,0) + 푊 퐻(4,0) = 0 * ∧ 푑퐻 퐻(4,0) = 0

where the indicial equation is

(푠 3)( 6푁) 푑푥/푥 6푑푥/푥 푁 = 0, − − * ∧ − ∧

with indicial roots 휃± = 3 6푖. 1 ± This corresponds to a perturbation of the 4-form on hyperbolic space.

2. The roots corresponding to functions / closed 1-forms / coclosed 3-forms / closed 4-forms

49 (a) The equations for 7휎 = 푇 푟퐻 (푘), 4휏 = 푇 푟푆(푘), 푘(1,1), 퐻(1,3), 퐻(0,4) are

푐푙 7 푐푙 푐푐 6푑퐻 퐻 푘 + 푑푆(3푇 푟퐻 (푘) 3푇 푟푆(푘)) 푉 + 푑푆 퐻 + 푑퐻 퐻 = 0 * (1,1) − ∧ * (0,4) * (1,3) 푐푙 푐푐 푑퐻 퐻(0,4) + 푑푆퐻(1,3) = 0 푐푐 푑퐻 퐻(1,3) = 0 푐푙 푐푙 푐푙 푐푐 푠푘 + 퐻 푘 + 12푘 6 푆 퐻 = 0 △ (1,1) △ (1,1) (1,1) − * (1,3) 푐푙 훥푆휏 + 훥퐻 휏 + 72휏 8 푆 퐻 = 0 − * 0,4 푐푙 훥푆휎 + 훥퐻 휎 + 12휎 + 4 푆 퐻 48휏 = 0 * 0,4 −

The indicial equations are

휆4 4푆2휆3 + 24푆 휆3 90휆3 + 6푆4휆2 72푆3휆2 − * − − + 342푆2휆2 756푆 휆2 + 1152휆2 4푆6휆 + 72푆5휆 414푆4휆 − * − − + 648푆3휆 + 1152푆2휆 3024푆 휆 + 10368휆 − * +푆8 24푆7 +162푆6 +108푆5 6192푆4 +31536푆3 33696푆2 155520푆 = 0 − − − − (2.43)

When 휆 = 16 there is a pair of roots with real part 3

푠 = 휃± = 3 푖√21116145/1655 (2.44) 2 ±

and when 휆 = 40 there is a pair of roots with real part 3

휃± = 3 푖3√582842/20098 3 ±

And here the five variables are related by

푐푙 푐푐 푐푙 퐻 = 푑푠 푠 푑푠휉, 퐻 = 푑퐻 푠 푑푠휉, 푘 = 푑푠훿퐻 휉, 4휎 = 7휏 = 휉 (0,4) * (1,3) − * (1,1) −

where

1 4 휉 훿푠 푆 ∈ ∧16

50 similarly we have another indicial kernel corresponding to 휃± with 휉 3 ∈ 1 4 훿푠 푆 . ∧40

(b) The equations for 퐻(3,1), 퐻(4,0) are

푐푙 푐푐 4 푐푐 푑푆 퐻 + 푑퐻 퐻 + 6 푉 퐻 = 0 * (3,1) * (4,0) ∧ (4,0) 푐푙 푐푐 푑퐻 퐻(3,1) + 푑푆퐻(4,0) = 0

where the indicial equations are

(푠 3)2 6푖(푠 3) 16 = 0 − ± − −

with indicial roots 푆± = 3 √7 3푖. 6 ± ±

3. The roots corresponding to coclosed 1-forms / closed 2-forms / coclosed 2-forms / closed 3-forms

(a) The equations for 푘(1,1), 퐻(1,3), 퐻(2,2) are

푐푐 푐푙 6푑퐻 퐻 푘 + 푑퐻 퐻 = 0 * (1,1) * (1,3) 푐푙 푐푐 푐푐 푑푆 퐻 + 푑퐻 퐻 + 6푑푆 퐻 푘 = 0 * (1,3) * (2,2) * (1,1) 푐푙 푐푐 푑퐻 퐻(1,3) + 푑푆퐻(2,2) = 0 1 푐푐 1 푐푐 푐푐 1 푐푙 푠푘 + 퐻 푘 + 6푘 푆 퐻 = 0 2 △ (1,1) 2 △ (1,1) (1,1) − 2 * (1,3)

The indicial equation is

휆2 (36 + (푠 1)(푠 5) + 푠2 6푠 1)휆 (푠 1)(푠 5)( 푠2 + 6푠 + 1) = 0. − − − − − − − − −

With the smallest eigenvalue for coclosed 1-forms being 휆 = 24, the indicial roots are √︁ 푆± = 3 3√97 + 31. 3 ± ±

51 (b) The equations for 퐻(2,2), 퐻(3,1) are

푐푙 푐푐 푑푆 퐻 + 푑퐻 퐻 = 0 * (2,2) * (3,1) 푐푙 푐푐 푑퐻 퐻(2,2) + 푑푆퐻(3,1) = 0

The indicial equations are

퐻표푑푔푒 (훥 (2 푠)(4 푠))퐻(3,1) = 0, 푆 − − −

and for 휆 = 24 we have 푆± = 3 √17. 5 ±

With respect to the volume form on H7 S4, there is an inclusion of weighted × functions and forms. For Re(휆) > 3,

⋀︁푝 ⋀︁푝 푥휆 ∞(푀; 푒 푇 *푀) 퐿2(푒 푇 *푀). 풞 ⊂ 푒

And this Re(휆) = 3 line is the 퐿2 cutoff line. The first thing to notice about the indicial roots results is that those indicial roots appear in pairs symmetric to Re(푠) = 3, which is the 퐿2 line. Most of the indicial roots are bounded away from Re(푠) = 3, and as the sperical eigenvalues become larger, they are bounded further. However, there are three pairs of roots that are on the 퐿2 line, which correponds to the kernel space.

2.3.3 Fredholm property

Now we discuss the behavior of this operator on different eigenspaces, according to whether the pair of indicial roots appear on the 퐿2 line or off of it. We will use the edge calculus to deal with the large eigenvalues and the 0-calculus to deal with the individual small eigenvalues. Specifically, we will construct two right inverses of this operator to deal with the fact that the kernel appears when the domain becomes

52 larger than 퐿2. We will show that for any weight 훿 off a discrete set of indicial roots, the operator 훿 푠,푘 푑푄 acting on 푥 퐻푒,푏 (푀; 푊 ) is Fredholm. Moreover, it is injective when 훿 > 1 and surjective when 훿 < 1. In the range of 훿 ( 1, 0), the kernel is finite dimensional, − ∈ − characterized by the three subspaces corresponding to the three indicial roots. First of all, we define the domain for the linearized operator:

Definition 2.3. Fix 훿 (0, 1), define the domain as ∈

3−훿 2,푘 3+훿 0,푘 퐷푘(훿) = 푢 푥 퐻 (푀; 푊 ):(푑푄 + 푖휖)푢 푥 퐻 (푀; 푊 ) . { ∈ 푒,푏 ∈ 푒,푏 }

Using the projection operator 휋휆 above, the domain can be decomposed in terms of the spherical harmonic decomposition on S4, as

퐷푘(훿) = 휆∈Λ퐷푘(휆, 훿), ⊕

where Λ is the set of eigenvalues on the 4-sphere. This set is divided into the following three subsets:

Eigenvalues on functions: 4푘(푘 + 3); ∙ Eigenvalues on closed one-forms: 4(푘 + 1)(푘 + 4); ∙ Eigenvalues on coclosed one-forms: 4(푘 + 2)(푘 + 3). ∙ We will separately discuss two parts. One part is the infinite dimensional subspace formed by large eigenvalues

휆>푀 퐷푘(휆, 훿), ⊕

on which the operators 푑푄 푖휖 are isomorphisms, and approach two limits 퐷± uni- ± formly as 휖 goes to zero. This is shown by using ellipticity and a parametrix construction. The other part is discussed for each small eigenvalue since there are only

finitely many. For most of the 휆푖’s, the operator has the same behavior as the "large" part. The rest correspond to indicial roots lying on the 퐿2 line, and we use scattering

theory to construct resolvents 푅± and discuss their null spaces.

53 2.3.4 Large eigenvalues

2 ⋀︀4 7 4 Consider the bundle 푊 = Sym (푀) (푀) over 푀 = B S which carries a ⊕ × unitary linear action of SO(5) covering the action on S4. There is an induced action of SO(5) on ˙∞(B S4; 푊 ), which extends to all the weighted hybrid Sobolev spaces 풞 × 푠 푘,푙 7 4 푥 퐻e,b(B S ; 푊 ) since the group acts through diffeomorphisms. The linearized × 2 7 4 operator 푑푄 Diffe(B S ; 푊 ) is an elliptic edge operator for the product edge ∈ × structure and we have shown that 푑푄 commutes with the induced action of SO(5) on ˙∞(B7 S4; 푊 ). 풞 × The Sobolev spaces of sections of 푊 decompose according to the irreducible representations of SO(5), all finite dimensional and forming a discrete set. In particular these may be labelled by the eigenvalues, 휆, of the Casimir operator for SO(5) with a finite dimensional span when 휆 is bounded above. The SO(7, 1) action on H7 commutes with the SO(5) action on 푊 and acts transitively on H7, so the multiplicity of the SO(5) representation does not vary over H7. The individual representations of SO(5) in the decomposition of 푊 therefore form bundles over H7. Therefore we have the following lemma:

훿 푠,푘 Lemma 2.9. The group SO(5) acts on 푥 퐻푒,푏 (푀; 푊 ) transitively, and the bundle decomposes to subbundles on H7.

And we are going to show the following proposition for projection off finitely many small eigenvalues.

Proposition 2.5. There is a sifficiently large 푀 > 0, such that for 휆 > 푀 and any 휖 > 0 the two operators 푑푄휆 푖휖 acting on 퐷휆(휖) are both isomorphisms. And their ± 푘 inverses approaches two operators uniformly as 휖 0. →

To prove this proposition, we will bundle all the large eigenvalues together.

Definition 2.4. For 휆 [0, ), let 휋≥휆 : 푊 푊 be defined as the the projection ∈ ∞ → off the span of the eigenspaces of the Casimir operator for SO(5) with eigenvalues ∑︀ smaller than 휆, i.e. 휋≥휆 := Id ′ 휋휆′ . − 휆 <휆

54 Proposition 2.6. For any weight 푠 R and any orders 푘, 푙, the bounded operator ∈ defined as 푑푄 : 푥푠퐻푘+푚,푙(퐻 푆; 푊 ) 푥푠퐻푘,푙(퐻 푆; 푊 ) 푒,푏 × → 푒,푏 × is such that 휋≥휆푑푄 is an isomorphism onto the range of 휋≥휆 for some 휆 [0, ) ∈ ∞ ∞ (depending on 푠 but not on 푘 and 푙). Moreover, the range of Id 휋≥휆 on 퐶 (H − × 4 ∞ S ; 푊 ) is the space 퐶 (푀; 휆′<휆푊휆′ ) of sections of a smooth vector bundle over 푀 ⊕ 푚 and 푑푄 restricts to it as an elliptic element of Diff (푀; 휆′<휆푊휆). 0 ⊕ To prove this proposition, we first construct an SO(5)-invariant parametrix in the small edge calculus by finding a appropriate kernel on the edge streched product space

2 2 푋푒 which is defined from 푋 by blowing up the fiber diagonal.

2 Definition 2.5. The edge stretched product 푋푒 for an edge manifold 푋 is defined as the blow up [푋2; 푆] where 푆 is consists of all fibres of the product fibration 휋2 : (휕푋)2 푌 2 which intersect the diagonal of (휕푋)2. → Notice that from the definition of fiber diagonal, the blow up actually preserves the product structure of H7 S4, i.e. the fibred diagonal contained in 푋2 is just the × 4 4 7 2 4 2 product of 훥푒 푆 푆 , and the manifold after the blow up is [(퐻 ) ; 휕훥] (푆 ) . × × × Lemma 2.10. For 푀 = H7 S4, the edge stretched product is actually a product: × 2 7 2 4 2 푋푒 = [(H ) , 휕Δ] (S ) . × The elliptic element 푑푄 is transversely elliptic to the fiber diagonal. Therefore we have a parametrix construction in the small edge calculus as follows.

Lemma 2.11. Any SO(5)-invariant elliptic operator 푑푄 Diff푚(푀; 푊 ) has an ∈ 푒 ˜ −푚 SO(5)-invariant parametrix 퐸 in Ψ푒 (푀; 푊 ), such that

Id 푑푄 퐸,˜ Id 퐸˜ 푑푄 Ψ−∞(푀; 푊 ) − ∘ − ∘ ∈ 푒 are also SO(5)-invariant.

Proof. Any elliptic edge differential operator has a parameterix in the small edge calculus, following Mazzeo [25]. The construction gives the kernel of 퐸 as a classical

55 conormal distribution with respect to the ‘lifted diagonal’ of the stretched edge pro-

2 duce 푀푒 . This latter manifold is constructed by blow-up of the fibre diagonal (for the product fibtration 푀 = H7 S4) over the boundary of 푀. In fact, globally in terms × of the product this is just the diagonal of hyperbolic space over the boundary, i.e.

2 7 2 4 2 푀푒 = (H )0 (S ) × where the first space is the zero-stretched product for hyperbolic space. Thus infact the action of SO(5) on the kernel 퐸, through the product action on 푀 2, lifts smoothly 2 to 푀푒 and preserves the lifted diagonal (which is the closure of the diagonal in the interior). So we may average under the product action and define

∫︁ 퐸˜ = 푔 퐸. 푔∈푆푂(5) ·

Since 푑푄 is 푆푂(5) invariant by assumption (which is verified for the supergravity operator), 퐸˜ is also a parametrix,

푑푄 퐸˜ = Id +푅,˜ ∘ and the average remainder 푅˜ is also SO(5) invariant.

As a consequence, now 퐸˜ and 푅˜ both commute with the spherical eigenvalue projection 휋≥휆. The remainder 푅˜ can be characterized as:

˜ ∞ 4 2 −∞ 7 Lemma 2.12. The Schwartz kernel of 푅 is in 퐶 ((S ) , Ψ0 (H ) Hom(푊 )) ⊗ ⊂ ∞ 2 4 2 퐶 (푀푒 , 푊 ). In consequence it is a smooth map from (S ) to bounded operators on 푠 푝 푘 푥 퐻0 (퐻; 푊 ) for any 푠, 푝, with a norm depending on some 퐶 norm for any bounded range of 푠.

−∞ ˜ Proof. As an element in Ψ푒 (푀; 푊 ), the Schwartz kernel 푅 is smooth on the double edge space 푀 2, with values in the bundle Hom(푊 ) 퐾 where K is the kernel density 푒 ⊗ bundle. From the properties of the small calculus, 푅˜ vanishes to infinite order at the 2 4 2 2 left and right boundary faces. Because 푀푒 has the product structure (S ) H0, the × 56 ∞ 4 2 ∞ 2 ∞ 2 Schwartz kernel is in 퐶 ((S ) , 퐶 (H0, Hom(푊 ) 퐾)) where 퐶 (H0, Hom(W) K) ⊗ ⊗ −∞ is the Ψ0 (H; 푊 )) operators acting on 푊 . −∞ 푠 푘 Consider the map from Ψ0 (퐻; 푊 ) to bounded operators on 푥 퐻0 (H; 푊 ): since it is a continuous map from a Fréchet space to a normed space, the norm is bounded −∞ ˜ 푠 푘 by some norm on Ψ0 (H, 푊 ), i.e. the operator norm of 푅 on 푥 퐻0 (H; 푊 ) is bounded

by a constant 퐶(푠) 푅˜ 푘 . For any bounded interval 푠 [ 푆, 푆], the bound of ‖ ‖퐶 (퐻;푊 ) ∈ − ˜ 푅 푥푠퐻푘(퐻;푊 ) is uniform. ‖ ‖ 0

We can use the following interpolation result to show that 휋휆푅˜ rapidly decays as 휆 tends to infinity.

Lemma 2.13. 푥푠퐻푝,푘(푀) = 퐿2(푆4; 푥푠퐻푝(퐻)) 퐻푝+푘(푆4; 푥푠퐿2(퐻)). 푒,푏 0 ∩ 0

2 푝 Proof. We only prove the case 푠 = 0. If a function 푓 is in 퐿 (푆, 퐻0 (퐻)), that is 푉 푝(푓) 퐿2(푀), then applying an elliptic k-th order differential b-operator to 푓 we 푒 ∈ obtain an element in 퐻푝(푆4, 퐿2(퐻)), therefore by elliptic regularity 푓 퐻푝,푘(푀). ∈ 푒,푏

Lemma 2.14. As 휆 tends to infinity, the bounded operators 휋≥휆푅˜ decay in any 푠 푝,푙 Sobolev norm 푥 퐻푒,푏(푀; 푊 ), i.e.

˜ lim 휋≥휆푅 푥푠퐻푝,푙(퐻×푆;푊 ) = 0. 휆→∞ ‖ ‖ 푒,푏

Proof. Using Plancherel it follows that the Schwartz kernel of 휋≥휆푅˜ rapidly con- ∞ 4 2 푠 2 2 4 2 푠 푝 verges to 0 in 퐶 ((푆 ) , 푥 퐿0(퐻; 푊 )) and 퐿 ((푆 ) , 푥 퐻0 (퐻; 푊 )). Then we obtain 푠 푝,푙 휋≥휆푅˜ 0 as bounded operators on 푥 퐻 (푀; 푊 ) by the above lemma. ‖ ‖ → 푒,푏 ˜ As a consequence, for any fixed 푠, 푘, 푙, there is a 휆0 such that 휋≥휆0 푅 푥푠퐻푘,푙(푀;푊 ) ‖ ‖ 푒,푏 ≤ 1 푘 ˜ 2 , and this 휆0 only depends on some 퐶 norm. In the case that 휋≥휆0 푅 is small, we ˜ get that 휋≥휆0 푑푄휋≥휆0 퐸 is a perturbation of the identity, which is still an isomorphism, that is,

Lemma 2.15. For any 푠, 푘, 푙, there is a 휆0 depending only on s, such that

휋 푑푄휋 퐸˜ = 퐼푑 + 휋 푅˜ ≥휆0 ≥휆0 휋≥휆0 푊 ≥휆0

57 푠 푘,푙 where the right hand side is an isomorphism from 푥 퐻푒,푏 (푀; 푊 ) to itself.

푠 푘,푙 Proof. The norm of the operator on the right hand side acting on 푥 퐻푒,푏 (푀; 푊 ) is bounded from 0.

From the above lemma, we get that 휋≥휆0 푑푄 is an isomorphism mapping from 푠 푘,푙 푠 푘−푚,푙 휋≥휆0 푥 퐻푒,푏 (푀; 푊 ) to 휋≥휆0 푥 퐻푒,푏 (푀; 푊 ), proving the first part of proposition 2.6.

2.3.5 Individual eigenvalues with 휆 = 0, 16, 40 ̸

Now we consider those eigenvalues smaller than 휆0. Consider the projected operator 7 휋휆푑푄휋휆, which is viewed as a 0-problem on the tensor bundles on H (tensored with 4 훿 2,푘 fixed eigenforms on S ). Consider the operator 푑푄 on the space 휋휆(푥 퐻푒,푏 )(푀; 푊 ): computation shows that for a fixed 휆, except for 휆 = 0, 16, 40, the indicial roots of 푑푄 are contained in the range ( , 훿] [훿,¯ ) for some 훿, 훿¯, , so they are separated apart. −∞ ∪ ∞ Moreover the indicial roots are separated further when 휆 is bigger. With this infor- 휆 훿 푠,푘 훿 푠−2,푘 mation, we will show that, For 휆 > 40, 푑푄 : 휋휆푥 퐻 (푀; 푊 ) 휋휆푥 퐻 (푀; 푊 ) 푒,푏 → 푒,푏 is Fredholm for any 훿. We consider two operator related to 푑푄휆: the normal operator and reduced normal operator.

Definition 2.6 (Mazzeo, [25]). For 퐿 Diff*(푋) the normal operator 푁(퐿)is defined ∈ 푒 2 to be the restriction to the front face 퐵11 of the lift of 퐿 to 푋푒 . In terms of the local coordinate, if

∑︁ 푗 훼 훽 퐿 = 푎푗,훼,훽(푥, 푦, 푧)(푥휕푥) (푥휕푦) 휕푧 푗+|훼|+|훽|≤푚 then

∑︁ 푗 훼 훽 푁(퐿) = 푎(0, 푦,˜ 푧)(푠휕푠) (푠휕푢) 휕푧 , 푗+|훼|+|훽|≤푚

2 where 푠, 푢, 푥,˜ 푦,˜ 푧, 푧˜ is the lifted coordinate system on 푋푒 .

푘+1 Here 푁(퐿) acts on the product of R+ 퐹 and is invariant under the linear × translations and dilations on the first factor. This may be further reduced to bethe reduced normal operator which is a family of differential b-operators.

58 Definition 2.7. The reduced normal operator 푁0(퐿) is defined by applying the Fourier transform in the R푘 direction to 푁(퐿) then doing a rescaling. Specifically,

∑︁ 푗 훼 훽 * 푁0(퐿) = 푎푗,훼,훽(푡휕푡) (푖푡휂^) 휕푧 , 푡 R+, 휂^ 푆푦˜푌. ∈ ∈ 푗+|훼|+|훽|≤푚

Note that in our case, we are interested in the reduced operator of 푑푄휆 which is independent of the spherical variables 푧, and therefore is an ordinary differential operator.

Lemma 2.16. For 휆 > 40, there are 훿 < 0 < 훿¯, such that the reduced normal operator 휆 훿 2 + ¯ 푁0(푑푄 ) is an isomorphism on 푥 퐿 (R푠 ) for any 훿 < 훿 < 훿.

휆 Proof. 푁0(푑푄 ) has a pair of indicial roots for each 휆, which tend to as 휆 goes ±∞ to infinity. For 휆 = 0, 16, 40, the pair of indicial roots have different real parts and ̸ 훿 2 + thus can be separated. An ODE operator is not injective on 푥 퐿 (R푠 )when 훿 is less than the bottom indicial root and not surjective when 훿 is bigger than the top indicial root. Since there is a gap between the pairs of indicial roots for 휆 > 40, we can find ¯ 휆 훿 2 + ¯ 훿, 훿 such that 푁0(푑푄 ) is an isomorphism on 푥 퐿 (R푠 ) for 훿 < 훿 < 훿.

Lemma 2.17. For 휆 > 40 and 훿 < 훿 < 훿¯, the normal operator 푁(푑푄휆) is Fredholm 훿 푘 7 on 푥 퐻푏 (H ; 휋휆푊 ).

Proof. The reduced normal operator is obtained by Fourier transform and normal- ization of the operator 푁(푃휆), so we may do an inverse Fourier transform to get back to 푁 from 푁0.

From Mazzeo [25], there exists a parametrix 퐺 and two projectors 푃푖, such that 2 the Schwartz kernels 푘(퐺) and 푘(푃푖) all lift to distributions on 푋푒 polyhomogeneous conormal at all boundary faces. The exponents in the expansions are determined by the indicial roots.

Remark 1. On 푀 we have the following inclusion:

푥3+훿퐶∞(푀; 푊 ) 푥훿퐿2(푀; 푊 ). ⊂ 푒 59 The reason why the line of indicial roots with Re(푠) = 3 is important is that there is a symmetry with respect to this line which is related to the self-adjointness of the operator.

휆 훿 푘 7 Lemma 2.18. The kernel of the normal operator 푁(푑푄 ) on 푥 퐻푏 (H ; 휋휆푊 ) is zero for 훿 > 0.

Proof. This follows from the fact that there are no finite dimensional 퐿2 eigenspaces for functions and tensors on H7.Indeed, consider the representation of 푆푂(7, 1) on tensor bundles on H7. There are no finite dimensional 퐿2 invariant subspace of forms on H7. For tensors, from Delay’s result [7], there are no 퐿2 eigentensors.

As a result we have the following:

휆 훿 푘 Lemma 2.19. For any 훿 > 0, the normal operator 푁(푑푄 ) is injective on 푥 퐻0 (푀; −훿 푘 휋휆푊 ) and surjective on 푥 퐻0 (푀; 휋휆푊 ).

훿 푘 2 Proof. The kernel of this map on 푥 퐻0 (푀; 휋휆푊 ) is contained in 퐿 eigenspace of forms and tensors on H7, which from the lemma above does not have any nontrivial elements. Therefore it is injective. By duality, it’s surjective on the bigger space

−훿 푘 푥 퐻0 (푀; 휋휆푊 ).

We now return to the original operator 푑푄휆 and show that it is Fredholm.

휆 3+훿 2,푘 3+훿 Proposition 2.7. For any 훿 > 0, the operator 푑푄 : 푥 퐻 (푀; 휋휆푊 ) 푥 푒,푏 → 0,푘 3−훿 2,푘 퐻푒,푏 (푀; 휋휆푊 ) is injective. Likewise, it is surjective on 푥 퐻푒,푏 (푀; 휋휆푊 ).

Proof. The normal operator is an isomorphism at each point of the boundary S6. Thus for a general kernel element of 푑푄휆, we decompose it using 푆푂(7, 1) action, so it falls into the kernel space of the normal operator 푁(푑푄휆) for which there is not any. Therefore the kernel is also trivial for the operator 푑푄휆. So it is injective on the smaller space, and by duality surjective on the bigger space.

60 2.3.6 Individual eigenvalues with 휆 = 0, 16, 40

For those eigenvalues corresponding to indicial roots with real part 3, we consider each

3−훿 2,푘 subspace 휋휆푥 퐻푒,푏 (푀; 푊 ) separately. Restricted to these subspaces, the linearized operator is a 0-operator on hyperbolic space, of which the main part is the hyperbolic laplacian 훥 . From Guillarmou [18], the resolvent of 훥 휆, denoted as 푅(휆), extends H H− to a meromorphic family with finite degree poles. Similarly, we want to show that 푑푄 −1 has two generalized inverses 푅±, which is the extension of the resolvent (푑푄 푖휖) ± when 휖 approaches the real axis. More specifically, we will prove the following result 휆 −훿 푠 7 훿 푠−2 7 that, for 휆 = 0, 16, 40, 푑푄 : 푥 퐻0 (H ; 휋휆푊 ) 푥 퐻0 (H ; 휋휆푊 ) is bounded and → has two generalized inverses. We will be using indicial roots analysis again here, but first we will need to show that the indicial roots may be separated from the 퐿2 line by perturbing the operator.

Lemma 2.20. For 휆 = 0, 16, 40, the two indicial roots of operator 푑푄휆 푖휖 lie off ± the Re(푠) = 3 line.

Proof. Suppose 푠 C is an indicial root for an operator 푃 on a point 푝 at the ∈ boundary, then we have 푃 (푥푠) = 푂(푥푠+1) by definition. For 휖 = 0, the following ̸ computation shows that 푠 is no longer an indicial root: (푃 +푖휖)(푥푠) = 푖휖푥푠 +푂(푥푠+1) / ∈ 푂(푥푠+1). Instead, take the harmonic 4-form part which has inidicial roots 3 6푖 which ± in the indicial root computation is 푃 (푥3+푠) = (푠2 +36)푥3+푠 +푂(푥4), after perturbation it becomes (푃 + 푖휖)(푥3+푠) = (푠2 + 푖휖 + 36)푥3+푠 + 푂(푥푠+1)

2 so the indicial equation becomes 푠 = 푖휖 36 which moves the two roots 3 + 푠± off − − the line of 푅푒(푠) = 3. A similar argument applies to other two pairs of roots.

휆 −1 −훿 0,푘 훿 2,푘 Lemma 2.21. For 휖 = 0, the inverse (푑푄 푖휖) : 푥 퐻 (푀; 휋휆푊 ) 푥 퐻 (푀; ̸ ± 푒,푏 → 푒,푏 휋휆푊 ) exists as a bounded operator.

Proof. Using the indicial roots separation and same argument as before for those

휆 훿 2,푘 eigenvalues greater than 40, the operator 푑푄 푖휖 is Fredholm on 푥 퐻 (푀; 휋휆푊 ), ± 푒,푏 injective on the smaller space and surjective on the larger space.

61 We need the following limiting absorption principle:

Proposition 2.8. For small weight 0 < 훿 < 1, and number 푠, 푘, the operators (푑푄휆 ± 푖휖)−1 converges uniformly to bounded operators on weighted space,

−1 휆 lim (푑푄휆 푖휖) 푅± = 0. 휖→0 ‖ ± − ‖

휆 훿 푠,푘 훿 푠−2,푘 where 푅 : 푥 퐻 (푀; 휋휆푊 ) 휋휆푥 퐻 (푀; 휋휆푊 ). ± 푒,푏 → 푒,푏

Proof. To prove this, we will consider the reduced normal operator of 푑푄휆 푖휖, which − is a differential operator (parametrized by 푦 and 휖), is injective from 푥훿퐻2(R+) → 푥훿퐿2(R+) for any fixed 훿. This ODE operator may be extended holomorphically as 휖 passes through zero from above, and the solution of the ODE extends holomorphically as well. After extending it past zero, the smaller indicial root moves into the larger

3+훿 푠,푘 3+훿 푠−2,푘 one. 푑푄휆 푖휖 is injective on 푥 퐻 (푀, 푊 ) 푥 퐻 (푀, 푊 ) (as it excludes half ± 푒,푏 → 푒,푏 −1 the roots), and the resolvent 푅+ := lim휖→0(푑푄휆 푖휖) is an right inverse. Similarly − for 푅−.

2.3.7 Boundary data for the linear operator

Combining the analysis for 휆 off the 퐿2 line and on the 퐿2 line, we conclude the following for 푑푄:

훿 푠,푘 Proposition 2.9. For 훿 (0, 1), there are two generalized inverses 푅± : 푥 퐻 (푀; 푊 ) ∈ 푒,푏 푥−훿퐻푠+2,푘(푀; 푊 ) for operator 푑푄, such that → 푒,푏

훿 푠,푘 훿 푠,푘 푃 푅+ = 퐼푑, 푃 푅− = 퐼푑 : 푥 퐻 (푀; 푊 ) 푥 퐻 (푀; 푊 ). ∘ ∘ 푒,푏 → 푒,푏

As a consequence, we find the following right inverse which is real:

1 (푑푄)−1 := (푅 + 푅 ). 2 + −

To get the main theorem, we will parametrize the domain by the boundary data

below to get a family of operators 푄푣. To show 푄푣 is a local isomorphism, we use an

62 −1 implicit function theorem argument that, for small boundary data 푣, 푄푣 (푑푄0) , ∘ which acts on a fixed space independent of 푣, is a perturbation of the identity. First we define three bundles on S6 that parametrize the incoming and outgoing boundary data for the linear operator.

± Definition 2.8. Let 푉 to be the space of sections of the bundle of 3-forms with 푆 1 * eigenvalue 푖: ± 3 ∞ 6 ⋀︁ * 6 푉1 := 푣1 퐶 (S ; 푇 S ): 6 푣1 = 푖푣1 . { ∈ *S }

± ± Similarly let 푉2 and 푉3 be the smooth functions on the 6-sphere tensored with eigenforms on 4-sphere:

∞ 6 푐푙 4 푉2 := 푣2 휉16 : 푣2 퐶 (S ; R), 휉16 퐸16(S ) , { ⊗ ∈ ∈ }

∞ 6 푐푙 4 푉3 := 푣3 휉40 : 푣3 퐶 (S ; R), 휉40 퐸40(S ) . { ⊗ ∈ ∈ } Remark 2. Note the dimension of the closed 1-form with the first and second eigenvalues are determined by the degree 2 and 3 spherical harmonics with 4 variables, which, repectively, are 5 and 14 dimensional vector spaces.

To save some space we will use the following abbreviation for the leading expansion given by the three parameters.

Definition 2.9 (Leading expansion for the linear operator). When we say the leading ∑︀3 + expansion is given by 푖=1 푣푖 휉푖, we will mean

+ − + 휃1 + 휃1 3+휖 퐻(4,0) = 푣1 휉1푥 + 푆1(푣1 )휉1푥 + 푂(푥 )

+ 휃+ + 휃− + 휃+ + 휃− 3+휖 Tr 7 푔 = Tr 7 ℎ + 7 푠 (푣 휉2푥 2 + 푆2(푣 )휉2푥 2 + 푣 휉3푥 3 + 푆3(푣 )휉3푥 3 ) + 푂(푥 ) H H * 2 2 3 3 + 휃+ + 휃− + 휃+ + 휃− 3+휖 Tr 4 푔 = Tr 4 ℎ + 4 푠 (푣 휉2푥 2 + 푆2(푣 )휉2푥 2 + 푣 휉3푥 3 + 푆3(푣 )휉3푥 3 ) + 푂(푥 ) S S * 2 2 3 3 + − + − + 휃2 + 휃2 + 휃3 + 휃3 3+휖 푔(1,1) = ℎ1,1 + (푣2 휉2푥 + 푆2(푣2 )휉2푥 + 푣3 휉3푥 + 푆3(푣3 )휉3푥 ) + 푂(푥 )

+ 휃+ + 휃− + 휃+ + 휃− 3+휖 퐻(1,3) = 푑퐻 (푣 휉2푥 2 + 푆2(푣 )휉2푥 2 + 푣 휉3푥 3 + 푆3(푣 )휉3푥 3 ) + 푂(푥 ) − 2 2 3 3 + 휃+ + 휃− + 휃+ + 휃− 3+휖 퐻(0,4) = 6 Vol 4 +푑푠 푠 (푣 휉2푥 2 + 푆2(푣 )휉2푥 2 + 푣 휉3푥 3 + 푆3(푣 )휉3푥 3 ) + 푂(푥 ) S * 2 2 3 3 (2.45)

63 Theorem 2.3. For any Poincaré–Einstein metric ℎ that is close to the background

metric 푔0, the solution to the linearized equations 푑푄(푔, 퐻) = 0 is parametrized by the data of the three bundles 푉푖. ⊕

To prove the theorem above, we will first work on hyperbolic space, and use a perturbation argument to show that, for a nearby Poincaré–Einstein metric, the parameter space is also transversal to the kernel. For the weight 3 훿 where 훿 > 0 − and is small, the operator 푑푄 is surjective but not injective, and there is a null space which corresponds to the indicial roots with real part 3. The scattering matrix relates the incoming and outgoing data of eigenfunctions corresponding to a point in the continuous spectrum. Once we fix the incoming data in the expansion, the outgoing data is determined by the scattering matrix. In the hyperbolic metric case, the scattering matrix 푆푖(푠), 푖 = 1, 2, 3 is defined for each pair of special indicial roots. The scattering matrix in the hyperbolic case is

∞ 7 + ∞ 7 − 푆푖(푠): 퐶 (휕H ; 푉푖 ) 퐶 (휕H ; 푉푖 ) (2.46) →

with property that if

3 ∑︁ 휃+ 휃− 3+휖 푑푄(푢) = 0, 푢 = 푓푖푥 푖 + 푔푖푥 푖 + 푂(푥 ), 푖=1 then

푔푖 휕푀 = 푆푖(푠)푓푖. |

Proposition 2.10. For the base case with hyperbolic metric, the kernel of operator

푑푄 is parametrized by the sections of 푉푖. More specifically, for any small incoming ⊕ + + + + real data 푣 = (푣1 , 푣2 , 푣3 ), there is a unique solution to the linearized equations with 3 + − ∑︀ + 휃푖 + 휃푖 푖=1 휉푖(푣푖 푥 + 푆푖(푣푖 )푥 ) as the leading expansion.

Proof. We use the result of Graham–Zworski [16] and Guillarmou–Naud [19] about

64 the description of scattering matrix in hyperbolic space:

(︁√︁ )︁ 푛 푛−1 2 1−푛 Γ ΔS푛 + ( ) + + 푠 푛−2푠 Γ( 2 푠) 2 2 푆(푠) = 2 − 푛 (︁√︁ )︁, Γ(푠 ) 푛−1 2 푛+1 2 Γ Δ 푛 + ( ) + 푠 − S 2 2 −

+ where if we put in 푠 = 휃1 = 3 + 6푖 we get

√︁ 25 1 Γ( 6푖) Γ( 훥S6 + 4 + 2 + 6푖) 푆(3 + 6푖) = 2−12푖 − . Γ(6푖) √︁ 25 1 Γ( 훥 6 + + 6푖) S 4 2 −

Since the scattering matrix is a function of the laplacian on the boundary 푆6 we can

take the eigenvalue expansion on 6-sphere with real eigenform 푓휆, we would consider the following expression, which is real and forms the leading order of the actual solution:

3+6푖 3−6푖 3+6푖 3−6푖 −12푖 푖2휃 6푖 푥 푓휆 + 푥 푆(3 + 6푖)푓휆 = 푥 푓휆 + 푥 (2 푒 휆 )푓휆.

Here 휃 is a real number determined by

(︁√︁ )︁ Γ 휆 + 25 + 1 + 6푖 2푖휃(휆) Γ( 6푖) 4 2 푒 = − (︁√︁ )︁, (2.47) Γ(6푖) Γ 휆 + 25 + 1 6푖 4 2 −

by using the relation of Γ(¯푧) = Γ(푧)

so that the right hand side of (2.47) is a complex number with norm 1 and 휃 is a real number determined by 휆.

Rearranging the expression, the solution in the eigenvalue 휆 component is

3+6푖 3−6푖 −12푖 2푖휃 휋휆푢 = 푥 푓휆 + 푥 2 푒 푓휆 (2.48)

3 −6푖 푖휃 (︀ 6푖 푖휃 −6푖 푖휃)︀ = 푥 2 푒 (2푥) 푒 + (2푥) 푒 푓휆 (2.49)

3 1−6푖 푖휃 (︀ 6푖 푖휃(휆))︀ = 푥 2 푒 Re (2푥) 푒 푓휆 (2.50)

65 which is a product of a real 3-form with complex constant 푥321−6푖푒푖휃. Therefore in this case,

푖푡 푓 = 푓 푒 , 푡 R, | | ∈ and 푓 is determined by a real 3-form. ± ± A similar computation shows that 휃2 , 휃3 are parametrized by real functions.

With the computation above, we have shown that the leading expansion of the solution has the form

± 휃± ± 휃± ± 휃± 3+훿 푢 = 푣 푑푥/푥휉1푥 1 + 휉2푣 푥 2 + 휉3푣 푥 3 + 푂(푥 ) 1 ∧ 2 3

where

± ± − + 푣 푉 , 푣 = 푆푖푣 . 푖 ∈ 푖 푖 푖 Definition 2.10. We define the Poisson operator 푃 for a Poincaré–Einstein metric ℎ: if we denote the operator that maps from the space of real solutions to the incoming + + + boundary data (푣1 , 푣2 , 푣3 ) by 푓, then the inverse of this map is denoted by 푃 , which is the Poisson operator:

−1 ∑︁ + 3+푖푠 + 3−푖푠 3−훿 ∞ 푃 = 푓 : 푉푖 퐷푣,ℎ, 푣푖 Re(푣푖 푥 + 푆푖(푠)푣푖 푥 ) 푥 퐻푏 (푀; 푊 ). ⊕ → { } ↦→ 푖,푠 ∈

For the hyperbolic case, it maps to the actual solution with leading expansion

(푣푖). For nearby Poincaré–Einstein metrics, it maps to a real element in the domain which is not necesssarily an element in the null space, however it is very close to a null element with the same leading expansion. We show below that it is transversal

to the range of right inverse 푅+ + 푅−. We also remark here that, the domain and range of this operator 푃 are real, and for nearby Poincaré–Einstein metrics, the composition 푄 푃 (푢) also maps into real ∘ space.

Proposition 2.11. For the base case, the real null space of 푑푄 is the range of 푖(푅+ − 푅−).

66 Proof. This is Stone’s theorem. For any element 푢, 푑푄 (푅+ 푅−)푢 = 0, and any ∘ − element in the kernel of 푑푄 is parametrized by 푢. We multiply 푖 to make it real since

푅+ 푅− is completely imaginary. −

Lemma 2.22. For a Poincaré–Einstein metric ℎ that is closed to the background

metric 푔0, the range space of the sum of two generalized inverses 푅± is transversal to

the range of their difference: Range(푅+ + 푅−) is transversal to Range(푅+ 푅−). −

Proof. For the base case: the range of P is the kernel of 푑푄, which is also range of

푅+ 푅−. However, the range of 푅+ + 푅− doesn’t contain any element of the kernel, − since 푑푄 푅+, 푑푄 푅− = 0. ∘ ∘ ̸ Since transversality is stable under small perturbations, the result follows.

Then with the two lemmas above, we conclude:

Lemma 2.23. The range space of the Poisson operator 푃 is transversal to Range(푅++

푅−).

2.4 Nonlinear equations: application of the implicit function theorem

From the discussion of the linear operator 푑푄 above, we now can apply the implicit function theorem to get results for the nonlinear operator. The nonlinear terms include two parts, one from the linearization of the curvature operator, the other from the product type terms. We will use a perturbation argument to show that for each Poincaré–Einstein metric, the nearby solutions are parametrized by the three parameters on S6 as in the linear case. To deal with the fact that the domain changes with the base metric and the parameters we put in, we will use an implicit function theorem, that is, constructing a map from range space to itself, and show that this map is a perturbation of identity, therefore an isomorphism.

67 2.4.1 Domain defined with parametrization

First of all we define the domain for all the product type metrics of a nearby Poincaré–

3 Einstein metric ℎ and each parameter set 푣 = (푣1, 푣2, 푣3) 푉푖. From the discus- ∈ ⊕푖=1 sion of the generalized inverses of the linearized operator, we know that the image of

1 2 (푅+ + 푅−) is transversal to the image of the Poisson operator which is close to the kernel of the linearized operator for a nearby Poincaré–Einstein metric. In the nonlinear case, we define the domain so that, for each parameter 푣, it is an affine section translated by 푃 푣, where 푃 is the Poisson operator defined above. The domain has the property that, in the linearized case with base hyperbolic metric, it is mapped by

훿 0,푘 푑푄 isomorphically back to the range space 푥 퐻푒,푏 (푀; 푊 ).

Definition 2.11. (Domain of nonlinear operator) For a Poincaré–Einstein metric ℎ

that is close to the base hyperbolic metric and a set of parameters 푣 = (푣1, 푣2, 푣3) in

bundle 푉 , the domain 퐷ℎ,푣 of the nonlinear operator is defined as

{︁1 ⃒ 훿 0,푘 }︁ 퐷ℎ,푣 := (푅+ + 푅−)푓 + 푃 푣 ⃒ 푓 푥 퐻 (푀; 푊 ) . 2 ⃒ ∈ 푒,푏

Note that the domain depends on the choice of ℎ and 푣, where the dependence of 1 −1 ℎ comes from the construction 2 (푅+ + 푅−) = (푑푄ℎ) . One important property of 훿 0,푘 this domain is that 퐷ℎ,푣 is mapped surjectively to the range space 푥 퐻푒,푏 (푀; 푊 ) by

the linear operator 푑푄ℎ.

훿 0,푘 Lemma 2.24. The range space of the linear map 푑푄ℎ acting on 퐷ℎ,푣 is 푥 퐻푒,푏 .

Proof. Since all the operations are linear,

1 푑푄 ( (푅 + 푅 )푓 + 푃 푣) = 푑푄 (푑푄 )−1푓 + 푑푄 (푃 푣) = 푓 + 푂(푥4). ℎ 2 + − ℎ ℎ ℎ

Here we used the fact that 푅+ and 푅− are both generalized inverses for 푑푄ℎ, and the Poisson operator maps into a space with extra decay in 푥. Since 푓 can be any element 훿 0,푘 in 푥 퐻푒,푏 (푀; 푊 ), it follows that 푑푄ℎ maps the domain 퐷ℎ,푣 onto the range.

Now with the domain we can define a nonlinear operator 푄ℎ,푣 which is parametrized

68 by the background Poincaré–Einstein metric ℎ and the set of parameters 푣, which

can be viewed as a translation of the original operator 푄ℎ,0.

Definition 2.12. We define the parametrized nonlinear operator 푄ℎ,푣 on domain

퐷ℎ,푣:

푄ℎ,푣푢 := 푄ℎ,0(푢 + 푃 푣)

As a translation of the original operator, the linearization of 푄ℎ,푣 at the point (0, 0) is the same for the original nonlinear supergravity operator.

Lemma 2.25. At any boundary point on S6, the linearization of the parametrized

nonlinear operator 푄ℎ,푣 at point 푢 = (푘, 퐻) = (0, 0) equals to 푑푄0,0.

Proof. Since 푄ℎ,푣 is defined as a translation of 푄푔0,0 by 푃 푣, then near the boundary S6,

푑푄ℎ,푣(0, 0)푢 = 푑(푄ℎ,0(푢 + 푃 푣)) = 푑푄ℎ,0푢 + 푑푄 푃 푣 = 푑푄푔 ,0푢 + 푂(푥) ∘ 0

Therefore on any boundary point, we have the same linearization as 푑푄0,0.

Next we show that the nonlinear terms are well controlled, i.e. mapped into the

훿 2,푘 smaller space 푥 퐻푒,푏 (푀; 푊 ).

Lemma 2.26. For 푘 sufficiently large, the product type nonlinear terms: 퐹 퐹 ∘ − 푑(퐹 퐹 ), and 퐹 ⋀︀ 퐹 푑(퐹 퐹 ) are both contained in 푥훿퐻2,푘(푀; 푒 4 (푀)). ∘ − ∧ 푒,푏 ∧ Proof. The nonlinear parts are 퐹 퐹 and 퐹 퐹 which are products of two elements ∧ ∘ in the range space 푥훿퐻2,푘(푀, 푆2(푒푇 *푀)) 푥훿퐻1,푘(푀, ⋀︀4 (푒푇 *푀)). With respect to 푒,푏 ⊕ 푒,푏 푐푙 훿 2,푘 basis of the edge bundles, these may be considered locally as functions in 푥 퐻푒,푏 (푀). Using the algebra property included in the appendix, we know that for 푟 > 3, and − 푠, 푘, and any 푓, 푔 푥푟퐻푠,푘(푀), the product 푓푔 is also in 푥푟퐻푠,푘(푀). Since in our ∈ 푒,푏 푒,푏 case 훿 > 0, the result follows.

The last nonlinear term is the remainder from the linearization of Ric, for which we show below that it is also contained in the range space.

69 Lemma 2.27. The nonlinear remainder of Ric, Ric 푑(Ric) is contained in 푥3+훿퐻0,푘 − 푒,푏 (푀; Sym2(푇 *푀)).

Proof. We compute the linearization 푑(Ric), which acting on a 2-tensor ℎ can be written as

1 푚푙 푑(푅푖푐)[ℎ] = − 푔 ( 푚 푙ℎ푗푘 푚 푘ℎ푗푙 푙 푗ℎ푚푘 푗 푘ℎ푚푙). 2 ∇ ∇ − ∇ ∇ − ∇ ∇ − ∇ ∇

Comparing Ric and 푑(Ric), the difference is a 3rd order polynomial of 푔, 푔−1 and first order derivatives of these with smooth coefficients. Since the metric component 푔 and −1 0 푠,푘 푔 are smooth, hence in 푥 퐻푒,푏 (푀), it follows again by the algebra property that 훿 푠,푘 2 their product is contained in 푥 퐻푒,푏 (푀; Sym (푀)).

−1 The composed operator 푄ℎ,푣 (푑푄0,0) is this well-defined operator as a map on ∘ the following space:

−1 훿 0,푘 훿 0,푘 푄ℎ,푣 (푑푄0,0) : 푥 퐻 (푀; 푊 ) 푥 퐻 (푀; 푊 ). ∘ 푒,푏 → 푒,푏 (︁1 )︁ 푓 푄ℎ,0 (푅+ + 푅−)푓 + 푃 푣 ↦→ 2 We now discuss the properties of this operator using the implicit function theorem.

Lemma 2.28 (Implicit function theorem). Consider the following smooth map 푓 :

푉 푀 푀 near a point (푣0, 푚0) 푉 푀 with 푓(푣0, 푚0) = 푐, if the linearization of × → ∈ × the map with respect to the second variable 푑푓2(푣0, 푚0): 푀 푀 is an isomorphism, → then there is neighborhood 푣0 푈 푉 and a smooth map 푔 : 푉 푀, such that ∈ ⊂ → 푓(푣, 푔(푣)) = 푐, 푣 푈. ∀ ∈

Theorem 2.4. For any 푠 2, 푘 0 there exists 훿 > 0, 휌 > 0, such that, for a ≥ ≫ Poincaré–Einstein metric ℎ that is sufficiently close to the base metric 푔0, for each 3 + small boundary value perturbation 푣 = 푖=1푣푖 with 푣 퐻푘(푀;⊕푉 ) < 휌, there is a unique ⊕ ‖ ‖ 푏 푖 −훿 푠,푘 solution 푢 = (푔, 퐻) 퐷푣,ℎ 푥 퐻 (푀; 푊 ) satisfying the supergravity equations ∈ ⊂ 푒,푏 70 푄(푢) = 0 with the following leading expansion

3 ∑︁ + 휃푖 (푔, 퐻) = (ℎ, 6 VolS4 ) + 푣푖 휉푖푥 (2.51) 푖=1

To prove the theorem, we will apply the implicit function theorem to the following operator:

−1 3 3+훿 0,푘 3+훿 0,푘 푄ℎ,푣 (푑푄0,0) : 푉푖 푥 퐻 (푀; 푊 ) 푥 퐻 (푀; 푊 ) · ∘ ⊕푖=1 × 푒,푏 → 푒,푏

−1 (푣, 푓) 푄ℎ,푣 (푑푄0,0) (푓) ↦→ ∘

푘 6 훿 0,푘 This map is from a neghborhood of the Banach space 퐻 (S , 푉푖) 푥 퐻푒,푏 (푀; 푊 ) to ⊕ × 3+훿 0,푘 the Banach space 푥 퐻푒,푏 (푀; 푊 ). The following is a consequence of Lemma 2.25.

−1 Lemma 2.29. The linearization of 푄ℎ,푣 (푑푄0,0) at point (푣, 푓) = (0, 0) 푉푖 ∘ ∈ ⊕ × 훿 0,푘 푥 퐻푒,푏 (푀; 푊 ) is an isomorphism.

훿 0,푘 Proof. From Lemma 2.25 we know that at the point (푣, 푓) = (0, 0) 푉푖 푥 퐻 (푀; ∈ ⊕ × 푒,푏 푊 ) the linearization, which is the composition of linearization of the operators, is

−1 훿 0,푘 훿 0,푘 푑(푄ℎ,푣 푑푄0,0) )(0,0) = id : 푥 퐻 (푀; 푊 ) 푥 퐻 (푀; 푊 ). ∘ 푒,푏 → 푒,푏

−1 Lemma 2.30. For a given metric ℎ, the map 푄ℎ,푣 (푑푄0,0) as an edge operator ∘ varies smoothly with the parameter 푣 푉 . ∈

−1 Proof. From the construction of 푑푄0,0 we know it is an edge operator. And from the discussion for 푄ℎ,0, this nonlinear operator is also edge. Now we we only need to show that when the nonlinear operator 푄 applies to elements of type 푓 +푃 푣, it varies smoothly with the parameter 푣. This follows from the algebra property and the fact 푠 푠−2 that a second order elliptic edge operator maps from 퐻푒 (푀) to 퐻푒 (푀) smoothly as shown in the appendix.

We now obtain the following, as a direct result of the implicit function theorem.

71 Proposition 2.12. There are 휌1, 휌2 > 0, such that on the two neighborhoods 푈1 := 3+훿 0,푘 푣 푉푖 푣푖 퐻푘 < 휌1 and 푈2 := 푓 푥 퐻푒,푏 (푀; 푊 ) 푓 푥3+훿퐻0,푘(푀;푊 ) < 휌2 , { ∈ ⊕ |‖ ‖ } { ∈ |‖ ‖ 푒,푏 } there exists a continuous differentiable map 푔 : 푈1 푈2 such that →

−1 푄ℎ,푣 (푑푄0,0) (푔(푣)) = 0. ·

Proof. Now we use the implicit function theorem, we can find neighborhoods of 푣 = 0 −1 and 푓 = 0, in this case, 푈1 and 푈2 such that the nonlinear map 푄ℎ,푣 (푑푄0,0) is a ∘ bijective smooth map on 푈2 for any 푣 푈1 . And this gives us the parametrized map ∈ 푔 from 푈1 to 푈2.

With the proposition above, we find a solution for each set of parameter 푣푖 . { }

−1 Proof of Theorem 2.4 . Using the definition of 푄ℎ,푣 (푑푄0,0) (푔(푣)) = 0 with the ∘ map 푔 constructed above, we can rewrite it as

−1 푄ℎ(푑푄 (푔(푣)) + 푃 푣) = 0.

That is, for each parameter set 푣, 푢 = 푑푄−1(푔(푣)) + 푃 푣 is the unique solution in the −훿 푠,푘 space 퐷푣,ℎ 푥 퐻 (푀; 푊 ). ⊂ 푒,푏

2.4.2 Regularity of the solution

Next we show that the solution obtained above is smooth if the boundary data is smooth.

∞ 6 Proposition 2.13. If the boundary data 푣 퐶 (S , 푉푖), then the solution 푢 is in ∈ ⊕ ∞ 퐻푏 (푀; 푊 ).

Proof. This is done by elliptic regularity. We would like to prove that for any 푘,

푘+2 푘 6 0,푘 푢 퐻 (푀;푊 ) 퐶( 푣 퐻 (푆 ;푉푖) + 푄(푢) 퐻 (푀;푊 )). ‖ ‖ 푏 ≤ ‖ ‖ ‖ ‖ 푒,푏 72 Since the principal part of 푄 is the elliptic edge operator 휆푑푄휆 and for each 휆, we ⊕ have such elliptic estimates

푘+2 푘 6 0,푘 푢휆 퐻 (푀;푊 ) 퐶( 푣휆 퐻 (푆 ;푉푖) + 푑푄휆(푢) 퐻 (푀;푊 )). ‖ ‖ 푏 ≤ ‖ ‖ ‖ ‖ 푒,푏

The nonlinear parts are lower order:

∑︁ 푄 퐷푄휆 퐻푘+2(푀;푊 ) < 퐶, ‖ − ‖ 푏 leading to an elliptic estimate for 푄.

We can also obtain a classical expansion of the solution. The leading terms are

given by the combination of incoming and outgoing boundary data from 푉푖, and lower order terms are solved by iteration.

Proposition 2.14. The solution has a classical polyhomogeneous expansion, with leading term 3 ∑︁ ± ∑︁ ∑︁ (︁ ∑︁ )︁ ± 3+휃푖 푗 푘 푢 = 푣푖 푥푖푥 + 푣푗푥 (log 푥) 푓푘 푖=1 푗≥4 푘≤푗

∞ where 푣푗 are eigenforms, and 푓푘 (푀, 푊 ). For the lower order terms, the expo- ∈ 풞 nent of the logarithmic terms grows linearly with the order.

Proof. We solve the problem iteratively. For the first order problem, from the lin-

∑︀ ± 3±휃푖 earization and its inverse construction, we have 푢1 = 푣푖 푥 푥1,푖 with

3+훿 ∞ 푄(푢1) = 푥 푒1, 푒1 (푀, 푊 ). ∈ 풞

훿 Then we solve away the 푥 푒1 term and lower indicial roots appear here, which gives us

3 ∑︁ ± 3±휃푖 푢2 = 푥 ( 푣푖 푥 푥푖,2 + 푥 log 푥). 푖 The log terms appear because an order in the expansion of 푥훿푒 coincides with one of

73 the indicial roots. Then iteratively we obtain the terms

∑︁ ∑︁ 푗 ∑︁ 푘 푢푗 = 푣푗푥 ( (log 푥) 푓푘), 푗≥4 푘≤푗

where each time the power of log increases by at most one.

In terms of the explicit formulae, we may summarize the previous results as follows:

± Theorem 2.5. The solution as we get from given boundary data 푣푖 is polyhomege- neous and has the following expansion:

+ − + 휃1 + 휃1 3+휖 퐻(4,0) = 푣1 휉1푥 + 푆1(푣1 )휉1푥 + 푂(푥 )

+ 휃+ + 휃− + 휃+ + 휃− 3+휖 퐻(1,3) = 푑퐻 (푣 휉2푥 2 + 푆2(푣 )휉2푥 2 + 푣 휉3푥 3 + 푆3(푣 )휉3푥 3 ) + 푂(푥 ) − 2 2 3 3 + 휃+ + 휃− + 휃+ + 휃− 3+휖 퐻(0,4) = 6 Vol 4 +푑푠 푠 (푣 휉2푥 2 + 푆2(푣 )휉2푥 2 + 푣 휉3푥 3 + 푆3(푣 )휉3푥 3 ) + 푂(푥 ) S * 2 2 3 3

Then finally using elliptic regularity, we can extend the result to boundary data with Sobolev regularity.

푘 6 Proposition 2.15. For any 푘 > 0, given boundary data 푣푖 퐻 (S ), the solutions ∈ 푠,푘 we get is in 퐻푒,푏 (푀; 푊 ).

Proof. This follows from the elliptic estimate

푢 퐻푠,푘(푀;푊 ) 퐶 푣 퐻푘(S6;푉 ) + 푄(푢) 퐻푠−2,푘(푀;푊 ). ‖ ‖ 푒,푏 ≤ ‖ ‖ ‖ ‖ 푒,푏

74 2.5 Edge operators

2.5.1 Edge vector fields and edge differential operators

푠,푘 Proposition 2.16. 퐻푒,푏 (푀) is a well-defined space.

Proof. It is easy to see by using the commutator relation [ 푒, 푏] 푏. 풱 풱 ⊂ 풱 푠,푘 푠−푚,푘 Proposition 2.17. Any m-th order edge operator P maps 퐻푒,푏 (푀) to 퐻푒,푏 (푀), for 푚 푠. ≤ Proof. Locally, any m-th order edge operator P can be written in the following form

∑︁ 푗 훼 훽 푃 = 푎푗,훼.훽(푥, 푦, 푧)(푥휕푥) (푥휕푦) 휕푧 푗+|훼|+|훽|≤푚

푠,푘 푠−1,푘 If we can prove for m=1, P maps 퐻푒,푏 (푀) to 퐻푒,푏 (푀), then by induction, we can prove for any m. Therefore we restrict to the case 푚 = 1. We just need to check that, for a function 푢 퐻푠,푘(푀), 푃 푢 satisfies ∈ 푒,푏

푉 푖푃 푢 퐻푘(푀), 0 푖 푠 1. 푒 ∈ 푏 ≤ ≤ −

The we prove the proposition by induction on k. For k=1 case, since a boundary vector field 푉 푏(푀) satisfies the commutator relation 푉 푃 = 푃 푉 + [푉, 푃 ] where ∈ 풱 the Lie bracket [푉, 푃 ] 푏, then ∈ 풱

푉 푃 (푢) = 푃 푉 (푢) + 푉푏(푢)

푠,푘 푠 by definition of 푢 퐻 , both 푉 (푢) and 푉푏(푢) are in 퐻 (푀), therefore 푃 푉 (푢) ∈ 푒,푏 푒 ∈ 푠−1 퐻푒 (푀). If it holds for 푘 1, then by the relation −

푘 푘−1 푘 푉푏 푃 (푢) = 푉푏 푃 푉푏(푢) + 푉푏 (푢),

푠,푘−1 푠−1,푘−1 since 푉푏(푢) 퐻 and from induction assumption 푃 푉푏(푢) 퐻 , therefore ∈ 푒,푏 ∈ 푒,푏 75 푘−1 푠−1 푠 the first term 푉 푃 푉푏(푢) 퐻 (푀), and the second term is in 퐻 by definition. 푏 ∈ 푒 푒 Therefore 푃 푢 퐻푠−1,푘−1, which completes the induction. ∈ 푒,푏

2.5.2 Hybrid Sobolev space

Proposition 2.18. For k large enough and 푟 3, 푥푟퐻푠,푘(푀) is an algebra. ≥ − 푒,푏

Proof. We first prove that, for the case 푟 = 3, the boundary Sobolev space 푥−3퐻푘 − 푏 is an algebra for large k. Working in the upper half plane model with coordinates

−3 푘 (푥, 푦1, ...푦푛, 푧). For any element 푓 푥 퐻 (푀), by definition, its Sobolev norm is ∈ 푏 ∫︁ 푉 푘(푥3푓) 2푥−7푑푥푑푦푑푧 | 푏 |

3 3 Since the commutator [푉푏, 푥 ]푓 푥 푓 , therefore the definition of the Sobolev norm ⊂ { } is the same as ∫︁ 푥3(푉 푘푓) 2푥−7푑푥푑푦푑푧 | 푏 |

If we do a coordinate transformation to change the problem back to R푛: let 휌 = 푙푛(푥), then 푥휕푥 = 휕휌. Therefore under the new coordinates, the boundary vector fields are spanned by (휕휌, 휕푦, 휕푧). Let F be the function after coordinate transformation

퐹 (휌, 푦, 푧) = 푓(푒휌, 푦, 푧) then from the discussion above we can see

∫︁ ∫︁ 2 3 푘 2 −7 푘 2 푓 푥−3퐻푘 = 푥 (푉푏 푓) 푥 푑푥푑푦푑푧 = 푉푏 퐹 푑휌푑푦푑푧 < ‖ ‖ 푏 | | | | ∞ which means 퐹 퐻푘(R푛). From [], the usual Sobolev space in R푛 is closed under ∈ multiplication if and only if 푘 > 푛 . Therefore, take two elements 푓, 푔 푥−3퐻푘(푀), 2 ∈ 푏 then the corresponding functions in R푛 satisfy 퐹 퐺 퐻푘(R푛). It follows that 푓푔 ∈ ∈ −3 푘 푥 퐻푏 (푀) by taking the inverse coordinate transformation. Then it is easy to see that 푥푟퐻푘(푀) is an algebra for 푟 > 3. From the result 푏 − 76 above, (푥푟퐻푘) (푥푟퐻푘) = 푥3+푟(푥−3퐻푘) 푥3+푟(푥−3퐻푘) 푏 · 푏 푏 · 푏 푥6+2푟(푥−3퐻푘) 푥3+푟(푥−3퐻푘) = 푥푟퐻푘(푀). ⊂ 푏 ⊂ 푏 푏

푘 Now that we proved 퐻푏 (푀) is closed under multiplication, then we want to prove 퐻푠,푘(푀) is also an algebra. For any functions 푓, 푔 퐻푠,푘(푀), by Leibniz rule, 푒,푏 ∈ 푒,푏

푗 푗 ∑︁ 푖 푗−푖 푉푒 (푓푔) = 푉푒 (푓)푉푒 (푔) 푖=0

푖 푗−푖 푘 where by assumption, both 푉푒 (푓) and 푉푒 (푔) are in 퐻푏 (푀), therefore their product is also in 퐻푘(푀) from the above result.Hence we proved 푉 푗(푓푔) 퐻푘(푀) for 0 푏 푒 ∈ 푏 ≤ 푗 푠.which shows 푓푔 퐻푠,푘(푀). ≤ ∈ 푒,푏

2.6 Computation of the indicial roots

2.6.1 Hodge decomposition

The system contains the following equations, where the (푖, 푗) notations means the splitting of degrees of forms with respect to the product structure of B7 S4. ×

From the first order equation ∙

⋀︁ 7 6푑퐻 7 푘(1,1) + 3푑푆(푇 푟퐻7 (푘) 푇 푟푆4 (푘)) 푉 (7, 1) : * − (2.52) +푑푆 퐻(0,4) + 푑퐻 퐻(1,3) = 0 * *

(6, 2) : 푑푆 퐻(1,3) + 푑퐻 퐻(2,2) + 6푑푆 7 푘(1,1) = 0 (2.53) * * *

(5, 3) : 푑푆 퐻(2,2) + 푑퐻 퐻(3,1) = 0 (2.54) * *

(4, 4) : 푑푆 퐻(3,1) + 푑퐻 퐻(4,0) + 푊 퐻(4,0) = 0 (2.55) * * ∧

77 From 푑퐻 = 0 ∙

푑퐻 퐻(0,4) + 푑푆퐻(1,3) = 0 (2.56)

푑퐻 퐻(1,3) + 푑푆퐻(2,2) = 0 (2.57)

푑퐻 퐻(2,2) + 푑푆퐻(3,1) = 0 (2.58)

푑퐻 퐻(3,1) + 푑푆퐻(4,0) = 0 (2.59)

푑퐻 퐻(4,0) = 0 (2.60)

From the laplacian: ∙ 1 1 푠푘퐼푗 + 퐻 푘퐼푗 + 6푘퐼푗 3 푆 퐻(1,3) = 0 (2.61) 2△ 2△ − * 1 ( 푠 + 퐻 )푘퐼퐽 푘퐼퐽 6푇 푟푆(푘)푡퐼퐽 + 푇 푟퐻 (푘)푡퐼퐽 + 2퐻(0,4)푡퐼퐽 = 0 (2.62) 2 △ △ − − 1 ( 푆 + 퐻 )푘푖푗 + 4푘푖푗 + 8푇 푟푆(푘)푡푖푗 퐻(0,4)푡푖푗 = 0 (2.63) 2 △ △ −

2.6.2 Indicial roots

Then we decompose further with respect to Hodge theory on sphere, and compute the indicial roots for each part.

Harmonic functions on S4: ∙

1. Trace-free 2-tensor on 퐻7, where the equation is

(훥푆 + 훥퐻 2)푘^퐼퐽 = 0, −

and the indicial equation is

2 ( 푠 + 6푠)푘퐼퐽 = 0. −

we have indicial roots

+ − 푆1 = 0, 푆1 = 6.

78 This corresponds to the perturbation of hyperbolic metric to Poincaré– Einstein metric.

2. Trace-free 2-tensor on 푆4, where the equation is

푟표푢푔ℎ^ ^ ^ 훥푆 푘푖푗 + 훥퐻 푘푖푗 + 8푘푖푗 = 0

where indicial equation is

2 ( 푠 + 6푠 + 8)푘^푖푗 = 0, −

indicial roots 푆± = 3 √17. 2 ±

3. We have

푑퐻 퐻(4,0) + 푊 퐻(4,0) = 0 (2.64) * ∧

푑퐻 퐻(4,0) = 0 (2.65)

The second equation can be deduced from the first one. Since the indicial

operator for 푑퐻 is

퐼[푑](푠)푤 = ( 1)푘(푠 푘)푤 푑푥/푥 − − ∧

Let

퐻(4,0) = 푇 + 푑푥/푥 푁 ∧ be the decomposition with respect to tangential and normal decomposition, then the indicial equations are

(푠 3)( 6푁) 푑푥/푥 6푑푥/푥 푁 = 0 − − * ∧ − ∧

(푠 4)푇 푑푥/푥 = 0 − ∧ 79 where the first equation gives

(푠 3) 6 푁 6푁 = 0 − * −

i.e. N is an eigenform of 6 and the corresponding indicial roots are *

3 ⋀︁ 6 푠− = 3 6푖, 푁 (S ); 6푁 = 푖푁; − ∈ *

3 ⋀︁ 6 푠+ = 3 + 6푖 : 푁 (S ); 6푁 = 푖푁. ∈ * − And plugging into the second equation, we have the vanishing of tangential form 푇 = 0.

Therefore the kernel in this case is

3 ⋀︁ 6 퐻(4,0) = 푑푥/푥 푁, 푁 (S ), 6푁 = 푖푁 . ∧ ∈ { * ± }

Then we consider Exact 1-form, which includes function/exact 1-form/ coexact ∙ 3-form/ exact 4-form on the eigenspace 휆 = 4(푘 + 1)(푘 + 4) starting from 푘 = 0.

1 1 푖푗 1 1 퐼퐽 1. Denote 휏 = 4 푇 푟푆(푘) = 4 푡 푘푖푗, 휎 = 7 푇 푟퐻 (푘) = 7 푡 푘퐼퐽 to be the normal- ized trace, then we have the following equations:

푐푙 ⋀︁ 7 6푑퐻 퐻 푘(1,1) + 푑푆(3푇 푟퐻 (푘) 3푇 푟푆(푘)) 푉 * − (2.66) 푐푙 푐푐 +푑푆 퐻 + 푑퐻 퐻 = 0 * (0,4) * (1,3)

푐푙 푐푐 푑퐻 퐻(0,4) + 푑푆퐻(1,3) = 0 (2.67)

푐푐 푑퐻 퐻(1,3) = 0 (2.68)

푐푙 푐푙 푐푙 푐푐 푠푘 + 퐻 푘 + 12푘 6 푆 퐻 = 0 (2.69) △ (1,1) △ (1,1) (1,1) − * (1,3)

푐푙 훥푆휏 + 훥퐻 휏 + 72휏 8 푆 퐻 = 0 (2.70) − * 0,4 80 푐푙 훥푆휎 + 훥퐻 휎 + 12휎 + 4 푆 퐻 48휏 = 0 (2.71) * 0,4 −

푐푙 First note that 2.68 can be derived from 2.67. Let 퐻(0,4) = 푑푆휂, here 휂 is a 푐푐 푐푙 (0,3)-form. Then 퐻 = 푑퐻 휂 by 2.68. Let 푓 = 푆푑푆휂. Let 푘 = 푑푆푤, (1,3) − * (1,1) w is (1,0)-form. Put it back to 2.66 we get

6푑퐻 퐻 푑푆푤 + 푑푆 퐻 (21휎 12휏) + 퐻 푑푆 푆 푑푆휂 푆푑퐻 퐻 푑퐻 휂 = 0 (2.72) * * − * * − * *

2 Apply 퐻 ( = 1), we get * *퐻

6 퐻 푑퐻 퐻 푑푆푤 + 푑푆(21휎 12휏) + 푑푆 푆 푑푆휂 푆 퐻 푑퐻 퐻 푑퐻 휂 = 0 * * − * − * * *

Then let 휂 = 푆푑푆휉, 휉 be a function, and pull out 푑푆 *

6훿퐻 푤 + (21휎 12휏) 훥푆휉 훥퐻 휉 = 0 (2.73) − − − − and put the expression to 2.69,

훥푆푑푆푤 + 훥퐻 푑푆푤 + 12푑푆푤 + 6 푆 푑퐻 푆 푑푆휉 = 0 * *

Apply 훿퐻 and pull out 푑푆

훥푆훿퐻 푤 + 훥퐻 훿퐻 푤 + +12훿퐻 푤 + 6훥퐻 휉 = 0 (2.74)

Now 2.70 becomes

훥푆휏 + 훥퐻 휏 + 72휏 + 8훥푆휉 = 0 (2.75)

And 2.71 is

훥푆휎 + 훥퐻 휎 + 12휎 4훥푆휉 48휏 = 0 (2.76) − −

Putting the above four equations together, and suppose the eigenvalue of

81 훥푆 is 휆, we get

⎛ ⎞ ⎛ ⎞ 12 + 휆 + 훥퐻 48 4휆 0 휎 ⎜ − − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 72 + 휆 + 훥퐻 8휆 0 ⎟ ⎜ 휏 ⎟ ⎜ ⎟ ⎜ ⎟ = 0 ⎜ ⎟ ⎜ ⎟ ⎜ 21 12 휆 훥퐻 6 ⎟ ⎜ 휉 ⎟ ⎝ − − − − ⎠ ⎝ ⎠ 0 0 6훥퐻 12 + 휆 + 훥퐻 훿퐻 푤

The determinant, after putting in the indicial operator of 훥퐻 , is

휆4 4푆2휆3 + 24푆 휆3 90휆3 + 6푆4휆2 72푆3휆2 − * − − + 342푆2휆2 756푆 휆2 + 1152휆2 4푆6휆 + 72푆5휆 414푆4휆 − * − − + 648푆3휆 + 1152푆2휆 3024푆 휆 + 10368휆 (2.77) − * + 푆8 24푆7 + 162푆6 + 108푆5 6192푆4 − − + 31536푆3 33696푆2 155520푆 = 0 − −

Putting the lowest two eigenvalues for closed 1-form, we get the following

two pairs of roots: for 휆 = 16 the indicial roots are 휃2 = 3 푖√21116145/1655. ± with kernel 푐푙 ⋀︁ 휉16 (푆) ∈ 휆=16 which is the closed 1-form on 4-sphere with eigenvalue 16. and the other pair is for 휆 = 40 then

휃3 = 3 푖3√582842/20098, ±

with kernel 푐푙 ⋀︁ 휉40 (푆). ∈ 휆=40 2. We have

푐푙 푐푐 4 푐푐 푑푆 퐻 + 푑퐻 퐻 + 6 푉 퐻 = 0, (2.78) * (3,1) * (4,0) ∧ (4,0) 푐푙 푐푐 푑퐻 퐻(3,1) + 푑푆퐻(4,0) = 0. (2.79)

82 Let

푐푙 퐻(3,1) = 푑푆휂

where 휂 is (3,0), put into second equation to get

푐푐 퐻 = 푑퐻 휂 (4,0) −

Put everything back to first equation, we get

4 푑푆 푑푆휂 푑퐻 푑퐻 휂 6 푉 푑퐻 휂 = 0. * − * − ∧

2 푘(4−푘) 4(푘+1)+1 Apply 푆, and note = ( 1) = 1, 훿푆 = ( 1) 푆 푑푆 푆 = * *푆 − − * * 푆 푑푆 푆, 훥푆 = 푑훿 + 훿푑, − * *

퐻 ( 훿푆)푑푆휂 푑퐻 퐻 푑퐻 휂 + 푑퐻 휂 = 0 * − − *

2 Then apply 퐻 , note ( 퐻 ) = 1, get * *

훥푆휂 퐻 푑퐻 퐻 푑퐻 휂 + 6 퐻 푑퐻 휂 = 0. − − * * *

Let 훥푆휂 = 휆휂

휆휂 훥퐻 휂 + 6 퐻 푑퐻 휂 = 0 − − * The indicial equation: using 퐼[푑](푠)푤 = ( 1)푘(푠 푘)푤 푑푥 , − − ∧ 푥

2 휆휂 + (푠 3) 휂 + 6(푠 3) 6 휂 = 0 − − − *

that is (푠 3)2 6푖(푠 3) 16 = 0 − ± − − with roots 푠 = 3 √7 3푖. ± ±

Then we consider the co-exact 1-form, which contains coexact 1-form/ exact ∙ 83 2-form/ coexact 2-form/ exact 3-form for 휆 = 4(푘 + 2)(푘 + 3) starting from 푘 = 0.

1. We have

푐푐 푐푙 6푑퐻 퐻 푘 + 푑퐻 퐻 = 0 (2.80) * (1,1) * (1,3)

푐푙 푐푐 푐푐 푑푆 퐻 + 푑퐻 퐻 + 6푑푆 퐻 푘 = 0 (2.81) * (1,3) * (2,2) * (1,1)

푐푙 푐푐 푑퐻 퐻(1,3) + 푑푆퐻(2,2) = 0 (2.82)

1 푐푐 1 푐푐 푐푐 1 푐푙 푠푘 + 퐻 푘 + 6푘 푆 퐻 = 0 (2.83) 2△ (1,1) 2△ (1,1) (1,1) − 2 * (1,3)

푐푙 First note that (2.80) can be derived from (2.81) Let 퐻(1,3) = 푑푠휂, where 휂 푐푐 is (1,2)-form. Then 퐻 = 푑퐻 휂 from (2.82). Put it to (2.81), 푑푆 푑푆휂 2,2 − * − 푐푐 푐푐 푑퐻 푑퐻 휂 +6푑푆 퐻 푘 = 0. Apply 푆, 퐻 , get 훥푆휂 훥퐻 휂 +6 푆 푑푆푘 = * * 1,1 * * − − * (1,1) 푐푐 0. Apply 푆푑푆 again, get 훥푆( 푆푑푆휂) 훥퐻 ( 푆푑푆휂) 6훥푆푘 = 0. * − * − * − (1,1) Combining with (2.83), and let 휆 be the eigenvalue for 훥푆 on coclosed 1-form, we get

⎛ ⎞ ⎛ ⎞ 휆 훥퐻 6휆 푆푑푆휂 ⎝ − − − ⎠ ⎝ * ⎠ = 0 푐푐 1 휆 + 훥퐻 + 12 푘 − (1,1)

The indicial equation is

휆2 (36 + (푠 1)(푠 5) + 푠2 6푠 1)휆 (푠 1)(푠 5)( 푠2 + 6푠 + 1) = 0. − − − − − − − − −

With smallest eigenvalue for coclosed 1-form to be 휆 = 24, indicial roots are √︁ 푆± = 3 3√97 + 31 3 ± ±

푐푙 푐푐 푑푆 퐻 + 푑퐻 퐻 = 0 (2.84) * (2,2) * (3,1) 푐푙 푐푐 푑퐻 퐻(2,2) + 푑푆퐻(3,1) = 0 (2.85)

84 Apply 푑퐻 and 푑푆 to the equations, we have

푐푙 푐푙 푑퐻 푑푆 퐻 = 0, 푑푆푑퐻 퐻 = 0 (2.86) * (2,2) (2,2)

푐푙 let 퐻(2,2) = 푑푆휂 where 휂 is a coclosed (2,1)-form, Putting it back, and 푐푐 using 푑푆 is an isomorphism, 푑퐻 휂 = 퐻 . Then from first equation, − (3,1) 푑푆 푑푆휂 푑퐻 푑퐻 휂 = 0, which is 퐻 푆훥푆휂 푆 퐻 훥퐻 휂 = 0 then it * − * − * * − * * requires 훥퐻 휂 = 휆휂. Putting 휆 = 4(푘 + 2)(푘 + 3), the result is −

푠 = 3 √17. ±

85 86 Chapter 3

Resolution of the canonical fiber metrics for a Lefschetz fibration

In the setting of complex surfaces, a Lefschetz fibration is a holomorphic map toa curve, generalizing an elliptic fibration in that it has only a finite number of singular points near which it is holomorphically reducible to normal crossing. Donaldson [8] showed that a four-dimensional simply-connected compact symplectic manifold, possibly after stabilization by a finite number of blow-ups, admits a Lefschetz fibration, in an appropriately generalized sense, over the sphere; Gompf [13] showed the converse. The reader is referred to the book of Gompf and Stipsicz [14] for a description of the important role played by Lefschetz fibrations in the general theory of 4-manifolds. To cover these cases we consider a compact connected almost-complex 4-manifold 푀 and a smooth map, with complex fibers, to a Riemann surface 푍

휓 푀 / 푍. (3.1)

We then require that this map be pseudo-holomorphic, have surjective differential outside a finite set 퐹 푀, on which 휓 is injective, so 휓 : 퐹 푆 푀, and near ⊂ ←→ ⊂ each of these singular points be reducible to the normal crossing, or plumbing variety, model (3.2) below.

A curve of genus 푔 with 푏 punctures is stable if its automorphism group is finite,

87 which is the case when 3푔 3+푏 > 0. In this paper we discuss Lefschetz fibrations with − regular fibers having genus 푔 > 1 and hence stable. All fibers carry a unique metric of curvature 1, for the singular fibers with cusp points replacing the nodes. Inview − of uniqueness and stability, these metrics necessarily vary smoothly near a regular fiber. We discuss here the precise uniform behavior of this family of metrics near the singular fibers, showing that in terms of appropriate (logarithmic) resolutions, of both the total and parameter spaces, to manifolds with corners the resulting fiber metric is polyhomogeneous and more particularly log-smooth, i.e. essentially smooth except for the appearance of logarithmic terms in the expansions at boundary surfaces. This refines a result of Obitsu and Wolpert [39] who gave the first two termsin the expansion. In a forthcoming paper the universal case of the Deligne-Mumford compactification of the moduli space of Riemann surfaces, also treated by Obitsuand Wolpert, will be discussed.

The local model for degeneration for the complex structure on a Riemann surface to a surface with a node is the ‘plumbing variety’ with its projection to the parameter space. We add boundaries, away from the singularity at the origin, to make this into a manifold with corners:

{︁ 2 3 3 1}︁ 푃 = (푧, 푤) C ; 푡 C, 푧푤 = 푡, 푧 , 푤 , 푡 ∈ ∃ ∈ | | ≤ 4 | | ≤ 4 | | ≤ 2 (3.2) 휑 {︁ 1}︁ 푃 D 1 = 푡 C; 푡 . −→ 2 ∈ | | ≤ 2

Thus near each point of 퐹 we require that 휓 can be reduced to 휑 in (almost) holomorphic coordinates in 푀 and 푍.

∞ A (real) manifold with corners 푀 has a principal ideal 퐹 (푀) corresponding ℐ ⊂ 풞 to each boundary hypersurface (by assumption embedded and connected) generated by a boundary defining function 휌퐹 0 with 퐹 = 휌퐹 = 0 and 푑휌퐹 = 0 on ≥ { } ̸ 퐹. A smooth map between manifolds with corners 푓 : 푀 푌 is an interior b- −→ map if each of these ideals on 푌 pulls back to non-trivial finite products of the corresponding ideals on 푀, it is b-normal if there is no common factor in these product decompositions – this is always the case here since the range space is a

88 manifold with boundary. Such a map is a b-fibration if in addition every smooth vector field tangent to all boundaries on 푌 is locally (and hence globally) 푓-related to such a vector field on 푀; it is then surjective. There is a slightly weaker notion than a manifold with corners, a tied manifold, which has the same local structure but in which the boundary hypersurfaces need not be embedded, meaning that transversal self-intersection is allowed. This arises below, although not in any essential way. There is still a principal ideal associated to each boundary hypersurface and the notions above carry over. The assumptions above mean that each singular fiber of 휓 has one singular point at which it has a normal crossing in the (almost) complex sense as a subvariety of 푀. The first step in the resolution is the blow up, in the real sense, of the singular fibers; this is well-defined in view of the transversality of the self-instersection but results in a tied manifold since the boundary faces are not globally embedded. The second step is to replace the ∞ structure by its logarithmic weakening, i.e. replacing each 풞 (local) boundary defining function 푥 by

ilog 푥 = (log 푥−1)−1.

푀mr, with the two boundary hypersurfaces denoted 퐵I, resolving the singular fiber,

and 퐵II arising at the final stage of the resolution. The parameter space 푍 is similarly resolved to a manifold with corners by the logarithmic blow up of each of the singular points. It is shown below that the Lefschetz fibration lifts to a smooth map

휓mr 푀mr / 푍mr (3.3)

which is a b-fibration. In particular it follows from this that smooth vector fieldson

89 푀mr which are tangent to all boundaries and to the fibers of 휓mr form the sections b of a smooth vector subbundle of 푇 푀mr of rank two. The boundary hypersurface 퐵II has a preferred class of boundary defining functions, an element of which is denoted

휌II, arising from the logarithmic nature of the resolution, and this allows a Lie algebra of vector fields to be defined by

∞ b * ∞ 2 ∞ 푉 (푀mr; 푇 푀mr), 푉 휓 (푍mr) = 0, 푉 휌II 휌 (푀mr). (3.4) ∈ 풞 풞 ∈ II풞

Theorem 3.1. The fiber metrics of fixed constant curvature on a Lefschetz fibration,

퐿 in the sense discussed above, extend to a continuous Hermitian metric on 푇 푀mr which is related to a smooth Hermitian metric on this complex line bundle by a log- smooth conformal factor.

The notion of log-smoothness here, for a function, is the same as polyhomogeneous conormality with non-negative integral powers and linear multiplicity of slope one.

Conormality in this context for 푓 : 푀mr R can be interpreted as the ‘symbol −→ estimates’ that

* ∞ 푓 (푀mr) Diff (푀mr)푓 퐿 (푀mr) (3.5) ∈ 풜 ⇐⇒ b ⊂ which in fact implies that the space of these functions is stable under the action, * Diff (푀mr) (푀mr) (푀mr). Polyhomogeneity means the existence of appropriate b 풜 ⊂ 풜 expansions at the boundary. On a manifold with boundary, 푀, log-smoothness of a conormal function 푓 (푀) means the existence of an expansion at the boundary, ∈ 풜 generalizing the Taylor series of a smooth function, so for any product decomposition near the boundary with boundary defining function 푥, there exist coefficients 푎푗,푘 ∈ 90 ∞(휕푀), 푗 0, 푗 푘 0 such that for any finite 푁, 풞 ≥ ≥ ≥

∑︁ 푗 푘 푁 푓 푎푗,푘푥 (log 푥) 푥 ([0, 1) 휕푀), 푁. (3.6) − ∈ 풜 × ∀ 푗≤푁,0≤푘≤푗

We denote the linear space of such functions ∞ (푀), it is independent of choices. 풞log In the case of a manifold with corners the definition may be extended by iteration

∞ of boundary codimension. Thus 푓 (푀mr) if for any product decompositions of ∈ 풞log ∞ 푀mr near the two boundaries there are corresponding coefficients 푎푗,푘,푏 (퐵푏), ∈ 풞log 푏 = I, II, such that

∑︁ 푗 푘 푁 푓 푎푗,푘,푏푥 (log 푥푏) 푥 ([0, 1) 퐵푏), 푏 = I, II, 푁. (3.7) − 푏 ∈ 푏 풜 × ∀ 푗≤푁,0≤푘≤푗

There are necessarily compatibility conditions between the two expansions at the corners, 퐵I 퐵II, and together they determine 푓 up to a smooth function on 푀mr ∩ vanishing to infinite order on both boundaries. In this sense the conformal factorin the main result above is ‘essentially smooth’.

In the model setting, (3.2), there is an explicit family of fiber metrics, the ‘plumbing metric’, of curvature 1, −

퐿 푇 푀mr which has curvature 푅 equal to 1 near 퐵II and to second order at 퐵I. We − prove the Theorem above by constructing the conformal factor 푒2푓 for this metric which satisfies the curvature equation, ensuring that the new metric has curvature 1 : − (훥 + 2)푓 + (푅 + 1) = 푒2푓 + 1 + 2푓 = 푂(푓 2). (3.9) −

This equation is first solved in the sense of formal power series (with logarithms)

91 at both boundaries, 퐵I and 퐵II, which gives us an approximate solution 푓0 with

2푓0 ∞ ∞ 훥푓0 = 푅 + 푒 + 푔, 푔 푠 (푀mr). − ∈ 푡 풞

Then a solution 푓 = 푓0 + 푓˜ to (3.9) amounts to solving

(︁ ˜ )︁ 푓˜ = (훥 + 2)−1 2푓˜(푒2푓0 1) + 푒2푓0 (푒2푓 1 2푓˜) 푔 = 퐾(푓˜). − − − − −

Here the non-linear operator 퐾 is at least quadratic in 푓˜ and the boundedness of 1 −1 − 2 푀 (훥 + 2) on 휌II 퐻b (푀mr) for all 푀 allow the Inverse Function Theorem to be ∞ ∞ applied to show that 푓˜ 푠 (푀mr) and hence that 푓 itself is log-smooth. ∈ 푡 풞 In §3.1 the model space and metric are analysed and in §3.2 the global resolution is described and the proof of the Theorem above is outlined. The linearized model involves the inverse of 훥+2 for the Laplacian on the fibers and the uniform behavior, at the singular fibers, of this operator is explained in §3.3. The solution of thecurva- ture problem in formal power series is discussed in §3.4 and using this the regularity of the fiber metric is shown in §3.5.

3.1 The plumbing model

We start with a description of the real resolution of the plumbing variety, given in (3.2), and the properties of the fiber metric, (3.8), on the resolved space. As noted above there are three steps in this resoluton, first the fiber complex structure is resolved, in a real sense, then two further steps are required to resolve the fiber metric.

The plumbing variety itself is smooth with 푧 and 푤 global complex coordinates – it is the model singular fibration 휑 which is to be ‘resolved’ in the real sense. The fibers above each 푡 = 0 are annuli ̸

푤=푡/푧 푡 푧 3 / 푡 푤 3 (3.10) {| | ≤ | | ≤ 4 } {| | ≤ | | ≤ 4 } 92 whereas the singular fiber above 푡 = 0 is the union of the two discs at 푧 = 0 and 푤 = 0 identified at their origins

{︁ 3}︁ {︁ 3}︁ 휑−1(0) = 푧 푤 /( 푧 = 0 푤 = 0 ). (3.11) | | ≤ 4 ∪ | | ≤ 4 { } ∼ { }

Note that the differential of 휑 vanishes at the singular point 푧 = 푤 = 0 so any smooth vector field on the range which lifts under it, i.e.is 휑-related to a smooth vector field

on 푃, vanishes at 푡 = 0. Conversely, 푡휕푡 is 휑-related to both 푧휕푧 and 푤휕푤 whereas the vector field

푉 = 푧휕푧 푤휕푤 (3.12) − annihilates 휑*푡 and so is everywhere tangent to the fibers of 휑.

The first step in the resolution of 휑 : 푃 D 1 consists in passing to the commu- −→ 2 tative square

휑휕 푃 / [ 1 , 0] (3.13) 휕 D 2

푃 / 1 . 휑 D 2

Here [ 1 , 0] is the space obtained by real blow up of the origin in the disk, which can D 2 be realized globally as

[︁ 1]︁ 푖휃 [D 1 , 0] 0, S (푟, 휃) 푡 = 푟푒 D 1 (3.14) 2 ≃ 2 × ∋ ↦−→ ∈ 2

if = /2휋 . As a real blow-up [ 1 , 0] is a well-defined manifold with boundary S R Z D 2 and any diffeomorphism of 1 fixing the origin lifts to a global diffeomorphism. The D 2 b complex structure on 1 lifts to a complex structure on 푇 [ 1 , 0] generated by 푡휕푡 = D 2 D 2

푟휕푟 + 푖휕휃 in terms of (3.14).

Proposition 3.1. The space

푃 = [푃 ; 푧 = 0 푤 = 0 ], (3.15) 휕 { } ∪ { }

obtained by the real blow-up of the two normally-intersecting divisors forming the

93 singular fiber of 휑, gives a commutative diagram (3.13) in which 휑휕 is a b-fibration with

* 휑 = I,L I,R (3.16) 휕ℐ휕 ℐ ℐ

where I,L and I,R correspond to the two boundary components introduced by the ℐ ℐ blow-up, forming the proper tranforms of 푧 = 0 and 푤 = 0 respectively.

Proof. The two divisors forming the singular fiber 휑−1(0) are each contained in a product product neighborhood D 1 D 3 푃 and D 3 D 1 푃. The transversality of 2 × 4 ⊂ 4 × 2 ⊂ their intersection is clear and it follows that the blow-up is well-defined independently of order with the new front faces being

퐵I,L = S [D 3 , 0 ] 푃휕, 퐵I,R = [D 3 , 0 ] S 푃휕. (3.17) × 4 { } ⊂ 4 { } × ⊂

Here each of the blown up disks corresponds to the introduction of polar coordinates,

so 푟푧 = 푧 is a defining function (globally) for 퐵I,L and 푟푤 = 푤 for 퐵I,R. Since | | | | 푟푡 = 푡 is a defining function for the blown-up disk in the range and | |

푟푡 = 푟푧푟푤 (3.18) the b-fibration condition follows from the behaviour of the corresponding angular variables 푒푖휃푡 = 푒푖휃푧 푒푖휃푤 . (3.19)

As a compact manifold with corners, 푃휕 is globally the product of an embedded manifold in R2 and a 2-torus

{︁ ⃒ }︁ ⃒ 3 1 푃휕 = (푟푧, 푟푤) 0 푟푧, 푟푤 , 푟푧푟푤 S푧 S푤. (3.20) ⃒ ≤ ≤ 4 ≤ 2 × ×

This first step in the resolution resolves the complex structure in a real sense.In

particular the vector fields tangent to the fibers of 휑휕 and to the boundaries form all

94 b the sections of a subbundle of 푇 푃휕 which has a complex structure, spanned by the lift of the single vector field (3.12).

Although the complex structure is effectively resolved, the plumbing metric in

(3.8) is not. That 푔푃 has curvature 1 on the fibers, away from the singular point, − can be seen by changing variables to 푠 = log 푟, 푟 = 푟푧 and 휃 = 휃푧 in terms of which

2 (︁ 휋/ log 푡 )︁ 2 2 푔푃 = | | (푑푠 + 푑휃 ). sin(휋푠/ log 푡 ) | | It then follows from the standard formula for the Gauss curvature that

(︂ )︂ 1 (︁ 휕푟푔 )︁ (︁ 휕휃푓 )︁ 푅 = 휕푟 + 휕휃 = 1. −2√푓푔 √푓푔 √푓푔 −

In view of the coefficients in 푔푃 it is natural to introduce the inverted logarithms of the new boundary defining functions, so replacing the radial by the logarithmic blow-up. Thus 1 푠푧 = ilog 푟푧 = , 푠푤 = ilog 푟푤 (3.21) log 1 푟푧

become new boundary defining functions in place of 푟푧 and 푟푤. The space with this new ∞ structure can be written 풞

[푃 ; 푧 = 0 log 푤 = 0 log]. (3.22) { } ∪ { }

However, even after this second step, the fiber metric does not have smooth coefficients: 2 2 (︂ 2 )︂ 휋 푠푡 푑푠푤 2 푔푃 = + 푑휃 . sin2( 휋푠푡 ) 푠4 푤 푠푤 푤 Indeed 푠 = 푠푧푠푤 is not a smooth function on the space (3.22). 푡 푠푧+푠푤

The final part of the metric resolution is to blow up, radially, the corner formed by the intersection of the two logarithmic boundary faces

푃mr = [[푃 ; 푧 = 0 log 푤 = 0 log]; 푠푧 = 푠푤 = 0 ]. (3.23) { } ∪ { } { } 95 In terms of the presentation (3.20) this preserves the torus factor and replaces the 2-manifold with corners by a new one with more smooth functions and an extra boundary hypersurface.

Proposition 3.2. The model Lefschetz fibration 휑 lifts to a b-fibration 휑mr giving a commutative diagram

휑mr 푃mr / [D 1 ; 0 log] (3.24) 2 { }

푃 / 1 . 휑 D 2

Proof. The radial variables on the spaces 푃휕 and [D 1 , 0 ] are related by 2 { }

푠푧푠푤 푡 = 푧 푤 = 푠푡 = , 푠푡 = ilog 푡 (3.25) | | | || | ⇒ 푠푧 + 푠푤 | | so 휑 does not lift to be smooth. However, consider the further introduction of the 2 2 1 radial variable 푅 = (푠푧 + 푠푤) 2 and the smooth defining functions 푅푧 = 푠푧/푅, 푅푤 =

푠푤/푅 for the lifts of the two boundary hypersurfaces. Then

푅푧푅푅푤 푠푡 = (3.26) 푅푧 + 푅푤 which is smooth since 푅푧 and 푅푤 have disjoint zero sets. It follows that 휑 lifts to a b-fibration as in (3.24) under which the boundary ideal lifts to the product ofthe three ideals

* 휑 푠 = 푅 푅 푅 . (3.27) mrℐ 푡 ℐ 푧 ℐ ℐ 푤

The generator 푉, in (3.12), of the fiber tangent space of 휑 lifts to 푃휕 as

푉 = 푟푧휕푟 푟푤휕푟 푖휕휃 + 푖휕휃 푧 − 푤 − 푧 푤 in terms of the coordinates in (3.19) and (3.18). Under the introduction of the loga-

96 rithmic variables in (3.21) it further lifts to

2 2 푉 = 푠 휕푠 푠 휕푠 푖휕휃 + 푖휕휃 . 푧 푧 − 푤 푤 − 푧 푤

In a neighborhood of the the lift of the face 푠푧 = 0 to 푃mr the variables 푠푤 (defining the new front face) and 휌푧 = 푠푧/푠푤 [0, ) (defining the lift of 푠푧 = 0) are valid and ∈ ∞

2 푉 = 푠푤(푠푤휕푠 휌푧휕휌 휌 휕휌 ) 푖휕휃 + 푖휕휃 . (3.28) − 푤 − 푧 − 푧 푧 − 푧 푤

Reviewing the three steps in the construction of 푃mr, notice that the two holomorphic defining functions 푧 and 푤 are well-defined up to constant multiples and addition of (holomorphic) terms 푂( 푧 2) and 푂( 푤 2) respectively. Under these two | | | | 2 ∞ changes, the logarithmic variables 푠푧 change to 푠푧 + 푠 퐺 with 퐺 (푃mr) smooth. 푧 ∈ 풞 The same is true of 푠푤 so it follows that the radial variable

2 2 1/2 ∞ 푅 = (푠 + 푠 ) (푃mr), (3.29) 푧 푤 ∈ 풞 which defines the front face, is also uniquely defined up to an additive term vanishing quadratically there. This determines a ‘cusp’ structure at 퐵II and from (3.28) we conclude that

−1 Lemma 3.1. The vector field 푅 푉 on 푃mr spans a smooth complex line bundle, 퐿 푇 푃mr over 푃mr with underlying real plane bundle having smooth sections precisely of the form 푅−1푊 where 푊 is a smooth vector field tangent to the boundaries, to the 2 fibers of 휑mr and satisfying 푊 푅 = 푂(푅 ) at 푅 = 0.

It is natural to consider this bundle, precisely because

퐿 Lemma 3.2. The plumbing metric defines an Hermitian metric on 푇 푃mr.

Proof. On 푃mr, in a neighborhood of the lift of 푠푧 = 0 as discussed above, { }

휋2푠2 (︂푑푠2 )︂ 휋2푠2 (︂ 푑휌2 )︂ 푔 = 푡 푤 + 푑휃2 = 푡 푧 + 푑휃2 . (3.30) 2 휋푠푡 4 푤 2 휋 2 4 푧 sin ( ) 푠 sin ( ) 푠 (1 + 휌푧) 푠푤 푤 1+휌푧 푡

This is Hermitian and the length of 푉 relative to it is a smooth positive multiple of 푅2.

3.2 Global resolution and outline

It is now straightforward to extend the resolution of the plumbing variety to a global resolution of any Lefschetz fibration as outlined in the Introduction. By hypothesis, the singular fibers of a Lefschetz fibration 휓, as in (3.1), are isolated and each contains precisely one singular point. Near the singular point the map 휓 is reduced to 휑 by local complex diffeomorphisms. Thus each singular fiber is a connected compact real manifold of dimension two with a trasversal self-intersection. The real blow-up of such a submanifold is well-defined, since it is locally well-defined away from the self-intersection and well-defined near the intersection in view of the transversality. Thus 휓 푀 = [푀, 휑−1(푆)] 휕 [푍, 푆] (3.31) 휕 −→ reduces to 휑 near the preimage of the finite singular set 퐹 푀. Similarly, the 휕 ⊂ logarithmic step can be extended globally since away from the singular set it corresponds to replacing 푧 , by ilog 푧 . Here 푧 is a local complex defining function with | | | | holomorphic differential along the singular fiber. Finally, the third step is withinthe preimage of the set of the singular points and so is precisely the same as for the plumbing variety.

Thus the resolved space 푀mr with its global b-fibration (3.3) is well-defined asis 퐿 퐿 the Hermitian bundle 푇 푀mr which reduces to 푇 푃mr near the singular points and is otherwise the bundle of fiber tangents to 푀mr with its inherited complex structure. To arrive at the description of the constant curvature fiber metric, as an Hermitian

퐿 metric on 푇 푀mr we start with the “grafting” construction of Obitsu and Wolpert

98 which we interpret as giving a good initial choice of Hermitian metric. Namely choose

퐿 any smooth Hermitian metric ℎ0 on 푇 푀mr; from Lemma 3.2

푓Pl 푔Pl = 푒 ℎ0 near 퐵II, 푓Pl smooth. (3.32)

Away from the singular set, near the singular fiber, 휓 is a fibration in the real sense. Thus, it has a product decomposition, with the fibration 휓 the projection, and this can be chosen to be consistent with the product structure on 푃 away from the singular point. Then the complex structure on the fibers is given by a smoothly varying tensor 퐽. The constant curvature metric 푔0 on the resolved singular fiber may therefore be extended trivially to a metric on the fibers nearby, away from the singular points. This has non-Hermitian part vanishing at the singular fiber, so removing this gives a smooth family of Hermitian metrics reducing to 푔0 and so with curvature equal to 1 at the singular fiber. After blow-up this remains true since the regular partof − the singular fiber is replaced by a trivial circle bundle over it. On the introduction of the logarithmic variables in the base and total space, the curvature of this smooth family, 푔I, is constant to infinite order at the singular fiber since it is equal tothe limiting metric 푔0 to infinite order. Comparing 푔I to the chosen Hermitian metric

푓I ∞ gives a conformal factor 푔I = 푒 ℎ, 푓I (푁) where 푁 is a neighborhood of 퐵I ∈ 풞 excluding a neighborhood of 퐵II. Moreover, 푔Pl is also equal to the trivial extension of 푔0 to second order in a compatible trivialization so the two conformal factors

푓I = 푓Pl to second order (3.33) in their common domain of definition.

The grafting construction of Obitsu and Wolpert interpreted in this setting is then

∞ to choose a cutoff 휒 (푀mr) equal to 1 in a neighborhood of 퐵II and supported ∈ 풞 near it and to set

휒푓Pl+(1−휒)푓I ℎ = 푒 ℎ0. (3.34)

퐿 It follows from the discussion above that ℎ is a smooth Hermitian metric on 푇 푀mr

99 near the preimage of the singular fibers and that its curvature

⎧ ⎨⎪ 1 near 퐵II 푅(ℎ) = − (3.35) 2 ⎩⎪ 1 + 푂(푠 ) near 퐵I. − 푡

We therefore use this in place of the initial choice of Hermitian metric.

Let 푔 be the unique Hermitian constant curvature metric on the regular fibers of 휓, so 푔 = 푒2푓 ℎ. The curvatures are related by

2푓 푅(푔)푒 = 훥ℎ푓 + 푅(ℎ),

which reduces to the curvature equation

2푓 훥푓 + 푅(ℎ) = 푒 , 훥 = 훥ℎ. (3.36) −

The linearization of this equation is

(훥 + 2)푓 = (푅(ℎ) + 1). (3.37) −

The uniform invertibility of 훥 + 2 with respect to the metric 퐿2 norm, shown below, implies that (3.36) has a unique small solution for small values of the parameter. The proof of the Theorem in the Introduction therefore reduces to the statement that (3.36) has a log-smooth solution vanishing at the boundary.

1 3.3 Bounds on (훥 + 2)−

In the linearization of the curvature equation (3.37), the operator 훥 + 2, for the fixed initial choice of smooth fiber hermitian metric, appears. For the Laplacian on a compact manifold, 훥 + 2 is an isomorphism of any Sobolev space 퐻푘+1 to 퐻푘−1, in particular this is the case for the map from the Dirichlet space to its dual, corresponding to the case 푘 = 0. For a smooth family of metrics on a fibration the

100 family of Dirichlet spaces forms the fiber 퐻1 space and its dual the fiber 퐻−1 space and 훥 + 2 is again an isomorphism between them. These spaces are modules over the ∞ functions of the total space and this, plus a simple commutation argument, shows 풞 that in this case of a fibration 훥 + 2 is an isomorphism for any 푘 1 between the ≥ space of functions with up to 푘 derivatives, in all directions, in the Dirichlet domain to the space with up to 푘 derivatives in the dual to the Dirichlet space. In particular it follows from this that 훥 + 2 is an isomorphism on functions supported away from the boundary:

∞ ∞ 훥 + 2 : (푀reg) (푀reg), 푀reg = 푀mr 휕푀mr. (3.38) 풞c ←→ 풞c ∖

We extend this result up to the boundary of the resolved space for the Lefschetz fibration in terms of tangential regularity.

Proposition 3.3. For the Laplacian of the grafted metric

1 1 −1 − 2 푘 − 2 푘 (훥 + 2) : 휌II 퐻b (푀mr) 휌II 퐻b (푀mr) 푘 N. (3.39) −→ ∀ ∈

The main complication in the proof arises from the fact that the Dirichlet space is not a ∞ module. 풞 First consider the following analog of Fubini’s theorem.

퐿 Lemma 3.3. For the fiber metrics corresponding to an Hermitian metric on 푇 푀mr, the metric density is of the form

푑푔 = 휌II휈b,fib (3.40) | | and the space of weighted 퐿2 functions with values in the 퐿2 spaces of the fibers can be realized as

1 2 * 2 2 − 2 2 퐿 (푀mr; 푑푔 휑 휈b(푍mr)) = 퐿 (푍mr; 퐿 ( 푑푔 )) = 휌 퐿 (푀mr). (3.41) | | mr b | | II b

퐿 Proof. Away from 퐵II 푀mr the resolved map 휓mr is a fibration, 푇 푀mr is the fiber ⊂ 101 tangent bundle and the boundary is in the base. Thus (3.40) and (3.41) reduce to the local product decomposition for a fibration and Fubini’s Theorem.

It therefore suffices to localize near 퐵II and to consider the plumbing metric since 퐿 all hermitian metrics on 푇 푀mr are quasi-conformal. The symmetry in 푧 and 푤

means that it suffices to consider the region in which 휌푧 = 푠푧/푠푤 and 푠푤 are defining

functions for the two boundary hypersurfaces 퐵I and 퐵II respectively. The plumbing metric may then be written

휋2푠2 (︂푑푠2 )︂ 휋2푠2 (︂ 푑휌2 )︂ 푔 = 푡 푤 + 푑휃2 = 푡 푧 + 푑휃2 . 2 휋푠푡 4 푤 2 휋 2 4 푧 sin ( ) 푠 sin ( ) 푠 (1 + 휌푧) 푠푤 푤 1+휌푧 푡

Thus the fiber area form,

2 2 휋 푠푡 푑휌푧 푠푡 푑휌푧 ˜ 푑휌푧 푑푔 = 2 휋 2 = 푓(휌푧) 푑휃푧 = 푓(휌푧)푠푤 푑휃푧, | | sin ( ) 푠푡(1 + 휌푧) 푑휃푧 휌푧 휌푧 휌푧 1+휌푧

is a positive multiple of 푠 푑휌푧 푑휃 from which (3.40) follows. 푤 휌푧 푧

The identication (3.41) holds after localization away from 퐵II and locally near it

∫︁ ∫︁ ∫︁ 2 2 푑푠푡 2 푓 2 2 = 푓 푑푔 푑휃푡 = 푓 휌II휈b. 퐿푏 (푍mr);퐿 (푑푔)) || || | | | | 푠푡 푍mr | |

Since (훥 + 2)−1 is a well-defined bounded operator on the metric 퐿2 space which depends continuously on the parameter in 푍 푆 with norm bounded by 1/2, it follows ∖ from (3.41) that 1 −1 − 2 2 (훥 + 2) is bounded on 휌II 퐿b(푀mr). (3.42)

We consider the ‘total’ Dirichlet space based on this 퐿2 space – we are free to choose the weighting in the parameter space. Thus, let 퐷 be the the completion of

the smooth functions on 푀mr supported in the interior with respect to

∫︁ 2 (︀ 2 2)︀ * 푢 = 푑fib푢 + 2 푢 푑푔 휑 휈b(푍mr). (3.43) ‖ ‖퐷 | |푔 | | | |

102 Note that 퐷 depends only on the quasi-isometry class of the fiber Hermitian metric but does depend on the induced fibration of the boundary 퐵II. The dual space, 퐷′, to 퐷 as an abstract Hilbert space, may be embedded in the * extendible distributions on 푀mr using the volume form 휑 휈b 푑푔 . As is clear from mr | | the discussion below, the image is independent of the choice of, 휈b, of a logarithmic ′ area form on 푍mr but the embedding itself depends on this choice. Thus, 푣˜ 퐷 is ∈ ˙∞ identified as a map 푣 : 푐 (푀mr) C by 풞 −→ ∫︁ * 푣휑 푑푔 휑 휈b(푍mr) =푣 ˜(휑). (3.44) | |

−1 We consider the space of vector fields 휌 b(푀mr) which are tangent to the 풲 ⊂ II 풱 fibers of 휓mr and to the fibers of 퐵II and which commute with 휕휃푧 and 휕휃푤 near 퐵II.

Proposition 3.4. For the grafted metric

′ −∞ 훥 + 2 : 퐷 퐷 (푀mr) → ⊂ 풞 is an isomorphism, where the elements of 퐷′ are precisely those extendible distributions which may be written as finite sums

1 ∑︁ − 2 2 푣 = 푊푗푢푗, 푊푗 , 푢푗 휌II 퐿b(푍mr) (3.45) 푗 ∈ 풲 ∈ and has the injectivity property that

−∞ ′ 푢 (푀mr), (훥 + 2)푢 퐷 = 푢 퐷. (3.46) ∈ 풞 ∈ ⇒ ∈

퐿 This result remains true for any Hermitian metric on 푇 푀mr but is only needed here for the grafted metric which is equal to the plumbing metric near 퐵II.

Proof. Although defined above by completion of the space of smooth functions supported away from the boundary of 푀mr with respect to the norm (3.43) the space 퐷

103 −∞ can be identified in the usual way with the subspace of (푀mr) consisting of those 풞

1 1 − 2 2 − 2 2 푢 휌 퐿 (푀mr) s.t. 푢 휌 퐿 (푀mr) (3.47) ∈ II b 풲 · ⊂ II b

with the derivatives taken in the sense of extendible distributions. Indeed, choosing

∞ a cutoff 휇 c (R) which is equal to 1 near 0 the sequence of multiplication opera- ∈ 풞 1 − 2 2 tors 1 휇(푛휌II) tends strongly to the identity on 휌 퐿 (푀mr). By assumption this − II b commutes with the elements of and it follows that elements with support in the 풲 interior of 푀mr, where 휓mr is a fibration, are dense in 퐷; for these approximation by smooth elements is standard.

′ −∞ That 훥+2 : 퐷 퐷 (푀mr) is the explicit form of the Riesz representation −→ ⊂ 풞 theorem in this setting. Then the identification, (3.45), of elements of 퐷′ follows from ∞ the form of 훥. Away from 퐵II, 퐷 is a module (since the elements of are smooth 풞 풲 −1 there) and then (3.45) is the identification of the fiber 퐻 space. Near 퐵II we may use the explicit form of the Laplacian for the plumbing metric.

Indeed, the local version of the Dirichlet form is

∫︁ (︀ )︀ 푑푠푤푑휃푤 퐷(휑, 휓) = 푉Re휑푉Re휑 + 푉Im휑푉Im휑 2 (3.48) 푠푤

where 푉 is given by (3.28) and it follows that the Laplacian acting on functions on the fibers can be written

sin2( 휋 ) 1+휌푧 (︀ 2 2)︀ 훥 = 2 2 푉R + (휕휃푧 휕휃푤 ) (3.49) − 휋 푠푡 −

in the coordinates 푠푤, 휌푧, 휃푤 and 휃푧.

−1 The vector fields 푉 and 휌 (휕휃 휕휃 ) generate near 퐵II over the functions R II 푧 − 푤 풲 푘 which are constant in 휃푤 and 휃푧. If we write Diff풲 (푀mr) for the differential operators which can be written as sums of products of elements of at most 푘 elements of 풲 with smooth coefficients which are independent of the angular variables near 퐵II then

2 훥 Diff (푀mr). (3.50) ∈ 풲 104 Moreover 1 1 − 2 2 Diff (푀mr): 퐷 휌 퐿 (푀mr) and 풲 −→ II b 1 (3.51) 1 − 2 2 ′ Diff (푀mr): 휌 퐿 (푀mr) 퐷 풲 II b −→ where the second statement follows by duality from the first. Together (3.50) and (3.51) imply (3.45).

Consider the space b(푀mr), defined analogously to , as consisting of the 풰 ⊂ 풱 풲 푘 vector fields which commute with 휕휃푧 and 휕휃푤 near 퐵II. Then let Diff풰 (푀mr) be the part of the enveloping algebra of up to order 푘, this just consists of the elements of 풰 푘 Diffb(푀mr) which commute with 휕휃푧 and 휕휃푤 near 퐵II. We may define ‘higher order’ versions of the spaces 퐷 and 퐷′ :

푘 퐷푘 = 푢 퐷; Diff (푀mr) 푢 퐷 , { ∈ 풰 · ⊂ } ′ ′ 푘 ′ 퐷푘 = 푢 퐷 ; Diff풰 (푀mr) 푢 퐷 , 푘 N. (3.52) { ∈ · ⊂ } ∈

∞ Since spans b(푀mr) over (푀mr) it follows that 풰 풱 풞

1 − 2 푘 ′ 퐷푘 휌 퐻 (푀mr) 퐷 푘. (3.53) ⊂ II b ⊂ 푘 ∀

∞ ′ Proposition 3.5. For any 푘, ˙ (푀mr) is dense in 퐷푘 and 퐷 and 풞 푘

′ 훥 + 2 : 퐷푘 퐷 (3.54) −→ 푘 is an isomorphism.

Proof. The density statement follows from the same argument as for 퐷 and 퐷′. Consider the commutator relation which follows directly from the definitions

푘 2 푘−1 [ , ] = [Diff풰 (푀mr), 훥] Diff풲 (푀mr) Diff풰 (푀mr), 푘 N. (3.55) 풰 풲 ⊂ 풲 ⇒ ⊂ · ∈

푘 To prove (3.54) we need to show that if 푢 퐷, 푄 Diff (푀mr) and 푓 = (훥+2)푢 ∈ ∈ 풰 ∈ ′ 푘−1 퐷 then 푄푢 퐷. Assuming the result for 푄 Diff (푀mr) it follows from (3.55) 푘 ∈ ∈ 풰 105 that

∑︁ 2 푘−1 훥푄푢 = 푄훥푢 + 퐿푝푄푝푢 with 퐿푝 Diff풲 (푀mr), 푄푝 Diff풰 (푀mr) 푝 ∈ ∈ = 훥푄푢 퐷′ = 푄푢 퐷 (3.56) ⇒ ∈ ⇒ ∈ by distributional uniqueness.

Proof of Proposition 3.3. The boundedness (3.39) follows directly from (3.54) and (3.53).

3.4 Formal solution of (훥 + 2)푢 = 푓

In the previous section the uniform invertibility of 훥 + 2 for the grafted metric was established. In particular the case 푘 = in (3.39) shows the invertibility on conormal ∞ functions. In this section we solve the same equation, (훥 + 2)푢 = 푓 in formal power series with logarithmic terms.

∞ ∞ Let (푀mr) (푀mr) denote the subspace annihilated to infinte order at 퐵II 풞퐹 ⊂ 풞 by the angular operators 퐷휃푧 and 퐷휃푤 .

Lemma 3.4. The restriction, 훥I, of the Laplacian to 퐵I satisfies

−1 (︀ 푘 )︀ (훥I + 2) 휌II(log 휌II) 푔푘

∑︁ 푝 ∞ ∞ = 휌II (log 휌II) 푢푝, 푢푝 (푀mr) 푔푘 (푀mr). (3.57) ∈ 풞퐹 ∀ ∈ 풞퐹 0≤푝≤푘+1

Proof. The fiber metric on 퐵I is a trivial family with respect to the product decomposition 퐵I = 퐴 S where 퐴 has the complete metric on the Riemann surface with × cusps arising from the ‘removal’ of the nodal points. The Laplacian is therefore essentially self-adjoint and non-negative, so 훥 + 2 is invertible. Either from the form of a parameterix or by Fourier expansion near the cusps it follows that rapid decay

−1 in the non-zero Fourier modes (in both angular variables) is preserved by (훥I + 2) . Near the boundary the zero Fourier mode satisfies a reduced, ordinary differential,

106 equation with regular singular points and having indicial roots 1 and 2 in terms of − a defining function for the (resolved) cusps. Then (3.57) follows directly.

∞ ∞ ⃒ Lemma 3.5. If 푢 퐹 (푀mr) then 훥푢 퐹 (푀mr) restricts to 퐵II to 훥̃︀II푣, 푣 = 푢⃒ ∈ 풞 ∈ 풞 퐵II where 훥̃︀II is an ordinary differential operator of order 2 elliptic in the interior with regular singular endpoints, with indicial roots 1, 2 such that −

−1 ∞ Nul(훥̃︀II + 2) 휌 (퐵II) (3.58) ⊂ I 풞

∞ has no smooth elements and for ℎ푗 (퐵II) ∈ 풞퐹

−1 푗 ∑︁ 푞 2 푗+1 (훥̃︀II + 2) (log 휌I) ℎ푗 = (log 휌I) 푣푞,푗 + 휌I (log 휌I) 푤푗 0≤푞≤푗 (3.59) ∞ with 푣푞,푗, 푤푗 (퐵II). ∈ 풞퐹

Proof. The form of the Laplacian in (3.49) shows that the reduced operator 훥̃︀II exists

and after the change coordinates on 퐵II to

1 휌 = (3.60) 1 + 휌II

becomes

sin(휋휌) 2 2 훥 + 2 = 2 ( ) [(휌휕휌) 휌휕휌]. (3.61) − 휋휌 − The indicial roots of this operator are 2 and 1 and its homoeneity shows that the − null space has no logarithmic terms. The absence of smooth elements in the null space follows by integration by parts and positivity.

The problem that we need to solve at 퐵II is

2 (1) ⃒ ⃒ (훥 + 2)(휌II푤) = 휌II푔 + 푂(휌II) = (훥̃︀II + 2)(푤⃒ ) = 푔⃒ . (3.62) ⇒ 퐵II 퐵II

Since the parameter, 푠푡, is the product of defining functions for 퐵I and 퐵II and (1) commutes through the problem this can be solved by dividing by it. Thus 훥̃︀II is

obtained from 훥̃︀II by conjugating by a boundary defining function on 퐵II so the

107 preceding Lemma can be applied after noting the shift of the indicial roots.

Lemma 3.6. For the conjugated operator on 퐵II,

(1) ∞ Nul(훥̃︀ + 2) (퐵II) (3.63) II ⊂ 풞 with the Dirichlet problem uniquely solvable and

−1 푗 ∑︁ 푞 3 푗+1 (훥̃︀II + 2) (log 휌I) ℎ푗 = (log 휌I) 푣푞,푗 + 휌I (log 휌I) 푤푗 0≤푞≤푗 (3.64) ∞ with 푣푞,푗, 푤푗 (퐵II). ∈ 풞퐹

To express the form of the expansion which occur below, consider the space of

∞ polynomials in log 휌I and log 휌II with coefficients in (푀mr) 풞퐹 {︃ }︃ 푘 ∑︁ 푙 푝 ∞ = 푢 = (log 휌I) (log 휌II) 푢푙,푝, 푢푙,푝 (푀mr) . (3.65) 풫 ∈ 풞퐹 0≤푙+푝≤푘

We also consider the filtration of these spaces by the maximal order in each ofthe variables:

{︃ }︃ 푘,푗 ∑︁ 푙 푝 ∞ = 푢 = (log 휌I) (log 휌II) 푢푙,푝, 푢푙,푝 (푀mr) , 푗 푘 풫I ∈ 풞퐹 ≤ 0≤푙+푝≤푘, 푙≤푗 {︃ }︃ (3.66) 푘,푚 ∑︁ 푙 푝 ∞ = 푢 = (log 휌I) (log 휌II) 푢푙,푝, 푢푙,푝 (푀mr) , 푚 푘. 풫II ∈ 풞퐹 ≤ 0≤푙+푝≤푘, 푝≤푚

∞ Since the coefficients are in (푀mr), 훥 acts as a smooth b-differential operator on 풞퐹 푘,푝 ′ ′ 푘,푝−1 푝 all of these spaces. If 푢 , then 푢 = 푢푝 + 푢 with 푢 and 푢푝 = 푣(log 휌I) ∈ 풫I ∈ 풫I 푘−푝,0 푝 ′ ′ 푘−1,푝−1 푘,푝−1 where 푣 . Then 훥푢 = (훥I푣)(log 휌I) + 푓 , 푓 + 휌I where the ∈ 풫I ∈ 풫I 풫I 푝 first error term corresponds to at least one derivation of (log 휌I) . Similar statements apply to 퐵II and 훥˜II. As a basis for iteration, to capture the somewhat complicated behavior of the logarthimic terms, we first consider a partial result.

108 Proposition 3.6. For each 푘

푘 푘+1 푘+1 2 푘+2,푘+1 푓 휌II + 휌I휌II = 푢 휌II + 휌 휌II (3.67) ∈ 풫 풫 ⇒ ∃ ∈ 풫 I 풫II such that (︀ 푘+1 푘+2)︀ (훥 + 2)푢 푓 푠푡 휌II + 휌I휌II . (3.68) − ∈ 풫 풫

Proof. We first solve on 퐵I, then on 퐵II. The second term in 푓 in (3.67) vanishes on 푘⃒ 퐵I so the restriction 푓I 휌II ⃒ . Proceeding iteratively, suppose ∈ 풫 퐵I

푘,푗 푘+1 푓 휌II + 휌I휌II ∈ 풫I 풫 with 푗 푘 and consider the term of order 푗 in log 휌I; this is a polynomial in log 휌II ≤ ∞ of degree at most 푘 푗 with coefficients in 휌II (퐵I). Applying Lemma 3.4 to the − 풞퐹 restriction to 퐵I gives a polynomial in log 휌II of degree at most 푘 푗 + 1 with coef- − ∞ ficients in 휌II (퐵I). Extending these coefficients off 퐵I and restoring the coefficient 풞퐹 푗 푘+1,푗 of (log 휌I) gives 푣푗 휌II such that ∈ 풫I

′ ′ 푘,푗−1 푘+1 (훥 + 2)푣푗 푓 = 푓 , 푓 휌II + 휌I휌II . − − ∈ 풫I 풫

푗 Here the first part of the error arises from differentiation of thefactor (log 휌I) in 푣푗 at least once. If we start with 푗 = 푘 and proceed iteratively over decreasing 푗 this 푘+1 allows us to find 푣 휌II such that ∈ 풫

푘+1 (훥 + 2)푣 푓 = 푔 휌I휌II . (3.69) − − ∈ 풫

Now we proceed similarly by solving on 퐵II using Lemma 3.6. So, suppose ℎ ∈ 푘+1,푝 푝 휌I휌II , for 푝 푘 + 1. Then the coefficient ℎ푝 of (log 휌II) is a polynomial of degee 풫II ≤ ∞ at most 푘 + 1 푝 in log 휌I with coefficients in 휌I휌II (푀mr). Conjugating away the − 풞퐹 factor of 휌II and applying Lemma 3.6 to the restriction to 퐵II and then extending the 푘+1,푝 2 푘+2,푝 coefficients off 퐵II allows us to find 푤푝 휌I휌II + 휌 휌II , where the second ∈ 풫II I 풫II term arises from the possible increase in multiplicity of the logarithmic coefficient of

109 2 휌I in the solution, satisfying

′ ′ 푘+1,푝−1 2 푘+1,푝 2 2 푘+2,푝 (훥 + 2)푤푝 푔 = 푔 + 푒, 푔 휌I휌II , 푒 휌I휌 + 휌 휌 (3.70) − − ∈ 풫II ∈ II풫II I II풫II

푝 where the first part of the error arises from differentiation of (log 휌II) at least once. Starting with 푝 = 푘 + 1 and iterating over decreasing 푝 allows us to find 푤 ∈ 푘+1 2 푘+2,푘+1 휌I휌II + 휌 휌II such that 풫 I 풫

2 푘+1 2 2 푘+2 (훥 + 2)푤 푔 휌I휌 + 휌 휌 . (3.71) − ∈ II풫 I II풫

Combining (3.69) and (3.71) gives (3.68) since 휌I휌II is a smooth multiple of 푠푡.

Proposition 3.6 allows iteration since 푠푡 commutes through 훥 + 2.

푘 푘+1 −1 −휖 ∞ Proposition 3.7. If 푓 휌II + 휌I휌II then 푢 = (훥 + 2) 푓 푠 퐻 (푀mr) for ∈ 풫 풫 ∈ 푡 b any 휖 > 0, has a complete asymptotic expansion of the form

∑︁ 푗 푘+푗 푘+푗+1,푘+푗 푢 푠푡 푢푗, 푢푗 휌II + 휌I휌II II . (3.72) ≃ 푗≥0 ∈ 풫 풫

1 휖 − 2 ∞ −휖 −1 Proof. For any 휖 > 0, 푔 = 푠 푓 휌 퐻 (푀mr) so 푢 = 푠 (훥 + 2) 푔 exists by (3.39). 푡 ∈ II b 푡 Comparing 푢 to the expansion cut off at a finite point gives (3.72).

This result can itself be iterated, asymototically summed and then the rapidly decaying remainder term again removed to show the polyhomogeneity of the solution for an asymptotically covergent sum over terms on the right in (3.72). For the solution of the curvature equation the leading term is smooth because of the special structure of the forcing term.

∞ −1 Lemma 3.7. If 푓 (푀mr) has support disjoint from 퐵II then 푢 = (훥 + 2) 푓 is ∈ 풞 log-smooth and has an asymptotic expansion of the form

∑︁ 푘 푘 푘+1,푘 푢 휌II푣0 + 푠 푣푘, 푣푘 휌II + 휌I휌II . (3.73) ≃ 푡 ∈ 풫 풫II 푘≥1

110 ∞ Note that log-smoothness follows from the fact that 푠푡 = 푎휌I휌II, 푎 (푀mr) so each ∈ 풞퐹 term in the expansion can be written as a polynomial in 휌I, 휌I log 휌I, 휌II and 휌II log 휌II of degree at least 2푘.

3.5 Polyhomogeneity for the curvature equation

Under a conformal change from the grafted metric ℎ with curvature 푅 to 푒2푓 ℎ the condition for the curvature of the new metric to be 1 given by (3.9). To construct − the canonical metrics on the fibers we proceed, as in the linear case discussed above, to solve (3.9) in the sense of formal power series at the two boundaries above 푠푡 = 0 and then, using the Implicit Function Theorem deduce that the actual solution has this asymptotic expansion.

Lemma 3.8. For the grafted metric there is a formal power series

∑︁ 푘 ∞ 푘−2 푘−1,푘−2 푠 푓푘, 푓2 (푀mr), 푓푘 휌II + 휌I휌II , 푘 3, (3.74) 푡 ∈ 풞퐹 ∈ 풫 풫II ≥ 푘≥2 solving (3.9).

푘 푘−1 The are defined in (3.65); in the last term there is no factorof (log 휌II) . 풫

2 ∞ Proof. Since 푅 + 1 푠 (푀mr) is supported away from 퐵II, Lemma 3.7 shows that ∈ 푡 풞 −1 푔1 = (훥 + 2) (푅 + 1) is of the form (3.74). We look for the formal power series − solution of the non-linear problem as

∑︁ 푓 푔푘 (3.75) ≃ 푘≥1

Inserting this sum into the equation gives

푗 ∑︁ ∑︁ 2 ∑︁ 푗 (훥 + 2)( 푔푖) = (푔1 + 푔푘) + 1 + 푅. (3.76) − 푗! 푖≥1 푗≥2 푘≥2

111 For each 푖 2 we fix 푔푖 by ≥

푗 푗 ∑︁ 2 ∑︁ 푗 2 ∑︁ 푗 (훥 + 2)푔푖 = (푔1 + 푔푘) (푔1 + 푔푘) − 푗! − 푗! 푗≥1 푖−1≥푘≥2 푖−2≥푘≥2

= 푔푖−1푃푖(푔1, 푔2, ...푔푖−1) (3.77)

where 푃푖 is a formal power series in 푔1, ...푔푖−1 without constant term. Proceeding by induction we claim that

∑︁ 푗 푗−2푖 푗−2푖+1,푗−2푖 푔푖 푠푡 푔푖,푗, 푔푖,푗 휌II + 휌I휌II II . (3.78) ≃ 푗≥2푖 ∈ 풫 풫

We have already seen that this holds for 푖 = 1 and using the obvious multiplicitivity properties 푘 푗 푗+푘, 푘 푗,푗−1 푗+푘,푗+푘−1 풫 · 풫 ⊂ 풫 풫 · 풫II ⊂ 풫II it follows from the inductive assumption, that (3.78) holds for all smaller indices, that

푔푖−1푃푖(푔1, 푔2, ...푔푖−1)

2푖 ∑︁ (︀ 푗−2푖+2 푗−2푖+3,푗−2푖+2)︀ (︁ 푘−2 푘−1,푘−2)︁ 푠 휌II + 휌I휌II 휌II + 휌I휌II ≃ 푡 풫 풫II 풫 풫II 푘≥2,푗≥2푖−2 (3.79)

∑︁ 푗 푘−2푖 푘−2푖+1,푘−2푖 푠 퐹푘, 퐹푘 휌II + 휌I휌II . ≃ 푘 ∈ 풫 풫II 푘≥2푖

Applying Proposition 3.7 we recover the inductive hypothesis at the next step. Then (3.74) follows from (3.75) and (3.78).

Summing the formal power series solution gives a polyhomogeneous function with

2푓0 ∞ 훥푓0 = 푅 + 푒 + 푔, 푔 푂(푠 ). (3.80) − ∈ 푡

Now we look for the solution as a perturbation 푓 = 푓0 + 푓,˜ so 푓˜ satisfies

˜ 훥푓˜ = 푔 + 푒2푓0 (푒2푓 1). (3.81) − − − 112 which can be rewritten as

(︁ ˜ )︁ 푓˜ = (훥 + 2)−1 2푓˜(푒2푓0 1) + 푒2푓0 (푒2푓 1 2푓˜) 푔 . − − − − −

So consider the nonlinear operator

(︁ ˜ )︁ 퐾 : 푓˜ (훥 + 2)−1 2푓˜(푒2푓0 1) + 푒2푓0 (푒2푓 1 2푓˜) 푔 (3.82) ↦→ − − − −

푁 푀 which acts on 푠 퐻 (푀mr) for all 푁 1 and 푀 > 2. Note that for 푀 > 2, the b- 푡 푏 ≥ 푀 space 퐻푏 (푀mr) is closed under multiplication, therefore this weighted Sobolev space is also an algebra. Since the nonlinear terms are at least quadratic, 퐾 is well-defined on this domain. The solution to (3.81) satisfies 푓˜ = 퐾(푓˜).

Proposition 3.8. For any 푀 > 1 and 푁 1 there is a unique solution 푓˜ ≥ ∈ 푁 푀 푠푡 퐻푏 (푀mr) to the equation (3.81).

˜ ˜ 푁 ∑︀ 푖 Proof. We construct the solution 푓 by iteration. Let 푓 = 푠푡 푖≥2 푠푡 푓푖, put it into 푁 equation (3.81), divide by the common factor 푠푡 on both sides and then we get

(︁ )︁ ∑︁ 푖 ∑︁ 푖 −1 2푓0 ∑︁ 푖 푁 ∑︁ 푖 2 −푁 푠푡푓푖 = 퐾( 푠푡푓푖) = (훥 + 2) (푒 1) 푠푡푓푖 + 푠푡 ( 푠푡푓푖) + 푠푡 푔 푖≥2 − (3.83)

−1 2 The right hand side belongs to (훥+2) (푂(푠푡 )) because of the quadratic structure and

2푓0 2 −1 the fact that 푒 1 푂(푠푡 ). Therefore the right hand side is the form (훥+2) (푠푡ℎ) 1 − ∈ − 2 푀 where 푠푡ℎ 휌 퐻 (푀mr) so this quantity is well-defined using Proposition 3.3. ∈ II b Now we proceed by induction. Assume that the first k terms in the expansion have been solved, then the equation for the next term 푓푘 is given by

−1 (︀ 2푓0 푁 )︀ 푓푘 = (훥 + 2) (푒 1)푓푘−2 + 푠 푄(푓0, ...푓푘−1) . − 푡 where the polynomial 푄 on the right hand side is a quadratic polynomial of order

푘 푁. By using the invertibility property in Proposition 3.3, we can now solve 푓푘. − Therefore the induction gives us the total expansion for 푓˜.

113 Proof of Theorem 3.1. From Proposition 3.8 we obtain the solution, 푓 = 푓0 + 푓˜, 2푓 to the curvature equation 푅(푒 ℎ) = 1. Since 푓0 is the formal power series and − ∞ ∞ 푓˜ 푠 (푀mr), we get the solution with required regularity. ∈ 푡 풞

114 Chapter 4

Group resolution

4.1 Introduction

For a general compact Lie group G acting on a smooth compact manifold with corners 푀, Albin and Melrose [1] showed that there is a canonical full resolution such that the group action lifts to the blow-up space 푌 (푀) to have a unique isotropy type. This generalized the result of Borel [4] that if all the isotropy groups of a compact group action are conjugate then the orbit space 퐺 푀 is smooth. ∖ In this paper, we give an explicit construction of such a resolution of the unitary group action on the space of self-adjoint matrices

* 푆 = 푆(푛) = 푋 푀푛(C) 푋 = 푋 { ∈ | } with the unitary group 푈(푛) acting by conjugation: for 푢 푈(푛), 푋 푆, ∈ ∈

푢 푋 := 푢푋푢−1. ·

The orbit of an element 푋 푆, denoted by 푈(푛) 푋, consists of the matrices with ∈ · the same eigenvalues including multiplicities. For a matrix 푋 푆 with 푚 distinct ∈ 푚 eigenvalues 휆푗 , each with multiplicity 푖푘, 푘 = 1, 2, .., 푚, the isotropy group of 푋 { }푗=1 115 is isomorphic to a direct sum of smaller unitary groups:

푋 푚 푈(푛) = 푢 푈(푛) 푢 푋 = 푋 = 푈(푖푘). { ∈ | · } ∼ ⊕푘=1

Thus the matrices with the same multiplicities 푖푘 , have conjugate isotropy groups. { } The isotropy types are therefore parametrized by the partition of 푛 into integers. Note here that the partition contains information about ordering of the eigenvalues, for example, the two partitions of 3, 푖1 = 1, 푖2 = 2 and 푖1 = 2, 푖2 = 1 , are not the { } { } same type. For 푛 > 1, the eigenvalues are not smooth functions on 푆, but are singular where the multiplicities change. We will show that, by doing an iterative blow up, the singularities are resolved and the eigenvalues become smooth functions on the resolved space. Recall the lemma of group action resolution in [1]:

Lemma 4.1 ([1]). A compact manifold (with corners), M, with a smooth, boundary intersection free, action by a compact Lie group, G, has a canonical full resolution, 푌 (푀), obtained by iterative blow-up of minimal isotropy types.

Consider the trivial bundle over 푆,

푛 푀 := 푆 C , × the fiber of which can be decomposed into 푛 eigenspaces of the self-adjoint matrix at the base point. This decomposition is not unique at matrices with multiple eigenvalues and in general the eigenspaces are not smooth. There are two basic kinds of real blow up, namely radial and projective, which give different results; radial blow up of a hypersurface produces a new boundary while projective blow up does not. As pointed out in [1], projective blow up usually requires an extra step of reflection in the iterative scheme in order to obtain smoothness. We will show that, after radial blow up, the trivial bundle 푀 decomposes into the direct sum of 푛 1-dimensional eigenspaces. In contrast, after the projective blow up, though

116 the eigenvalues are still smooth on the resolved space and locally this is a smooth decomposition into simple eigenspaces, but the trivial bundle doesn’t split into global line bundles. Next we recall the resolution in the sense of Albin and Melrose.

Definition 4.1 (eigenresolution). By an eigenresolution of S, we mean a manifold with corners 푆^, with a surjective smooth map 훽 : 푆^ 푆 such that the self-adjoint → matrices have a smooth (local) diagonalization when lifted to 푆^, with eigenvalues lifted to smooth functions on 푆^.

Note in the definition we only require the the diagonalization exists locally. To encompass the information of global decomposition of eigenvectors, we introduce the full resolution below.

Definition 4.2 (full eigenresolution). A full eigenresolution is an eigenresolution

with global eigenbundles. The eigenvalues are lifted to 푛 smooth functions 푓푖 on 푆^, and 푀, which is the trivial n-dimensional complex vector bundle on 푆^, is decomposed into 푛 smooth line bundles: 푛 ؁ 푛 ^ 푆 C = 퐸푖 × 푖=1 such that

훽(푥)푣푖 = 푓푖(푥)푣푖, 푣푖 퐸푖(푥), 푥 푆.^ ∀ ∈ ∀ ∈ We use the blow-up constructions introduced by Melrose in the book [34, Chap- ter 5] and show that we can obtain resolutions in this way and, in particular, full resolution if we use radial blow-up.

Theorem 4.1. The iterative blow up of the isotropy types in 푆, in an order compatible with inclusion of the conjugation class of the isotropy group, yields an eigenresolution. In particular, radial blow up gives a full eigenresolution.

4.2 Proof of the theorem

The proof proceeds through induction on dimension. We begin the proof by discussing the first example which is the 2 2 matrices. × 117 Lemma 4.2 (2 2 case). For the 2 2 matrices 푆(2), the eigenvalues and eigenvec- × × tors are smooth except at the scalar matrices. After radial blow up, the singularities are resolved and the trivial 2-dim bundle splits into the direct sum of two line bundles. The projective blow up also gives smooth eigenvalues, but does not give two global line bundles.. ⎧⎛ ⎞⃒ ⎫ ⃒ ⎨ 푎11 푧12 ⃒ ⎬ 4 Proof. In this case 푆 = 푆(2) = ⎝ ⎠⃒ 푎푖푖 R, 푧12 C = R . Thus S is ⃒ ∈ ∈ ∼ ⎩ 푧¯12 푎22 ⃒ ⎭ isomorphic to the product of R and the trace-free subspace

⎧⎛ ⎞⃒ ⎫ ⃒ ⎨ 푎11 푧12 ⃒ ⎬ 푆0 = ⎝ ⎠⃒ 푎11 + 푎22 = 0 , (4.1) ⃒ ⎩ 푧¯12 푎22 ⃒ ⎭ i.e. there is a bijective linear map:

휑 : 푆 푆0 R →⎛ × ⎞ 푎11 푧12 (4.2) 퐴 = ⎝ ⎠ (퐴0 := 퐴 (푎11 + 푎22)퐼, 푎11 + 푎22) ↦→ − 푧¯12 푎22

The eigenvalues 휆푖 and eigenvectors 푣푖 of 퐴 are related to those 퐴0 by 휆푖(퐴) =

휆푖(퐴0) + 푡푟(퐴), 푣푖(퐴) = 푣푖(퐴0), 푖 = 1, 2. Therefore, we can restrict the discussion of resolution to the subspace 푆0, since the smoothness of eigenvalues and eigenvectors on 푆 follows.

3 Let 푧12 = 푐 + 푑푖. The space 푆0 can be identified with R = (푎11, 푐, 푑) . The { } eigenvalues of this matrix are:

√︁ 2 2 2 휆± = 푎 + 푐 + 푑 . (4.3) ± 11

Hence the only singularity of the eigenvalues on 푆0 is at the point 푎11 = 푐 = 푑 = 0 which is the zero matrix. Based on the resolution formula in [34], the radial blow up can be realized as

^ + 2 푆0 = [푆0, 0 ] = 푆 푁 0 (푆0 0 ) S [0, )+ (4.4) { } { } ⊔ ∖ { } ≃ × ∞ 118 + 2 √︀ 2 2 2 where the front face 푆 푁 0 S . Here the radial variable r is 푎11 + 푐 + 푑 . The { } ≃ blow-down map is

2 훽 :[푆0, 0 ] 푆0, (푟, 휃) 푟휃, 푟 R+, 휃 S . (4.5) { } → ↦→ ∈ ∈

The radial variable 푟 lifts to be a smooth on the blown up space, therefore the two eigenvalues 휆± = 푟 become smooth functions. ±

Now we consider the eigenvectors of the corresponding eigenvalues 휆±

√︁ 2 2 2 2 푣± = (푐 + 푑푖, 푎11 + 푐 + 푑 푎11) C . (4.6) ± − ∈

Similar to the discussion of the eigenvalues, the only singularity is at 푟 = 0, which becomes a smooth function on [푆0, 0 ], it follows that 푣+ and 푣− span two smooth { } line bundles on [푆0, 0 ]. { } If we do the projective blow up instead, which identifies the antipodal points in 2 the front face of S2 to get RP , namely

˜ 3 2 푆0 = (푥, 푙) 푥 푙 R RP (4.7) { | ∈ } ⊂ × for which we will cover it with three coordinate patches

푑 푎11 3 (푥1, 푦1, 푧1) = (푐, , ) R 푐 푐 ∈

푐 푎11 푐 푑 and the other two (푥2, 푦2, 푧2), (푥3, 푦3, 푧3)=(푑, , ), (푎11, , ) are similar. The two 푑 푑 푎11 푎11 eigenvalues we get from here are

√︁ √︁ 2 2 2 2 2 푣± = 푎 + 푐 + 푑 = 푥1 (1 + 푦 + 푧 ). ± 11 ±| | 1 1 which is smooth at 푥1 > 0 . Similar discussions hold for the other two coordinate { } patches.

However, the trivial bundle does not decompose into two line bundles as in the

119 2 radial case. The nontriviality of eigenbundles can be seen by taking a loop in RP

푙 = 훽−1( 푟 = 1 ) 푆˜ { } ⊂

which is a curve that winds twice around origin. This curve intersects the line 푐 = 푑 = 0 twice, which hits at two different places thus both 푎± = 1 are on the 11 ± + curve, and (4.6) shows that starting from 푣− = (0, 2) = (0, 2푎 ), this turns into − − 11 − 푣+ = (0, 2) = (0, 2푎 ), which means they are not separated by projective blow up. − 11 Now that we have done the radial resolution for the trace free slice 푆0, the resolution of S follows. Consider S as a 3-dim vector bundle on R with trace being the

projection map, then at each base point 휆, the fiber is 푆0 + 휆퐼. The resolution is

[푆0 + 휆퐼; 휆퐼] ∼= [푆0; 0]. Since the trace direction is transversal to the blow up, and therefore

[푆; R퐼] = [푆0; 0 ] R. (4.8) { } × And because the trace don’t change the eigenvectors, the smoothness follows.

To proceed to higher dimensions, we first discuss the partition of eigenvalues into clusters. The basic case is when the eigenvalues are divided into two clusters, then the 푈(푛) action of the matrices can be decomposed to two commuting actions.

Definition 4.3 (spectral gap). A connected neighborhood 푈 푆 has a spectral gap ⊂ at 푐 R, if c is not an eigenvalue of X, for any 푋 푈. ∈ ∈ Note here that since U is connected, the number of eigenvalues less than c stays the same for all 푋 푈, denoted by k. ∈ Lemma 4.3 (local eigenspace decomposition). If a neighborhood 푈 푆(푛) has a ⊂ spectral gap at 푐, then the matrices in U can be decomposed into two self-adjoint commuting matrices smoothly:

푋 = 퐿푋 + 푅푋 , 퐿푋 푅푋 = 푅푋 퐿푋 .

with rank( 퐿푋 ) = 푘, rank(푅푋 ) = 푛 푘. − 120 Proof. Let 훾 be a simple closed curve on C such that it intersects with R only at -R and c, where R is a sufficiently large number. In this way, for any matrix 푋 푈, the ∈ k smallest eigenvalues are all contained inside 훾. We consider the operator

푛 푛 푃푋 : C C → ∮︁ 1 −1 푃푋 := (푋 푠퐼) 푑푠 (4.9) −2휋푖 훾 −

Since the resolvent is nonsingular on 훾, 푃푋 is a well-defined operator and varies smoothly with X. And the integral is independent of choice of 훾 up to homotopy.

First we show that 푃푋 is a projection operator, i.e.

2 푃푋 = 푃푋

Let 훾푠 and 훾푡 be two curves satisfying the above condition with 훾푠 completely inside

훾푡, then

2 1 ∮︀ −1 ∮︀ −1 푃 = 2 (푋 푡퐼) 푑푡( (푋 푠퐼) 푑푠) 푋 − 4휋 훾푡 − 훾푠 − 1 ∮︀ ∮︀ 1 −1 ∮︀ 1 −1 = 2 푑푡[ (푋 푠퐼) 푑푠 (푋 푡퐼) 푑푠] − 4휋 훾푡 훾푠 푠−푡 − − 훾푠 푠−푡 − = 퐼 퐼퐼 −

where using the fact that 푠 is completely inside 훾푡

1 ∮︁ 1 ∮︁ 1 1 ∮︁ 1 퐼 = 푑푠 푑푡 = ( 2휋푖) 푑푠 = 푃푋 −4휋2 푋 푠퐼 푠 푡 −4휋2 − 푋 푠퐼 훾푠 − 훾푡 − 훾푠 − and any t on 훾푡 is outside of the loop 훾푠

∮︁ 1 푑푠 = 0 푠 푡 훾푠 − we have 1 ∮︁ ∮︁ 1 퐼퐼 = (푋 푡퐼)−1푑푡 푑푠 = 0 −4휋2 − 푠 푡 훾푡 훾푠 − 2 Therefore 푃푋 = 푃푋 .

121 Then we show that 푃푋 is self-adjoint. This is because

∫︁ ∫︁ * 1 −1 * 1 푃푋 = ((푋 푠퐼) ) 푑푠¯ = (푋 푠퐼)푑푠 = 푃푋 . 2휋푖 훾 − 2휋푖 −훾¯ −

푛 푃푋 maps R to the invariant subspace spanned by the eigenvectors corresponding to eigenvalues that are less than c. We denote this invariant subspace by 퐿 and its orthogonal complement by 푅. Write X as the diagonalization 푋 = 푉 Λ푉 −1 where Λ is the eigenvalue matrix and 푉 consists its eigenvectors as columns. Then 퐿 is spanned by the first 푘 columns of 푉 . Take one of the eigenvectors 푣푗 퐿, 푗 = 1, 2, ..., 푘, ∈ ∮︁ 1 −1 푃푋 푣푗 = (푋 푠퐼) 푣푗푑푠 −2휋푖 훾 − ∮︁ 1 −1 −1 = 푉 (Λ 푠퐼) 푉 푣푗 −2휋푖 − 1 ∮︁ 1 = 푣푗 푑푠 = 푣푗. −2휋푖 휆푗 푠 −

Similarly for 푣푗 푅 that corresponds to an eigenvalue greater than c (therefore 휆푗 is ∈ outside the loop), 1 ∮︁ 1 푃푋 푣푗 = 푣푗 푑푠 = 0, −2휋푖 휆푗 푠 − therefore

(퐼 푃푋 )푣푗 = 푣푗, 푣푗 푅. − ∀ ∈

Then using the projection 푃푋 we define two operators 퐿푋 and 푅푋 as

퐿푋 := 푃푋 푋푃푋 (4.10)

and

푅푋 := (퐼 푃푋 )푋(퐼 푃푋 ). (4.11) − −

Since 푃푋 is smooth, the two operators are also smooth. Moreover, using the fact that

푃푋 is a projection onto the invariant subspace 퐿, we have

(퐼 푃푋 )푋푃푋 = 푃푋 푋(퐼 푃푋 ) = 0, − − 122 therefore

푋 = 퐿푋 + 푅푋 .

For an eigenvector 푣 퐿, ∈

퐿푋 푣 = 푋푣, 푅푋 푣 = 0, (4.12)

* i.e. 퐿푋 equals to 푋 when restricted to 퐿, similarly 푅푋 푅 = 푋. Since 푃 = 푃푋 , 퐿푋 | 푋 and 푅푋 are also self-adjoint. In this way we get two commuting lower rank matrices

퐿푋 and 푅푋 .

It is natural to to have a finer decomposition when there is more than one spectral gap in the neighborhood, and we have the following corollary.

Corollary 2. If the eigenvalues of matrices in a neighborhood U can be grouped into k clusters, Then the matrices can be decomposed into k lower rank self-adjoint commuting matrices smoothly.

Proof. Do the decomposition inductively. If k=2, then it is the case in Lemma 4.3. Suppose the decomposition for 푘 = 푙 1 is defined. Then for 푘 = 푙, since the − eigenvalues can also be divided into 2 clusters (by combining the smallest l-1 groups

of eigenvalues together), then 푋 = 퐿푋 + 푅푋 , with 퐿푋 and 푅푋 corresponding to the

two intervals. Then 퐿푋 satisfies the separation condition for 푙 1 clusters, so by − induction, 퐿푋 = 퐿1 + ... + 퐿푙−1. Therefore, 푋 = 퐿1 + 퐿2 + ... + 퐿푙−1 + 푅푋 is the desired division.

Using the above Lemma 4.3 of decomposition of matrices in a neighborhood, we

can now show that locally the trivial bundle 푆 C푛 decomposes into two subspaces if × there is a spectral gap. And moreover, locally there is a product structure of two lower order matrices. In order to see this, we need to introduce the Grassmannian. Let

푛 퐺푟C(푛, 푘) denote the Grassmannian, i.e. the set of k-dim subspace in C . Consider the tautological vector bundle over Grassmannian:

−1 휋 : 푇푘 퐺푟 (푛, 푘), 휋 (푝) = 푉 (푝). → C 123 where each fibre is a k-dimensional subspace in C푛, with self-adjoint operators acting on it. Similarly, we define 푇푛−푘 to be the orthogonal complement of 푇푘:

−1 ⊥ 휋 : 푇푛−푘 퐺푟 (푛, 푘), 휋 (푝) = 푉 (푝) . → C

Definition 4.4 (operator bundle). Let 푃푘 (resp. 푃푛−푘) be the bundles over 퐺푟C(푛, 푘) of the fibre-wise self-adjoint operators on the tautological bundle 푇푘 (resp. 푇푛−푘).

Let 휋 : 푃푘 푃푛−푘 퐺푟 (푛, 푘) be the whitney sum of the two bundles. Each of its ⊕ → C fiber is isomorphic to 푆(푘) 푆(푛 푘) when we pick a basis. There is a U(n) action ⊕ − on this bundle:

−1 −1 푔 (푝, (푝푘, 푝푛−푘)) = (푔 푝, (푔 푝푘 푔 , 푔 푝푛−푘 푔 )), · · ∘ ∘ ∘ ∘ (4.13) 푝 퐺푟 (푛, 푘), 푝푘 푃푘(푝), 푝푛−푘 푃푛−푘(푝). ∈ C ∈ ∈

Suppose an open neighborhood 푈 푆 satisfies the spectral gap condition. Let ∈ 푈(푛) 푈 be the group invariant neighborhood generated by U, that is, ·

푈(푛) 푈 := 푔∈푈(푛)푔 푈. (4.14) · ∪ ·

Then 푈(푛) 푈 is open and connected, and also satisfies the spectral gap condition as 푈 · does, since U(n) action preserves the eigenvalues. From the proof of the Lemma 4.3, it is shown that in the neighborhood, the trivial C푛 bundle over 푈 naturally splits into two subbundles 퐸푘 퐸푛−푘. And this gives a local product structure. We will ⊕ prove that, for a U(n)-invariant neighborhood, there is actually a group equivariant isomorphism with the operator bundles defined above.

Lemma 4.4 (bundle map). If a point 푋0 푆 satisfies the spectral gap condition, ∈ then there is a neighborhood 푋0 푉 푆 such that it is isomorphic to a neighborhood ∈ ⊂ in the product of lower rank matrices and Grassmannian, i.e.

휑 : 푉 = 푉 (푘) 푉 (푛 푘) 푉퐺푟 푆(푘) 푆(푛 푘) 퐺푟 (푛, 푘) 푃푛 푃푛−푘. ∼ × − × ⊂ × − × C ⊂ ⊕ 124 Moreover, if we take the neighborhood 푈(푛) 푉 , it is isomorphic to a neighborhood · 푊 푃푘 푃푛−푘 such that 휋(푊 ) = 퐺푟 (푛, 푘) and the isomorphism 휑 is 푈(푛)-invariant. ⊂ ⊕ C

Proof. From the proof of Lemma 4.3, there is a neighborhood U of 푋0, such that each element 푋 푈 are decomposed into 퐿푋 + 푅푋 . Moreover, it induces a decomposition ∈ of the trivial bundle 푈 C푛 into two subbundles: ×

푛 푈 C = 퐸푘 퐸푛−푘 (4.15) × ⊕

where 퐸푘(푋) and 퐸푛−푘(푋) are determined by the projection operator 푃푋 defined in equation (4.9):

⊥ 퐸푘(푋) = 퐼푚(푃푋 ), 퐸푛−푘(푋) = 퐼푚(푃푋 ) (4.16)

Let (휉1, ...휉푘) be the basis for 퐸푘(푋0). 퐸푘 over 푈 is an open neighborhood in

퐺푟C(푛, 푘). We can find a neighborhood V of 푋0 (possibly smaller than 푈) such that,

for every point in V, the k-dimensional space 퐸푘 projects onto 퐸푘(푋0). And an

orthonormal basis of 퐸푘(푋) is uniquely determined by requiring the projection of the

first j vectors to 퐸푘(푋0) spans (휉1, ...휉푗) for every j smaller than k. In this way, we

picked a basis for each fiber of 퐸푘 and 퐸푘 is trivialized to be a k-dimensional vector 푛 bundle on V. Since the action of X on C has been decomposed to 퐿푋 and 푅푋 , then

with the choice of basis, the action of 퐿푋 on 퐸푘(푋) gives a 푘 푘 self-adjoint matrix, × and by continuity, these matrices form a neighborhood 푉푘 in S(k). And the same

argument works for 푅푋 . Therefore, we have the following map 휑:

휑 : 푉 푃푘 푃푛−푘 → ⊕ (4.17) 푋 (퐸푘(푋), (퐿푋 퐸 (푋), 푅푋 퐸 (푋))) ↦→ | 푘 | 푛−푘

This map is an isomorphism between 푉 and 휑(푉 ). It’s injective, since the action of the two invariant subspace uniquely determines the action on C푛, therefore gives the unique operator X. The continuity of 휑 and 휑−1 comes from the continuity of the

projection operator defined in Lemma 4.1 therefore the continuity of 퐸푘, 퐿푋 and 푅푋

125 are continuous.

푛 Now take 푈(푛) 푉 , since 퐸푘 takes every possible k-subspace of C under the · action of 푈(푛), we know that the first entry of 휑(푈(푛) 푉 ) maps onto 퐺푟 (푛, 푘). · C Moreover, since the decomposition respects the action of 푈(푛), it is easily seen that, for 푔 푈(푛), 푋 퐺 푉 , ∈ ∈ ·

−1 −1 휑(푔 푋) = (푔 퐸푘(푋), (푔 퐿푋 푔 , 푔 푅푋 푔 )) = 푔 (휑(푋)) (4.18) · · ∘ ∘ ∘ ∘ · which means the isomorphism is group invariant.

To do the induction, we will need to define an index on the inclusion isotropy types, so the blow up procedure could be done in the partial order given by the index. Recall that two matrices have the same isotropy type if they have the same "clustering" of eigenvalues. Now we define the isotropy index of a matrix 푋 as follows.

Definition 4.5 (Isotropy index). Suppose the eigenvalues of a matrix 푋 are

휆1 = .. = 휆푖1 < 휆푖1+1 = .. = 휆푖2 < 휆푖2+1 = ... < 휆푖푘−1+1 = ... = 휆푛 then the isotropy index of 푋 is defined as the set

퐼(푋) = 푖0 = 0, 푖1, 푖2, ..., 푖푘−1, 푖푘 = 푛 . { }

There is a partial order of this index on 푆, given by the inclusion of isotropy types. That is, if for matrix 푋 and 푌 we have 퐼(푋) 퐼(푌 ) then we say that the order is ⊂ 푋 푌 . Note there is an inverse inclusion to isotropy group. The smallest isotropy ≤ index is 퐼(휆퐼) = 0, 푛 while the isotropy group is 푈(푛) which is the largest. And { } the largest index is 0, 1, 2, ..., 푛 1, 푛 which correspond to 푛 distinct eigenvalues, { − } where the isotropy group contains only identity. The last lemma we need before the induction is the comparability of conjugacy class inclusion and the decomposition to two submatrices, which shows the order of resolution in Lemma 4.1 is comparable with the decomposition.

126 Lemma 4.5 (Compatibility with conjugacy class). The partial order of conjugacy class inclusion is comparable with the decomposition in Lemma 4.3.

Proof. Suppose a neighborhood 푉 푆(푛) has a decomposition in Lemma 4.3. We ⊂ need to show that, if 푆(푛)퐼1 is the stratum of minimal isotropy type in V, then the decomposition of this stratum corresponds to the minimal isotropy type in 푈(푘) and 푈(푛 푘). − Since V satisfies the spectral gap condition, all the isotropy types in Vwould be subgroups of 푈(푘) 푈(푛 푘). Suppose the minimal stratum corresponds to ⊕ − the index 퐼 = 0, 푖1, ..., 푖푚 which must contain 푘 as one element because of the { } spectral gap condition. Then the isotropy type of two subgroups are 0, 푖1, ...푘 and { } 푖푗 푘 = 0, 푗푗+1, ..., 푛 푘 . They would still be the minimal in each subgroup, { − − } otherwise when the two smallest elements combined it’ll give a smaller index than 퐼 which is a contradiction.

Now we can finally prove Theorem 4.1 using the above lemmas.

Proof of Theorem 4.1. We prove the theorem by induction of the matrix size. The 2 2 case is shown in Lemma 4.2. Suppose the claim holds for all the cases up to × 푛 1. Now we claim that, by an iterative blow up, we can get 푆^(푛) for dimension=n, − with eigenvalues and eigenbundles lifted to satisfy the full eigenresolution properties.

As in the 2 2 example, we shall first consider the trace free slice 푆0(푛) since other × slices have the same behavior in terms of smoothness of eigenvalues and eigenbundles. Take the smallest index 퐼 = 0, 푛 with the largest possible isotropy group 푈(푛), and { } the stratum in 푆0(푛) with such an isotropy group is the zero matrix. After blowing up, we get [푆0; 0] as the first step. The next smallest index is 0, 푘, 푛 where 1 < 푘 < 푛. And the strata correspond- { } ing to different 푘 become disjoint in [푆0; 0] because if the eigenvalues of a matrix ′ ′ 푋 푆0 satisfy 푘1휆1 + 푘2휆2 = 0, 푘 휆1 + 푘 휆2 = 0, then 휆1 = 휆2 = 0, which has been ∈ 1 2 blown up in the previous step. Therefore we can blow up those strata separately. For any point 푋 푆0(푛) with 퐼(푋) = 0, 푘, 푛 , we can generate a neighborhood 푈(푛) 푉 ∈ { } · 127 as in Lemma 4.4, which is isomorphic to a neighborhood in the bundle 푃푘 푃푛−푘. Lo- ⊕ cally there is a product structure 푉 = 푉푘 푉푛−푘 푉퐺푟 푆(푘) 푆(푛 푘) 퐺푟 (푛, 푘). ∼ × × ⊂ × − × C For every base point 푝 푉퐺푟, since the fibre is isomorphic to a neighborhood in ∈ 푆(푘) 푆(푛 푘), the resolution can be done separately to 푉푘 and 푉푛−푘. And accord- × − ing to Lemma 4.5 the index order is preserved when decomposed into two subspaces, so the blow up construction indexed by isotropy type inclusion can be done on 푉푘 and 푉푛−푘. By induction, after the full resolution of the two subspaces, 퐸푘 and 퐸푛−푘 both split into line bundles, and eigenvalues also extend to the frontface smoothly. And since this local product structure is U(n)-invariant on 푈(푛) 푉 , the splitting of · eigenbundles are actually global. Therefore, after the resolution, we have iteratively blown up the stratum according to isotropy indices to get

푆^ = [푆; 0 ; 푆퐼1 ; 푆퐼2 ; ...푆퐼푛 ], (4.19) { } above which there are 푛 line bundles as eigenbundles and the corresponding eigenvalues are also smooth.

128 Bibliography

[1] Pierre Albin and Richard Melrose. Resolution of smooth group actions. volume 535 of Contemp. Math., pages 1–26. Amer. Math. Soc., Providence, RI, 2011.

[2] E Bergshoeff, Ergin Sezgin, and Paul K Townsend. Supermembranes and eleven- dimensional supergravity. Physics Letters B, 189(1):75–78, 1987.

[3] Matthias Blau, Jose Figueroa-O’Farrill, and George Papadopoulos. Penrose limits, supergravity and brane dynamics. Classical and Quantum Gravity, 19(18):4753, 2002.

[4] Armand Borel and Lizhen Ji. Compactifications of symmetric and locally symmetric spaces. Springer, 2006.

[5] L Castellani, Riccardo D’Auria, P Fre, K Pilch, and P Van Nieuwenhuizen. The bosonic mass formula for Freund–Rubin solutions of d= 11 supergravity on general coset manifolds. Classical and Quantum Gravity, 1(4):339, 1984.

[6] E. Cremmer and J. Scherk. Spontaneous compactification of extra space dimensions. Nuclear Phys. B, 118(1–2):61–75, 1977.

[7] Erwann Delay. Essential spectrum of the Lichnerowicz laplacian on two tensors on asymptotically hyperbolic manifolds. Journal of Geometry and Physics, 43(1):33–44, 2002.

[8] S. K. Donaldson. Lefschetz fibrations in symplectic geometry. In Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), number Extra Vol. II, pages 309–314, 1998.

[9] Joel Fine. Constant scalar curvature Kähler metrics on fibred complex surfaces. Journal of Differential Geometry, 68(3):397–432, 2004.

[10] Joel Fine. Constant scalar curvature Kähler metrics on fibred complex surfaces. PhD thesis, University of London, 2004.

[11] Peter GO Freund and Mark A Rubin. Dynamics of dimensional reduction. Physics Letters B, 97(2):233–235, 1980.

[12] Terry Fuller. Hyperelliptic Lefschetz fibrations and branched covering spaces. Pacific Journal of Mathematics, 196(2):369–393, 2000.

129 [13] Robert E Gompf. A new construction of symplectic manifolds. Annals of Math- ematics, pages 527–595, 1995.

[14] Robert E. Gompf and András Stipsicz. 4-manifolds and Kirby calculus. Num- ber 20. American Mathematical Soc., 1999.

[15] C. Robin Graham and John M. Lee. Einstein metrics with prescribed conformal infinity on the ball. Adv. Math., 87(2):186–225, 1991.

[16] C. Robin Graham and Maciej Zworski. Scattering matrix in conformal geometry. Invent. Math., 152(1):89–118, 2003.

[17] Daniel Grieser. Basics of the b-calculus. In Approaches to singular analysis, pages 30–84. Springer, 2001.

[18] Colin Guillarmou. Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds. Duke Math. J., 129(1):1–37, 2005.

[19] Colin Guillarmou and Frédéric Naud. Wave 0-trace and length spectrum on convex co-compact hyperbolic manifolds. Comm. Anal. Geom, 14(5):945–967, 2006.

[20] Joshua M Kantor. Eleven Dimensional Supergravity on Edge Manifolds. PhD thesis, University of Washington, 2009.

[21] Dietrich Kramer and Ernst Schmutzer. Exact solutions of Einstein’s field equations, volume 19. Cambridge University Press Cambridge, 1980.

[22] John M Lee. Fredholm operators and Einstein metrics on conformally compact manifolds. American Mathematical Soc., 2006.

[23] Rafe Mazzeo. The Hodge cohomology of a conformally compact metric. Journal of differential geometry, 28(2):309–339, 1988.

[24] Rafe Mazzeo. Elliptic theory of differential edge operators. I. Comm. Partial Differential Equations, 16(10):1615–1664, 1991.

[25] Rafe Mazzeo and Richard B Melrose. Analytic surgery and the eta invariant. Geometric & Functional Analysis GAFA, 5(1):14–75, 1995.

[26] Rafe Mazzeo and Richard B Melrose. Pseudodifferential operators on manifolds with fibred boundaries. arXiv preprint math/9812120, 1998.

[27] Rafe Mazzeo and Jan Swoboda. Asymptotics of the Weil–Petersson metric. arXiv preprint arXiv:1503.02365, 2015.

[28] Rafe Mazzeo and Boris Vertman. Elliptic theory of differential edge operators, II: boundary value problems. ArXiv e-prints, July 2013.

130 [29] Rafe R. Mazzeo and Richard B. Melrose. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal., 75(2):260–310, 1987.

[30] Richard Melrose. The Atiyah-Patodi-Singer index theorem, volume 4. AK Peters Wellesley, 1993.

[31] Richard Melrose and Xuwen Zhu. Resolution of the canonical fiber metrics for a lefschetz fibration. arXiv preprint arXiv:1501.04124, 2015.

[32] Richard B Melrose. Calculus of conormal distributions on manifolds with corners. International Mathematics Research Notices, 1992(3):51–61, 1992.

[33] Richard B. Melrose. Geometric scattering theory, volume 1. Cambridge Univer- sity Press, 1995.

[34] Richard B Melrose. Differential analysis on manifolds with corners, 1996.

[35] Richard B Melrose, Paolo Piazza, et al. Families of Dirac operators, boundaries and the b-calculus. J. Differential Geom, 46(1):99–180, 1997.

[36] Boris Moishezon. Complex surfaces and connected sums of complex projective planes. Springer, 1977.

[37] Werner Nahm. Supersymmetries and their representations. Nuclear Physics B, 135(1):149–166, 1978.

[38] Horatiu Nastase. Introduction to supergravity. arXiv preprint arXiv:1112.3502, 2011.

[39] Kunio Obitsu and Scott A. Wolpert. Grafting hyperbolic metrics and Eisenstein series. Math. Ann., 341(3):685–706, 2008.

[40] Joel W. Robbin and Dietmar A. Salamon. A construction of the Deligne- Mumford orbifold. J. Eur. Math. Soc. (JEMS), 8(4):611–699, 2006.

[41] Robert Seeley and Isadore M Singer. Extending 휕¯ to singular Riemann surfaces. Journal of Geometry and Physics, 5(1):121–136, 1988.

[42] Peter Van Nieuwenhuizen. Supergravity. Physics Reports, 68(4):189–398, 1981.

[43] Peter Van Nieuwenhuizen. The complete mass spectrum of d= 11 supergravity compactified on s4 and a general mass formula for arbitrary cosetsm4. Classical and Quantum Gravity, 2(1):1, 1985.

[44] Fang Wang. Dirichlet-to-Neumann map for Poincare–Einstein metrics in even dimensions. arXiv preprint arXiv:0905.2457, 2009.

[45] Edward Witten. On flux quantization in M theory and the effective action. J.Geom.Phys., 22:1–13, 1997.

131 [46] Scott A Wolpert. The hyperbolic metric and the geometry of the universal curve. Journal of Differential Geometry, 31(2):417–472, 1990.

[47] Xuwen Zhu. Eleven dimension supergravity equations on edge manifolds. preprint, 2014.

[48] Xuwen Zhu. Resolution of eigenvalues. preprint, 2014.

132