DOCSLIB.ORG

SOME PROGRESS in CONFORMAL GEOMETRY Sun-Yung A. Chang, Jie Qing and Paul Yang Dedicated to the Memory of Thomas Branson 1. Confo

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
SOME PROGRESS in CONFORMAL GEOMETRY Sun-Yung A. Chang, Jie Qing and Paul Yang Dedicated to the Memory of Thomas Branson 1. Confo SOME PROGRESS IN CONFORMAL GEOMETRY Sun-Yung A. Chang, Jie Qing and Paul Yang Dedicated to the memory of Thomas Branson Abstract. In this paper we describe our current research in the theory of partial differential equations in conformal geometry. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the σ2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes. 1. Conformal Gap and finiteness Theorem for a class of closed 4-manifolds 1.1 Introduction Suppose that (M 4; g) is a closed 4-manifold. It follows from the positive mass theorem that, for a 4-manifold with positive Yamabe constant, σ dv 16π2 2 ≤ ZM and equality holds if and only if (M 4; g) is conformally equivalent to the standard 4-sphere, where 1 1 σ [g] = R2 E 2; 2 24 − 2j j R is the scalar curvature of g and E is the traceless Ricci curvature of g. This is an interesting fact in conformal geometry because the above integral is a conformal invariant like the Yamabe constant. Typeset by AMS-TEX 1 2 CONFORMAL GEOMETRY One may ask, whether there is a constant 0 > 0 such that a closed 4-manifold M 4 has to be diffeomorphic to S4 if it admits a metric g with positive Yamabe constant and σ [g]dv (1 )16π2: 2 g ≥ − ZM for some < 0? Notice that here the Yamabe invariant for such [g] is automatically close to that for the round 4-sphere. There is an analogous gap theorem of Bray and Neves for Yamabe invariant in dimension 3 [BN]. One cannot expect the Yamabe invariant alone to isolate the sphere, and it is more plausible to consider the integral of σ2. We will answer the question affirmatively in the class of Bach flat 4-manifolds. Recall that Riemann curvature tensor decomposes into R = + (A g A g A g + A g ); ijkl Wijkl ik jl − il jk − jk il jl ik in dimension 4, where is the Weyl curvature, Wijkl 1 1 A = (R Rg ) ij 2 ij − 6 ij is Weyl-Schouten curvature tensor and Rij is the Ricci curvature tensor. Also recall that the Bach tensor is 1 B = + R W : ij Wkijl;lk 2 kl kijl We say that a metric g is Bach flat if Bij = 0. Bach flat metrics are critical metrics 2 for the functional M dv. Bach flatness is conformally invariant in dimension 4. It follows from Chern-Gauss-Bonnet,jWj R 8π2χ(M 4) = (σ + 2)dv; 2 jWj ZM that M σ2dv is conformally invariant. The gap theorem is as follows: R Theorem 1.1.1. Suppose that (M 4; [g]) is a Bach flat closed 4-manifold with pos- itive Yamabe constant and that ( 2dv)[g] Λ : jWj ≤ 0 ZM for some fixed positive number Λ0. Then there is a positive number 0 > 0 such that, if σ [g]dv (1 )16π2 2 g ≥ − ZM CHANG, QING AND YANG 3 4 holds for some constant < 0, then (M ; [g]) is conformally equivalent to the standard 4-sphere. Our approach is based on the recent work on the compactness of Bach flat metrics on 4-manifolds of Tian and Viaclovsky [TV1][TV2], and of Anderson [An]. Indeed our work relies on a more precise understanding of the bubbling process near points of curvature concentration. For that purpose we develop the bubble tree structure in a sequence of metrics that describes precisely the concentration of curvature. Our method to develop bubble tree structure is inspired by the work of Anderson and Cheeger [AC] on the bubble tree configurations of the degenerations of metrics of bounded Ricci curvature. Our construction is modeled after this work but differs in the way that our bubble tree is built from the bubbles at points with the smallest scale of concentration to bubbles with larger scale; while the bubble tree in [AC] is constructed from bubbles of large scale to bubbles with smaller scales. The inductive method of construction of our bubble tree is modeled on earlier work of [BC], [Q] and [St] on the study of concentrations of energies in harmonic maps and the scalar curvature equations. As a consequence of the bubble tree construction we are able to obtain the following finite diffeomorphism theorem: Theorem 1.1.2. Suppose that is a collection of Bach flat Riemannian manifolds (M 4; g) with positive Yamabe constant,A satisfying ( 2dv)[g] Λ ; jWj ≤ 0 ZM for some fixed positive number Λ0, and (σ dv)[g] σ ; 2 ≥ 0 ZM for some fixed positive number σ0. Then there are only finite many diffeomorphism types in . A It is known that in each conformal class of metrics belonging to the family , 2w A there is a metric g¯ = e g such that σ2(Ag¯) = 1, which we shall call the σ2 metric. The bubble tree structure in the degeneration of Yamabe metrics in is also helpful to understand the behavior of the σ -metrics in . For example: A 2 A Theorem 1.1.3. For the conformal classes [g0] the conformal metrics g = 2w 2 A e g0 satisfying the equation σ2(g) = 1 has a uniform bound for the diameter. 1.2 The Neck Theorem 4 CONFORMAL GEOMETRY The main tool we need to develop the bubble tree picture is the neck theorem which should be compared with the neck theorem in the work of Anderson and Cheeger [AC]. Due to the lack of point-wise bounds on Ricci curvature, our version of the neck theorem will have weaker conclusion. But it is sufficient to allow us to construct the bubble tree at each point of curvature concentration. Let (M 4; g) be a Riemannian manifold. For a point p M, denote by B (p) 2 r the geodesic ball with radius r centered at p, Sr(p) the geodesic sphere of radius r centered at p. Consider the geodesic annulus centered at p: A¯ (p) = q M : r dist(q; p) r : r1;r2 f 2 1 ≤ ≤ 2g ¯ In general, Ar1;r2 (p) may have more than one connected components. We will consider any one component A (p) Ar¯ ; r (p) r1;r2 ⊂ 1 2 that meets the geodesic sphere of radius r2: A (p) S (p) = r1;r2 r2 6 ; 3 \ Let H (Sr(p)) be the 3D-Hausdorff measure of the geodesic sphere Sr(p). Theoerm 1.2.1. Suppose (M 4; g) is a Bach flat and simply connected 4-manifold with a Yamabe metric of positive Yamabe constant. Let p M, α (0; 1), > 0, 2 2 v1 > 0, and a < dist(p; @M). Then there exist positive numbers δ0; c2; n depending on , α, Cs; v1; a such that the following holds. Let Ar1;r2 (p) be a connected compo- nent of the geodesic annulus in M such that r c a; r δ r ; 2 ≤ 2 1 ≤ 0 2 H3(S (p)) v r3; r [r ; 100r ]; r ≤ 1 8 2 1 1 and 2 Rm dv δ0: A (p) j j ≤ Z r1;r2 Then Ar1;r2 (p) is the only such component. In addition, for the only component A − 1 1 (p); 4 4 (δ )r1;(δ +)r2 0 − 0 3 which intersects with S 1 (p), there exist some Γ O(4), acting freely on S , 4 (δ0 +)r2 ⊂ with Γ n, and an quasi isometry , with j j ≤ A − 1 1 (p) 4 4 (δ +)r1;(δ )r2 0 0 − CHANG, QING AND YANG 5 3 Ψ(C − 1 1 (S =Γ)) 4 4 ⊂ δ0 r1,δ0 r2 A − 1 1 (p) 4 4 ⊂ (δ )r1;(δ +)r2 0 − 0 3 3 such that for all C 1 (S =Γ) C − 1 1 (S =Γ), in the cone coordinates, one 2 r;r 4 4 ⊂ δ0 r1,δ0 r2 has 2 (Ψ∗(r− g)) δ 1,α . j ij − ij jC ≤ The first step in the proof is to use the Sobolev inequality to show the uniqueness of the connected annulus Ar1;r2 (p). The second step is to establish the growth of volume of geodesic spheres H3(S (p) Cr3 r ≤ 1 for all r [r1; 2 r2]. Here we rely on the work of Tian and Viaclovsky [TV1] [TV2] where they2 analyzed the end structure of a Bach-flat, scalar flat manifolds with finite L2 total curvature. The last step is to use the Gromov and Cheeger compactness argument as in the work of Anderson and Cheeger [AC] to get the cone structure of the neck. 1.3 Bubble tree construction In this section we attempt to give a clear picture about what happen at curvature concentration points. We will detect and extract bubbles by locating the centers and scales of curvature concentration. We will assume here that (Mi; gi) are Bach flat 4-manifolds with positive scalar curvature Yamabe metrics, vanishing first homology, and finite L2 total curvature. Choose δ small enough according to the -estimates and the neck Theorem in the previous section. Suppose that X M contains a geodesic ball of a fixed radius i ⊂ i r0 and δ Rmi 2dvi ; T (@X ) j j ≤ 2 Z η0 i where T (@X ) = p M : dist(p; @X ) < η ; η0 i f 2 i i 0g for some fixed positive number 4η < r .
Load more

Recommended publications
  • Non-Linear Elliptic Equations in Conformal Geometry, by S.-Y
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 2, April 2007, Pages 323–330 S 0273-0979(07)01136-6 Article electronically published on January 5, 2007 Non-linear elliptic equations in conformal geometry, by S.-Y. Alice Chang, Z¨urich Lectures in Advanced Mathematics, European Mathematical Society, Z¨urich, 2004, viii+92 pp., US$28.00, ISBN 3-03719-006-X A conformal transformation is a diffeomorphism which preserves angles; the differential at each point is the composition of a rotation and a dilation. In its original sense, conformal geometry is the study of those geometric properties pre- served under transformations of this type. This subject is deeply intertwined with complex analysis for the simple reason that any holomorphic function f(z)ofone complex variable is conformal (away from points where f (z) = 0), and conversely, any conformal transformation from one neighbourhood of the plane to another is either holomorphic or antiholomorphic. This provides a rich supply of local conformal transformations in two dimensions. More globally, the (orientation pre- serving) conformal diffeomorphisms of the Riemann sphere S2 are its holomorphic automorphisms, and these in turn constitute the non-compact group of linear frac- tional transformations. By contrast, the group of conformal diffeomorphisms of any other compact Riemann surface is always compact (and finite when the genus is greater than 1). Implicit here is the notion that a Riemann surface is a smooth two-dimensional surface together with a conformal structure, i.e. a fixed way to measure angles on each tangent space. There is a nice finite dimensional structure on the set of all inequivalent conformal structures on a fixed compact surface; this is the starting point of Teichm¨uller theory.
    [Show full text]
  • WHAT IS Q-CURVATURE? 1. Introduction Throughout His Distinguished Research Career, Tom Branson Was Fascinated by Con- Formal
    WHAT IS Q-CURVATURE? S.-Y. ALICE CHANG, MICHAEL EASTWOOD, BENT ØRSTED, AND PAUL C. YANG In memory of Thomas P. Branson (1953–2006). Abstract. Branson’s Q-curvature is now recognized as a fundamental quantity in conformal geometry. We outline its construction and present its basic properties. 1. Introduction Throughout his distinguished research career, Tom Branson was fascinated by con- formal differential geometry and made several substantial contributions to this field. There is no doubt, however, that his favorite was the notion of Q-curvature. In this article we outline the construction and basic properties of Branson’s Q-curvature. As a Riemannian invariant, defined on even-dimensional manifolds, there is apparently nothing special about Q. On a surface Q is essentially the Gaussian curvature. In 4 dimensions there is a simple but unrevealing formula (4.1) for Q. In 6 dimensions an explicit formula is already quite difficult. What is truly remarkable, however, is how Q interacts with conformal, i.e. angle-preserving, transformations. We shall suppose that the reader is familiar with the basics of Riemannian differ- ential geometry but a few remarks on notation are in order. Sometimes, we shall write gab for a metric and ∇a for the corresponding connection. Let us write Rab and R for the Ricci and scalar curvatures, respectively. We shall use the metric to ‘raise and lower’ indices in the usual fashion and adopt the summation convention whereby one implicitly sums over repeated indices. Using these conventions, the Laplacian is ab a the differential operator ∆ ≡ g ∇a∇b = ∇ ∇a. A conformal structure on a smooth manifold is a metric defined only up to smoothly varying scale.
    [Show full text]
  • The Wave Equation on Asymptotically Anti-De Sitter Spaces
    THE WAVE EQUATION ON ASYMPTOTICALLY ANTI-DE SITTER SPACES ANDRAS´ VASY Abstract. In this paper we describe the behavior of solutions of the Klein- ◦ Gordon equation, (g + λ)u = f, on Lorentzian manifolds (X ,g) which are anti-de Sitter-like (AdS-like) at infinity. Such manifolds are Lorentzian ana- logues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces, in the sense that the metric is conformal to a smooth Lorentzian metricg ˆ on X, where X has a non-trivial boundary, in the sense that g = x−2gˆ, with x a boundary defining function. The boundary is confor- mally time-like for these spaces, unlike asymptotically de Sitter spaces studied in [38, 6], which are similar but with the boundary being conformally space- like. Here we show local well-posedness for the Klein-Gordon equation, and also global well-posedness under global assumptions on the (null)bicharacteristic flow, for λ below the Breitenlohner-Freedman bound, (n − 1)2/4. These have been known under additional assumptions, [8, 9, 18]. Further, we describe the propagation of singularities of solutions and obtain the asymptotic behavior (at ∂X) of regular solutions. We also define the scattering operator, which in this case is an analogue of the hyperbolic Dirichlet-to-Neumann map. Thus, it is shown that below the Breitenlohner-Freedman bound, the Klein-Gordon equation behaves much like it would for the conformally related metric,g ˆ, with Dirichlet boundary conditions, for which propagation of singularities was shown by Melrose, Sj¨ostrand and Taylor [25, 26, 31, 28], though the precise form of the asymptotics is different.
    [Show full text]
  • Math 865, Topics in Riemannian Geometry
    Math 865, Topics in Riemannian Geometry Jeff A. Viaclovsky Fall 2007 Contents 1 Introduction 3 2 Lecture 1: September 4, 2007 4 2.1 Metrics, vectors, and one-forms . 4 2.2 The musical isomorphisms . 4 2.3 Inner product on tensor bundles . 5 2.4 Connections on vector bundles . 6 2.5 Covariant derivatives of tensor fields . 7 2.6 Gradient and Hessian . 9 3 Lecture 2: September 6, 2007 9 3.1 Curvature in vector bundles . 9 3.2 Curvature in the tangent bundle . 10 3.3 Sectional curvature, Ricci tensor, and scalar curvature . 13 4 Lecture 3: September 11, 2007 14 4.1 Differential Bianchi Identity . 14 4.2 Algebraic study of the curvature tensor . 15 5 Lecture 4: September 13, 2007 19 5.1 Orthogonal decomposition of the curvature tensor . 19 5.2 The curvature operator . 20 5.3 Curvature in dimension three . 21 6 Lecture 5: September 18, 2007 22 6.1 Covariant derivatives redux . 22 6.2 Commuting covariant derivatives . 24 6.3 Rough Laplacian and gradient . 25 7 Lecture 6: September 20, 2007 26 7.1 Commuting Laplacian and Hessian . 26 7.2 An application to PDE . 28 1 8 Lecture 7: Tuesday, September 25. 29 8.1 Integration and adjoints . 29 9 Lecture 8: September 23, 2007 34 9.1 Bochner and Weitzenb¨ock formulas . 34 10 Lecture 9: October 2, 2007 38 10.1 Manifolds with positive curvature operator . 38 11 Lecture 10: October 4, 2007 41 11.1 Killing vector fields . 41 11.2 Isometries . 44 12 Lecture 11: October 9, 2007 45 12.1 Linearization of Ricci tensor .
    [Show full text]
  • CONFORMAL SYMMETRY BREAKING OPERATORS for ANTI-DE SITTER SPACES 1. Introduction Let X Be a Manifold Endowed with a Pseudo-Rieman
    CONFORMAL SYMMETRY BREAKING OPERATORS FOR ANTI-DE SITTER SPACES TOSHIYUKI KOBAYASHI, TOSHIHISA KUBO, AND MICHAEL PEVZNER Abstract. For a pseudo-Riemannian manifold X and a totally geodesic hypersur- face Y , we consider the problem of constructing and classifying all linear differential operators Ei(X) !Ej(Y ) between the spaces of differential forms that intertwine multiplier representations of the Lie algebra of conformal vector fields. Extending the recent results in the Riemannian setting by Kobayashi{Kubo{Pevzner [Lecture Notes in Math. 2170, (2016)], we construct such differential operators and give a classification of them in the pseudo-Riemannian setting where both X and Y are of constant sectional curvature, illustrated by the examples of anti-de Sitter spaces and hyperbolic spaces. Key words and phrases: conformal geometry, conformal group, symmetry breaking operator, branching law, holography, space form, pseudo-Riemannian geometry, hyperbolic manifold. 1. Introduction Let X be a manifold endowed with a pseudo-Riemannian metric g. A vector field Z on X is called conformal if there exists ρ(Z; ·) 2 C1(X)(conformal factor) such that LZ g = ρ(Z; ·)g; where LZ stands for the Lie derivative with respect to the vector field Z. We denote by conf(X) the Lie algebra of conformal vector fields on X. Let E i(X) be the space of (complex-valued) smooth i-forms on X. We define a family of multiplier representations of the Lie algebra conf(X) on E i(X) (0 ≤ i ≤ dim X) with parameter u 2 C by 1 (1.1) Π(i)(Z)α := L α + uρ(Z; ·)α for α 2 E i(X): u Z 2 i (i) i For simplicity, we write E (X)u for the representation Πu of conf(X) on E (X).
    [Show full text]
  • Computational Conformal Geometry: Theories and Applications
    ComputationalComputationalComputational ConformalConformalConformal Geometry:Geometry:Geometry: TheoriesTheoriesTheories andandand ApplicationsApplicationsApplications XianfengXianfeng GuGu,, Shing Shing-Tung-Tung Yau Yau UFL,UFL, HarvardHarvard UniversityUniversity AnAnAn importantimportantimportant goalgoalgoal ofofof classicalclassicalclassical geometrygeometrygeometry isisis tototo givegivegive aaa descriptiondescriptiondescription ofofof geometricgeometricgeometric figuresfiguresfigures thatthatthat wewewe seeseesee ininin nature.nature.nature. HighHighHigh powerpowerpower computercomputercomputer computationcomputationcomputation andandand threethreethree dimensiondimensiondimension cameracameracamera allowsallowsallows ususus tototo applyapplyapply classicalclassicalclassical andandand modernmodernmodern geometricgeometricgeometric technologytechnologytechnology tototo realrealreal featuresfeaturesfeatures ofofof objectsobjectsobjects thatthatthat wewewe seeseesee ininin daydayday tototo daydayday life.life.life. ForForFor aaa givengivengiven surfacesurfacesurface ininin threethreethree space,space,space, ititit isisis importantimportantimportant tototo realizerealizerealize thatthatthat theretherethere areareare severalseveralseveral geometricgeometricgeometric structuresstructuresstructures thatthatthat areareare inheritinheritinherit ininin theirtheirtheir description.description.description. TheTheThe mostmostmost importantimportantimportant oneoneone isisis thethethe conformalconformalconformal structure.structure.structure.
    [Show full text]
  • Space-Time Philosophy Reconstructed Via Massive Nordstr\" Om Scalar Gravities? Laws Vs. Geometry, Conventionality, and Underdetermination
    Space-time Philosophy Reconstructed via Massive Nordstr¨om Scalar Gravities? Laws vs. Geometry, Conventionality, and Underdetermination October 15, 2018 J. Brian Pitts Faculty of Philosophy, University of Cambridge John Templeton Foundation grant #38761 [email protected] arXiv:1509.03303v1 [physics.hist-ph] 9 Sep 2015 1 Abstract What if gravity satisfied the Klein-Gordon equation? Both particle physics from the 1920s-30s and the 1890s Neumann-Seeliger modification of Newtonian gravity with exponential decay suggest considering a “graviton mass term” for gravity, which is algebraic in the potential. Unlike Nordstr¨om’s “massless” theory, massive scalar grav- ity is strictly special relativistic in the sense of being invariant under the Poincar´e group but not the 15-parameter Bateman-Cunningham conformal group. It therefore exhibits the whole of Minkowski space-time structure, albeit only indirectly concerning volumes. Massive scalar gravity is plausible in terms of relativistic field theory, while violating most interesting versions of Einstein’s principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide to understanding massive scalar gravity(s): matter sees a conformally flat metric due to universal coupling, but gravity also sees the rest of the flat metric (barely or on long distances) in the mass term. What is the ‘true’ geometry, one might wonder, in line with Poincar´e’s modal conventionality argument? Infinitely many theories exhibit this bimetric ‘geometry,’ all with the total stress-energy’s trace as source; thus geometry does not explain the field equations. The irrelevance of the Ehlers-Pirani-Schild construction to a critique of conventionalism becomes evident when multi-geometry theories are contemplated.
    [Show full text]
  • Alexandr Medvedev: Conformal Geodesics and Geometry of the 3Rd Order Odes Systems
    Conformal geodesics and geometry of the 3rd order ODEs systems Alexandr Medvedev Masaryk University Trest, 2012 Supported by the project CZ.1.07/2.3.00/20.0003 of the Operational Programme Education for Competitiveness of the Ministry of Education, Youth and Sports of the Czech Republic. Alexandr Medvedev Geometry of the 3rd order ODEs systems The problem Characterize the geometry of the 3rd order ODEs system 000 00 0 yi (x) = fi (yj (x); yk (x); yl (x); x); 1 ≤ i; j; k; l ≤ m: (1) under point transformations. The number of equations will be at most 2. Historical remark I Single ODE of 3rd order was studied by S.Chern I System of 2nd order ODE was studied by M.Fels, D.Grossman a aa Order Numberaa 2 3 aa 1 E.Cartan S.Chern ≥ 2 M.Fels, D.Grossman ??? Alexandr Medvedev Geometry of the 3rd order ODEs systems ODE as a Surface in Jet Space Consider J3(M) be a 3 order jet space of the curves on M. Let's use the following coordinates on the jet space J3(M): 0 0 00 00 000 000 x; y1; :::; ym; p1 = y1; :::; pm = ym; q1 = y1 ; qm = ym; y1 ; :::; ym : We can represent equation as surface E in the J3(M). In coordinates we have the following equations that define E : 000 E = fyi = fi (qj ; pk ; yl ; x)g: (2) The lifts of the solutions of E are integral curves of 1-dimensional distribution E = T E\ C(J3(M)) @ @ @ i @ E = h + pi + qi + f i; @x @yi @pi @qi Alexandr Medvedev Geometry of the 3rd order ODEs systems Filtered Manifolds I We call a manifold M a filtered manifold if it is supplied with a filtration C −i of the tangent bundle: TM = C −k ⊃ C −k+1 ⊃ · · · ⊃ C −1 ⊃ C 0 = 0 (3) where [C −i ; C −1] ⊂ C −i−1: I With every filtered manifold we associate a filtration gr TM of the following form: m M gr TM = gr−i TM; (4) i=1 −i −i+1 where gr−i TM = C =C : I Space gr TM has a nilpotent Lie algebra structure.
    [Show full text]
  • Variational Principles for Conformal Geodesics
    VARIATIONAL PRINCIPLES FOR CONFORMAL GEODESICS MACIEJ DUNAJSKI AND WOJCIECH KRYNSKI´ Abstract. Conformal geodesics are solutions to a system of third order equations, which makes a Lagrangian formulation problematic. We show how enlarging the class of allowed variations leads to a variational formulation for this system with a third{order conformally invariant Lagrangian. We also discuss the conformally invariant system of fourth order ODEs arising from this Lagrangian, and show that some of its integral curves are spirals. 1. Introduction A geodesic on a (pseudo) Riemannian manifold is uniquely specified by a point and a tangent direction. Moreover, if two points are sufficiently close to each other, then there exists exactly one length minimising (or, in Lorentzian geometry, maximising) curve between these two points, which is geodesic. This formulation relies on the methods of the calculus of variations, and enables a construction of normal neighbourhoods, as well as the analysis of the Jacobi fields determining the existence of conjugate points. The variational formulation has been missing in conformal geometry, where conformal geodesics arise as solutions to a system of third order ODEs: A conformal geodesic is uniquely specified by a point, a tangent direction, and a perpendicular acceleration [19, 3, 18]. The odd order of the underlying system of equations is not amenable to the usual methods of calculus of variation, where the resulting Euler{Lagrange equations for non{ degenerate Lagrangians are of even order. One way around this difficulty, which we explore in this paper, is to consider a more gen- eral class of variations. As we shall see, this allows to terminate the procedure of integration by parts when the integrand depends on a derivative of a variation.
    [Show full text]
  • M-Eigenvalues of Riemann Curvature Tensor of Conformally Flat Manifolds
    M-eigenvalues of Riemann Curvature Tensor of Conformally Flat Manifolds Yun Miao∗ Liqun Qi† Yimin Wei‡ August 7, 2018 Abstract We generalized Xiang, Qi and Wei’s results on the M-eigenvalues of Riemann curvature tensor to higher dimensional conformal flat manifolds. The expression of M-eigenvalues and M-eigenvectors are found in our paper. As a special case, M-eigenvalues of conformal flat Einstein manifold have also been discussed, and the conformal the invariance of M- eigentriple has been found. We also discussed the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold. We proved that the M-eigenvalue can determine the Riemann curvature tensor uniquely and generalize the real M-eigenvalue to complex cases. In the last part of our paper, we give an example to compute the M-eigentriple of de Sitter spacetime which is well-known in general relativity. Keywords. M-eigenvalue, Riemann Curvature tensor, Ricci tensor, Conformal invariant, Canonical form. AMS subject classifications. 15A48, 15A69, 65F10, 65H10, 65N22. 1 Introduction The eigenvalue problem is a very important topic in tensor computation theories. Xiang, Qi and Wei [1, 2] considered the M-eigenvalue problem of elasticity tensor and discussed the relation between strong ellipticity condition of a tensor and it’s M-eigenvalues, and then extended the M-eigenvalue problem to Riemann curvature tensor. Qi, Dai and Han [12] introduced the M-eigenvalues for the elasticity tensor. The M-eigenvalues θ of the fourth-order tensor Eijkl are defined as follows. arXiv:1808.01882v1 [math.DG] 28 Jul 2018 j k l Eijkly x y = θxi, ∗E-mail: [email protected].
    [Show full text]
  • Conformal and Cr Mappings on Carnot Groups
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 7, Pages 67–81 (June 17, 2020) https://doi.org/10.1090/bproc/48 CONFORMAL AND CR MAPPINGS ON CARNOT GROUPS MICHAEL G. COWLING, JI LI, ALESSANDRO OTTAZZI, AND QINGYAN WU (Communicated by Jeremy Tyson) Abstract. We consider a class of stratified groups with a CR structure and a compatible control distance. For these Lie groups we show that the space of conformal maps coincide with the space of CR and anti-CR diffeomorphisms. Furthermore, we prove that on products of such groups, all CR and anti-CR maps are product maps, up to a permutation isomorphism, and affine in each component. As examples, we consider free groups on two generators, and show that these admit very simple polynomial embeddings in CN that induce their CR structure. 1. Introduction In this article, we consider the interplay between metric and complex geometry on some model manifolds. This is the first outcome of a larger project which aims to develop a unified theory of conformal and CR structures on the one hand, and to define explicit polynomial embeddings of certain CR manifolds into Cn on the other. This kind of embedding is what permits the detailed analysis of the ∂b operator carried out by Stein and his collaborators since the 1970s (see [6]). The analogy between CR and conformal geometry is well documented in the case of CR manifolds of hypersurface-type, see, e.g., [7,10–14,17]. The easiest and perhaps the most studied example in this setting is that of the Heisenberg group, taken with its sub-Riemannian structure.
    [Show full text]
  • Surface Classification Using Conformal Structures
    Surface Classification Using Conformal Structures Xianfeng Gu Shing-Tung Yau Department of CISE Department of Mathematics University of Florida Harvard University Gainesville, FL 32611 Cambridge, MA 02138 [email protected]fl.edu [email protected] Abstract novel viewpoint: treating them as Riemann surfaces with conformal structures, or namely one dimensional complex 3D surface classification is a fundamental problem in manifolds. A Riemann surface is a surface covered by holo- computer vision and computational geometry. Surfaces can morphic coordinate charts. The conformal structure is the be classified by different transformation groups. Traditional complete invariants under conformal transformations and classification methods mainly use topological transforma- can be represented as matrices. tion groups and Euclidean transformation groups. This pa- Compared to other surface classification methods, con- per introduces a novel method to classify surfaces by con- formal classification has some advantages. Theoretically, formal transformation groups. Conformal equivalent class conformal geometry has a sound foundation. Conformal is refiner than topological equivalent class and coarser than equivalent classes are much refiner than the topological isometric equivalent class, making it suitable for practical equivalent classes and much coarser than the isometric classification purposes. For general surfaces, the gradient classes. Each topological equivalent class has infinite con- fields of conformal maps form a vector space, which has formal equivalent classes, each conformal equivalent class a natural structure invariant under conformal transforma- has infinite isometric classes. The conformal structures tions. We present an algorithm to compute this conformal are intrinsic to the geometry, independent of triangulation, structure, which can be represented as matrices, and use it insensitive to resolution and local features, and robust to to classify surfaces.
    [Show full text]