Spectral Geometry and Asymptotically Conic Convergence, Ph.D

Home , Heat kernel

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g := dx2 + (ǫ2 + x2)g ǫ [0, 1). ǫ H ∈ As ǫ 0, g x2g +dx2, which has an isolated conic singularity at x = 0. Geometrically, → ǫ → H M is pinched along the hypersurface H as ǫ 0, and the resulting metric has a conic → singularity as x 0. The study of this metric collapse is the content of the 1990 thesis of → McDonald [25]. In 1995, Mazzeo and Melrose developed pseudodiﬀerential techniques to describe the behavior of the spectral geometry under another speciﬁc type of metric collapse known as analytic surgery similar to the conic collapse considered by McDonald. As ǫ 0 the → metrics dx 2 dx 2 g = | | + h | | + h = g ; ǫ x2 + ǫ2 → x2 0 g0 is an exact b-metric on the compact manifold with boundary M¯ obtained by cutting M along H and compactifying as a manifold with boundary, hence the name, analytic surgery. Under certain assumptions on the associated Dirac operators,1 Mazzeo and Melrose proved

lim η(∂ǫ)= ηb(∂M¯ ), ǫ→0

where ηb(∂M¯ ) is the b-version of the eta invariant introduced by Melrose. These results were proven by analyzing the resolvent family of the Dirac operators ∂ǫ uniformly near zero. This led to a precise description of the behaviour of the small eigenvalues. Our goal is to obtain precise analytic results like those of [15]. The convergence considered here, asymptotically conic (ac) convergence, is more restrictive than that of Cheeger, Colding and Ding, but more general than the conic degeneration

1In a later collaboration with Hassel [15] these hypotheses were removed.

2 of McDonald. We note, however, that ac convergence does not require Ricci curvature bounds. The conic collapse of McDonald, the analogous smooth collapse of a higher codimension submanifold and the collapse of an open neighborhood of the manifold with some restrictions on the local geometry all fit this new definition. Before stating our spectral convergence results for geometric Laplacians, we recall their definition.

Deﬁnition 1. Let (E, ) be a Hermitian vector bundle over a Riemannian manifold (M, g) ∇ with metric-compatible connection . A geometric Laplacian is an operator ∆ acting on ∇ sections of E which has the form

∆= ∗ + R, ∇ ∇ where R is a non-negative self-adjoint endomorphism of E. By the Weitzenb¨ock Theorem, the Laplacian on k-forms is a geometric Laplacian, as is the Hodge Laplacian and the conformal Laplacian; any geometric Laplace-type operator is a geometric Laplacian.

Our results are the following.

Spectral Convergence: Theorem 1. Let (M0, g0) be a compact Riemannian n-manifold with isolated conic singularity, and let (Z, g ) be an asymptotically conic space, with n 3. z ≥ Assume (M, g ) converges asymptotically conically to (M , g ). Let (E , ) and (E , ) ǫ 0 0 0 ∇0 z ∇z be Hermitian vector bundles over (M0, g0) and (Z, gz), respectively, so that each of these bundles in a neighborhood of the boundary is the pullback from a bundle over the cross section (Y, h). Let ∆0, ∆z be the corresponding Friedrich’s extensions of geometric Lapla- cians, and let ∆ be the induced geometric Laplacian on (M, g ). Assume ∆ has no 2 ǫ ǫ z L nullspace. Then the accumulation points of the spectrum of ∆ as ǫ 0 are precisely the ǫ → points of the spectrum of ∆0, counting multiplicity. The setting for our next result is the acc heat space, a manifold with corners constructed in section 7.

Heat Kernel Convergence: Theorem 2. Let (M0, g0) be a compact Riemannian n- manifold with isolated conic singularity, and let (Z, gz) be an asymptotically conic space, with n 3. Assume (M, g ) converges asymptotically conically to (M , g ). Let (E , ) ≥ ǫ 0 0 0 ∇0 and (E , ) be Hermitian vector bundles over (M , g ) and (Z, g ), respectively, so that z ∇z 0 0 z each of these bundles in a neighborhood of the boundary is the pullback from a bundle over the cross section (Y, h). Let ∆0, ∆z be the corresponding Friedrich’s extensions of geometric Laplacians, and let ∆ǫ be the induced geometric Laplacian on (M, gǫ). Then the associated heat kernels H have a full polyhomogeneous expansion as ǫ 0 on the asymptotically ǫ → conic convergence (acc) heat space with the following leading terms at the ǫ = 0 faces:

H (z, z′,t) H (z, z′,t′) at F ǫ → 0 0101 H (z, z′,t) (ρ )2H (z, z′,τ) at F , ǫ → 1010 b 1010

3 ′ 2 2 Hǫ(z, z ,t)= O (ρ1001) (ρ0110) at F1001, F0110.

Above, ρwxyz is the deﬁning function for boundary face Fwxyz,H0 is the heat kernel for ′ ∆0,Hb is the b-heat kernel for ∆z and t and τ are rescaled time variables with

′ t t t = 2 , τ = 2 . (ρ1010) (ρ1001ρ0110) This convergence is uniform in ǫ for all time and moreover, the error term is bounded by C ǫtN , for any N N . N ∈ 0 ′ + Remark: The convergence of the heat kernels Hǫ(z, z ,t) on M M R to the ′ ′ × × heat kernel H0(z, z ,t ) on M0 follows immediately from Theorem 2. The error term is ǫ2(H (z, z′,τ)+ O(1)) + O(ǫt∞) as ǫ,t 0 and ǫ2(H (z, z′,τ)+ O(1)) + O(ǫ) as ǫ 0 b → b → for all t. However, Theorem 2 is stronger than this statement: the theorem speciﬁes the leading order behavior of Hǫ precisely at the ǫ = 0 faces of the acc heat space and gives the existence of a full polyhomogeneous expansion as ǫ 0. → These theorems are proven in sections 5 and 7, respectively. In section 2 we deﬁne the resolution blowup and ac convergence. Sections 3 and 4 contain a brief review of geometric and analytic results and terminology on manifolds with corners. In section 6 we construct the heat spaces and heat operator calculi that will be used to prove the main theorem in section 7. This work is based on the author’s doctoral dissertation completed at Stanford Univer- sity in June, 2006 under the supervision of Rafe Mazzeo. The author wishes to thank Rafe Mazzeo for excellent advising, Andras Vasy for many helpful conversations and suggestions, and Richard Melrose for insightful comments.

2 Asymptotically Conic Convergence

The deﬁnition of asymptotically conic convergence involves three geometries: a family of smooth metrics on a compact manifold, a conic metric on a compact (incomplete) manifold, and an asymptotically conic or ac scattering metric on a compact (complete) manifold with boundary. First, we deﬁne ac scattering metric.

Deﬁnition 2. An ac scattering metric (Z,¯ gz) is a smooth metric on a compact n-manifold with boundary, ∂Z¯ = (Y, h) a smooth compact (n 1) manifold. Z¯ has a product decom- − position (0, r ) Y in a neighborhood of the boundary deﬁned by r = 0, so that on this 1 r × neighborhood, dr2 h(r) g = + , h(r) h as r 0, z r4 r2 → → in other words, h extends to a ∞ tensor on [0, r ). Uniquely associated to the ac scattering C 1 metric (Z,¯ gz) is the complete non-compact manifold Z with asymptotically conic end; Z is

4 known as an ac space.2 Letting ρ = 1 , there is a compact subset K Z so that r z ⊂ Z K = (1/r , ) Y, g = dρ2 + ρ2h(1/ρ). − z ∼ 1 ∞ ρ × z|Z−Kz A familiar example of an ac scattering metric is the standard metric on the radial compactification of Rn with boundary Sn−1 at infinity. Next, we define the conic metric on a compact manifold.

Definition 3. Let Mc be a compact metric space with Riemannian metric g. Then, (Mc, g) has an isolated conic singularity at the point p and g is called a conic metric if the following hold. 1. (M p , g) is a smooth, open manifold. c − { } 2. There is a neighborhood N of p and a function x : N p (0,x ] for some x > 0, −{ }→ 1 1 such that N p is diffeomorphic to (0,x ] Y with metric g = dx2 + x2h(x) − { } 1 x × where (Y, h) is a compact, smooth n 1 manifold and h(x) is a smooth family of − { } metrics on Y converging to h as x 0, in other words, h extends to a ∞ tensor on → C [0,x ) Y. 1 x × Associated to a manifold with isolated conic singularity is the manifold with boundary obtained by blowing up the cone point, adding a copy of (Y, h) at this point. Then, x is a 0 boundary defining function for the boundary, (Y, h). We use M0 = Mc p to denote the − { } 0 smooth incomplete conic manifold, Mc to denote the metric space closure of M0 , and M0 to denote the associated manifold with boundary. We work almost exclusively with M0. The resolution blowup is a nonstandard blowup that resolves an isolated conic singularity in a compact manifold using an asymptotically conic space. In the definition of resolution blowup we use the notation M N for a smooth manifold constructed from the ∪φ smooth manifolds M and N with a diffeomorphism φ from V N to U M that gives ⊂ ⊂ the equivalence relation, V p φ(p) U. M N is the disjoint union of M and N ∋ ∼ ∈ ∪φ modulo the equivalence relation of φ. The smooth structure on M N and the topology ∪φ is induced by that of M and N.

Definition 4. Let (M0, g0) be a compact n manifold with isolated conic singularity and let (Z, gz) be an asymptotically conic space of dimension n, so that ∂M0 = ∂Z¯ = (Y, h). Then, M 0 = K V , 0 0 ∪ 0 where V = (0,x ) Y, and K is compact. With this diffeomorphism 0 ∼ 1 x × 0 g = dx2 + x2h˜(x) on (0,x ) Y. 0 1 × 2Note that asymptotically conic spaces are sometimes called “asymptotically locally Euclidean,” or ALE. However, that term is often used for the more restrictive class of spaces that are asymptotic at infinity to a cone over a quotient of the sphere by a finite group, so to avoid confusion, we use the term asymptotically conic.

5 We may assume the boundary of K in M 0 is of the form ∂K = x = x = Y, and we 0 0 0 { 1} ∼ may extend x smoothly to K so that 2x >x x on K . Similarly, 0 1 ≥ 1 0 Z = K V , z ∪ z where V = (ρ , ) Y, and K is compact. With this diﬀeomorphism z ∼ 1 ∞ ρ × z g = dρ2 + ρ2h(ρ, y) on (ρ , ) Y. z 1 ∞ × We may assume that ∂K is of the form ∂K = ρ = ρ = Y, and we similarly extend ρ z z { 1} ∼ smoothly to K so that ρ ρ > ρ /2 on K . z 1 ≥ 1 z Let δ = min x , 1/ρ . Then for 0 < ǫ < δ, and R > ρ , let { 1 1} 1 M = (x,y) M : x>ǫ ,Z = (ρ, y) Z : ρ < R . 0,ǫ { ∈ 0 } R { ∈ }

The resolution blowup of (M0, g0) by (Z, gz) is,

M := M Z , ǫ 0,ǫ ∪φǫ 1/ǫ where the joining map φ is defined for each ǫ by x φ : M M Z Z , φ (x,y)= ,y . ǫ 0,ǫ − 0,δ → 1/ǫ − 1/δ ǫ δǫ ′ For δ>ǫ>ǫ > 0, the manifolds Mǫ and Mǫ′ are diffeomorphic, and so the resolution blowup of M0 by Z which we call M is unique up to diffeomorphism. Remark: The resolution blowup resolves the singularity in a conic manifold using an ac scattering space. Instead of resolving the singularity in M0 using Z, we may equivalently 3 define the resolution blowup to resolve the boundary of Z¯ using M0. The ac single space, analogous to the analytic surgery single space in [24], is the setting for the definition of ac convergence.

Definition 5. Let (M0, g0) be a conic metric and let (Z,¯ gz) be a scattering metric; assume both are dimension n with the same cross section (Y, h) at the boundary, and assume δ = 1 (definition 4). Then, M = ((0, 1) Y ) K and Z¯ = ((0, 1) Y ) K with 0 ∼ x × ∪ 0 ∼ r × ∪ z ∂K = Y = ∂K . The asymptotically conic convergence (acc) single space is defined as 0 ∼ ∼ z S follows,

:= [0, 1) [0, 1) Y (K x = 1, r = 1 ) (K r = 1,x = 1 ). S x × r × ∪ 0 × { 6 } ∪ z × { 6 } 3 Let r = 1/ρ be the defining function for ∂Z.¯ The resolution blowup of Z¯ by M0 is ǫδ M0 ∪ ǫ Z,¯ ψǫ(x,y)= ,y , ψ „ x « where the patching map ψǫ is defined on M0 − K0 with image (0, δ)r × Y a neighborhood of ∂Z.¯ The resulting smooth compact resolution space is diffeomorphic to M.

6 The smooth structure of is induced by that of M and Z.¯ Namely, smooth functions S 0 on are functions which are smooth jointly in x and r on (0, 1) (0, 1) Y, smoothly S x × r × extend to a smooth function on K0 at x = 1, on Kz at r = 1, on M0 at r = 0, and on Z¯ at x = 0. To give a precise description of the metrics considered here, we deﬁne below the acc tensor, a smooth, polyhomogeneous, symmetric 2-cotensor on the acc single space.

Definition 6. Let be the acc single space associated to M and Z¯ as in definition 5. S 0 Let ǫ(p) = x(p)r(p) : [0, 1), where x, r are extended to K and K respectively to be S → 0 z identically 1. We define the acc tensor as follows: G 2 2 2 dr2 h(r) dx + x h(x) + (xr) 4 + 2 x, r [0, 1) r r ∈ 2 2 2 =  dx + x h(x) + (xr) (gz Kz ) r = 1, x [0, 1) G 2 | ∈  2 dr h(r)  g + (xr) 4 + 2 x = 1, r [0, 1) 0|K0 r r ∈  For 0 <ǫ< 1,let M = xr = ǫ ; note that this M is diffeomorphic to the ǫ { } ⊂ S ǫ resolution blowup M of M by Z.4 The family of metrics g = on M is said to 0 { ǫ G|Mǫ } converge asymptotically conically to (M0, g0). Remarks

1. The acc single space has two boundary hypersurfaces at ǫ = 0; these are diffeomorphic to M0 at r = 0and Z¯ at x = 0, and they meet in a codimension 2 corner diffeomorphic to Y. There are also boundary hypersurfaces at ǫ = 1 which we ignore since we are interested in ǫ 0. → 2. Since g is a smooth metric on M = ((0, 1) Y ) K and g is a smooth metric 0 0 ∼ x × ∪ 0 z on Z¯ = ((0, 1) Y ) K , is a smooth symmetric 2-cotensor on which extends ∼ r × ∪ z G S smoothly across neighborhoods where it is piecewise defined.

3. At r = 0, restricts to = g ; as x 0, vanishes to order 2. G G|{r=0} 0 → G 4. On M where 0 < r(p),x(p) < 1, ǫ ⊂ S ǫ ǫ2 r = = dr2 = dx2, x ⇒ x4 so dr2 h(r) g = dx2 + x2h(x)= ǫ2 + . ǫ r4 r2 4In the definition of the acc single space we have assumed δ from the definition of resolution blowup is ǫ ǫ 1. This is to simplify our calculations. On {xr = ǫ} we then have r = x , equivalently x = r , so letting δ = 1 in the definition of resolution blowup the identification of {xr = ǫ}⊂S and the resolution blowup Mǫ follows immediately.

7 5. On Mǫ for x ǫ, r 1 and this subset is diﬀeomorphic to Kz with metric gǫ = 2 ≡ ≡ ǫ gz Kz . On Mǫ for x 1, r ǫ and this subset is diﬀeomorphic to K0 with metric | 2 ≡ ≡ gǫ = g0 + O(ǫ) . The following lemma is useful both for visualizing ac convergence and for proving spectral convergence.

Lemma 1. Let (M0, g0) and (Z, gz) be as in definitions 4, 5, 6 and let (M, gǫ) converge asymptotically conically to (M , g ). Then, there exists a family of diffeomorphisms φ 0 0 { ǫ} from a fixed open proper subset U M to increasing neighborhoods Z Z such that ⊂ 1/ǫ ⊂ g = ǫ2(φ )∗g . Moreover, on M U, g g smoothly as ǫ 0 and any ǫ|U ∼ ǫ |Z1/ǫ |U − ǫ → 0 → K M0 is diffeomorphic to some fixed K′ M so that g g smoothly and uniformly ⊂⊂ 0 ⊂ ǫ → 0 on K′.

Proof The existence of φ and U M follows immediately from the definition of resolution ǫ ⊂ blowup and the diffeomorphism between the resolution blowup M and xr = ǫ . By { } ⊂ S the above remarks, on the neighborhood U M where this diffeomorphism is defined, ⊂ g = (ǫ2)(φ )∗g . ǫ|U ǫ z|Z1/ǫ Since g = g + (ǫ2), the smooth convergence of g to g on M U follows immediately. ǫ 0 O ǫ 0 − Any compact subset K M is contained in M for some ǫ> 0 and so is diffeomorphic ⊂⊂ 0 0,ǫ to K M and to K′ (M U). Conversely, any K (M U) is diffeomorphic to ǫ ⊂⊂ ǫ ⊂⊂ − ⊂⊂ − K M and to K′ M M . ǫ ⊂ ǫ ⊂⊂ 0,ǫ ⊂ 0

♥ 3 Geometric Preliminaries

This section is a brief review of the theory and terminology of manifolds with corners, b maps, and blowups. A complete reference is [26], see also [23].

3.1 Manifolds with Corners Let X be a manifold with corners. This means that near any of its points, X is modeled on a product [0, )k Rn−k where k depends on the point and is the maximal codimension ∞ × of the boundary face containing that point. We also assume that all boundary faces of X are embedded so they too are manifolds with corners. The space (X) of all smooth V vector ﬁelds on X is a Lie algebra under the standard bracket operation. It contains the Lie subalgebra

(X) := V (X) ; V is tangent to each boundary face of X. (1) Vb { ∈V }

8 Then (X) is itself the space of all smooth sections of a vector bundle, Vb (X)= ∞(X; bT X), Vb C where bT X is the bundle deﬁned so that the above holds and is called the b-tangent bundle.

3.1.1 Blowing Up An embedded codimension k submanifold Y of a manifold with corners X is called a p- submanifold (p for product) if near each point of Y there are local product coordinates so that Y is defined by the vanishing of some subset of them. In other words, X and Y must have consistent local product decompositions. Then one can define a new manifold with corners [X; Y ] to be the normal blowup of X around Y. This is obtained by replacing Y by its inward-pointing spherical normal bundle. The union of this normal bundle and X Y has a unique minimal differential structure as a manifold with corners so that the − lifts of smooth functions on X and polar coordinates around Y are smooth. One can also consider iterated blowups, written [[X; Y1]; Y2]] , where Y1 is a p-submanifold of X and Y2 is a p-submanifold of Y1. We may consider any finite sequence of such blowups. If we have such a sequence of embedded p-submanifolds,

X Y Y Y . . . Y ⊃ 1 ⊃ 2 ⊃ 3 ⊃ n then the iterated blowup [[X; Y1]; Y2]; . . . ; Yn] can be performed in any order with the same result [24]. Blowups may also be deﬁned using equivalence classes of curves [26]. Let r be a deﬁning function for the p submanifold and consider the family of curves γ(t) = (r(t),y(t)) such that

γ(t) Y t = 0, ∈ ⇐⇒ r(t)= O(t). Let E be the set of equivalence classes of all such curves with

γ γ′ (y y′)(t)= O(t), (r r′)(t)= O(t2). ∼ ⇐⇒ − − There is a natural R+ action on E given by

R+ a : γ(t) γ(at). ∋ → Then E modulo this equivalence relation is naturally diﬀeomorphic to N +(Y ), the inward pointing spherical normal bundle of Y, so we can deﬁne [X; Y ] by

[X; Y ] = (X Y ) E/(R+ 0 ). − ∪ − { }

9 We can also deﬁne parabolic blowups in certain contexts [9]. Let Y be a p-submanifold of codimension k so that there exist local coordinates (r, y) = (r1, . . . , rk,y1,...,yn−k) in a neighborhood of Y with ri vanishing precisely at Y, and so that dr1 induces a sub-budle of the tangent bundle T X. Instead of the above equivalence classes of curves we consider γ(t) such that

γ(t) = (r (t), . . . , r (t),y (t),...y (t)) Y t = 0, 1 k 1 n−k ∈ ⇐⇒ r (t)= O(t), i = 1, r (t)= O(t2). i 6 1 Two such curves are equivalent if

γ γ′ (y y′ )(t)= O(t), (r r′)(t)= O(t2) i = 1, (r (t) r′ (t)) = O(t3). ∼ ⇐⇒ j − j i − i 6 1 − 1 + Since dr1 is a sub-bundle of T X there is a natural R action on the set of equivalence classes E2 of all such curves,

R+ a : γ(t) (r (a2t), r (at),...,y (at)). ∋ → 1 i j The set of equivalence classes of all such curves modulo this R+ action is naturally diﬀeo- morphic to the inward pointing r1-parabolic normal bundle of Y,

E /(R+ 0 ) = PN +(Y ). 2 − { } ∼ r1 We deﬁne the r -parabolic blowup of X around Y as the union of X Y and this inward 1 − pointing r1-parabolic bundle,

[X; Y, dr ] := (X Y ) PN +(Y ). 1 − ∪ r1 The union of this r -parabolic bundle and X Y again has a unique minimal diﬀerential 1 − structure as a manifold with corners so that the lifts of smooth functions on X and r1- parabolic coordinates around Y are smooth. By r1-parabolic coordinates around Y, we mean the coordinates,

1 ρ = (r2 + r4 + . . . + r4) 4 , θ = (θ ,...θ ) Sn−1, 1 2 k 1 n ∈

with local coordinates (r1, . . . rk,y1,...,yn−k) in a neighborhood of Y satisfying

r = ρθ , i = 1, r = ρ2θ , y = ρθ . i i 6 1 1 j j For any parabolic or spherical blowup there is a natural blow-down map β : [X; Y ] X ∗ → and corresponding blow-up map β∗ : X [X; Y ], so that the image of Y under β∗ → is a boundary hypersurface of [X; Y ] diﬀeomorphic to the inward pointing spherical (or parabolic) normal bundle of Y. As such,

[X; Y ] = (X Y ) β∗(Y ). − ∪

10 3.1.2 b-Maps and b-Fibrations Definition 7. Let M be a manifold with boundary hypersurfaces, N k , and defining 1 { j}j=1 functions r . Let M be a manifold with boundary hypersurfaces, L l , and defining j 2 { i}i=1 functions ρi. Then f : M1 M2 is called a b-map if for every i there exist nonnegative → ∗ k e(i,j) integers e(i, j) and a smooth nonvanishing function h such that f (ρi)= h j=1 rj .

The image under a b-map of the interior of each boundary hypersurfaceQ of M1 is either contained in or disjoint from each boundary hypersurface of M2 and the order of vanishing of the differential of f is constant along each boundary hypersurface of M1. The matrix (e(i, j)) is called the lifting matrix for f. In order for the map, f, to preserve polyhomogeneity, stronger conditions are required. Associated to a manifold with corners are the b-tangent and cotangent bundles, bT M (1) and bT ∗M.5 The map f may be extended to induce the map bf :b T M b T M . ∗ 1 → 2 Definition 8. The b-map, f : M M , is called a b-fibration if the associated maps bf 1 → 2 ∗ at each p ∂M are surjective at each p ∂M and the lifting matrix (e(i, j)) has the ∈ 1 ∈ 1 property that for each j there is at most one i such that (e(i, j)) = 0. In other words, f 6 does not map any boundary hypersurface of M1 to a corner of M2.

3.1.3 b-manifolds and the b-blowup A b-manifold is a manifold with corners that is closely related to conic manifolds and ac scattering metrics. Definition 9. Let (X, g) be a smooth Riemannian manifold with boundary (Y, h) and boundary defining function x such that in a collared neighborhood N of the boundary X has a product decomposition, N = [0,x ) Y and in this neighborhood ∼ 1 x × dx2 g = + h(x), x2 where h(x) is a smoothly varying family of metrics on Y that converges smoothly to h as x 0. Then (X, g) is said to be a b-manifold. → Equivalently, a b-manifold is a complete manifold with asymptotically cylindrical ends. The Schwartz kernels of operators on a b-manifold with reasonable regularity lift to a blown up manifold called the b-double space. This space is obtained from X2 by performing a radial blowup called the b-blowup along the codimension 2 corner at the boundary in each 2 copy of X and it is written Xb , X2 = [X X; ∂X ∂X] = [X X; Y Y ]. (2) b × × × × For any manifold M with boundary having a product structure in a neighborhood of the boundary, we may define the b-blowup in the analogous way, M 2 := [M M; ∂M ∂M]. b × × 5These are also called the totally characteristic tangent and cotangent bundles.

11 3.2 Asymptotically conic convergence double space The acc double space is an instructive model for the more sophisticated acc heat space construction in section 7. Let 2 := [ 2; C C ], Sb S 1 × 1 where we recall C = Y is the codimension two corner in each copy of . The acc dou- 1 ∼ S ble space is the submanifold of 2 defined by the vanishing set of f(p) = x(p)r(p) D Sb − x′(p)r′(p)= ǫ(p) ǫ′(p) : − = p 2 : f(p)= x(p)r(p) x′(p)r′(p) = 0 = p 2 : ǫ(p)= ǫ′(p) . D { ∈ Sb − } { ∈ Sb } The acc double space has various boundary faces but we are only interested in those at ǫ = 0. There are four boundary faces at ǫ = 0, described in the following table. Here and throughout, we label each face Fwxyz where the subscript indicates the order to which each of the scalar variables x,r,x′, r′ vanishes at that face. Arising from face geometry ′ x = 0,x = 0, F1010 [Z¯ Z¯; Y Y ] ′ × × x = 0, r = 0 F1001 [Z¯ M0; Y Y ] ′ × × r = 0,x = 0 F0110 [M0 Z¯; Y Y ] ′ × × r = 0, r = 0 F0101 [M0 M0; Y Y ] These faces meet in the following× corners.× Arising from corner geometry in faces x = 0, r = 0,x′ = 0 C Y Z¯ F , F 1110 × 1010 0110 x = 0, r = 0, r′ = 0 C Y M F , F 1101 × 0 1001 0101 x = 0,x′ = 0, r′ = 0 C Z¯ Y F , F 1011 × 1010 1001 r = 0,x′ = 0, r′ = 0 C M Y F , F 0111 0 × 0110 0101 x = 0,x′ = 0, r = 0, r′ = 0 blowup C SN +(Y Y ) F , F , F , F . 1111 × 1010 1001 0110 0101 To see that is a smooth submanifold of 2 we consider the function D Sb f(p)= x(p)r(p) x′(p)r′(p). (3) − Away from the boundary faces f is smooth with non-vanishing differential. In a neighborhood of S11 F1001, let − x(p) r′(p) f (p)= . 0110 x′(p) − r(p) ′ Since x rf0110 = f, we see that f0110 is smooth near the ǫ = 0 boundary faces away from where those faces meet F0110. Moreover, wherever defined, f0110 has nonvanishing differential and the zero set of f0110 coincides with that of f away from F0110. Similarly, let x(p) x′(p) f (p)= . 1001 r′(p) − r(p)

12 f1001 is smooth with nonvanishing diﬀerential and has the same vanishing set as f in a neighborhood of ǫ = 0 F . This shows that is a smooth submanifold of 2. While { }− 1001 D Sb the acc double space will not be used here, we note that the acc double space, with an additional blowup along the diagonal for ǫ 0, would be the natural space on which to ≥ study the resolvent behavior under ac convergence.

4 Analytic Preliminaries

Since we are working on manifolds with singularities, corners and boundaries, we brieﬂy review some key features of the analysis in these settings.

4.1 Polyhomogeneous conormal functions On manifolds M with corners having a consistent local product structure near each boundary and corner, a natural class of functions (or sections) with good regularity near the boundary and corners are the polyhomogeneous conormal functions (or sections). For a complete reference on polyhomogeneity on manifolds with corners, see [23]. In a neighborhood of a corner, we have coordinates (x1,...,xk,y1,...,yn−k) where x1,...,xk vanish at this corner and (y ,...,y ) are smooth local coordinates on a smooth compact n k 1 n−k − manifold Y. The edge tangent bundle in a neighborhood of this corner is spanned over Ve ∞(M) by the vector ﬁelds, C x ∂ , ∂ α . { i xi y } The basic conormal space of sections is

0(M )= φ : V ...V φ L∞(M ), V , and l . A 0 { 1 l ∈ 0 ∀ i ∈Ve ∀ } Let α and p be multi indices with α C and p N . Then we deﬁne j ∈ j ∈ 0 α,p(M )= xα(log x)p 0. A 0 A The space ∗ is the union of all these spaces, for all α and p. The space ∗ (M ) consists A Aphg 0 of all conormal distributional sections which have an expansion of the form

pj φ xαj (log x)pa (x,y), a ∞. ∼ j,p j,p ∈C Re(αXj )→∞ Xp=0 We deﬁne an index set to be a discrete subset E C N such that ⊂ × 0 (α ,p ) E, (α ,p ) = Re(α ) . j j ∈ | j j |→∞ ⇒ j →∞ Then, the space E (M ) consists of those distributional sections φ ∗ having poly- Aphg 0 ∈ Aphg homogeneous expansions with (α ,p ) E. j j ∈

13 4.2 Conic diﬀerential operators and b-operators

Let (M0, g0) be a Riemannian manifold with isolated conic singularity, deﬁned by x = 0, so that in a neighborhood of the singularity,

M = (0,x ) Y, g = dx2 + x2h(x) 0 ∼ 1 x × 0 with h(x) h, where (Y, h) is a smooth, n 1 dimensional compact manifold. A conic → − diﬀerential operator of order m is a smooth diﬀerential operator on M0 such that in a neighborhood of the singularity it can be expressed

m A = x−m B (x)( x∂ )k k − x Xk=0 with B ∞((0,x ), Diffm−k(Y )), where Diffj(Y ) denotes the space of differential oper- k ∈ C 1 ators of order j N on Y with smooth coefficients. The cone differential operators are ∈ 0 elements of the cone operator calculus; for a detailed description, see [19]. These cone operators are closely related to b-operators. A b-operator of order m is a smooth differential operator such that near the boundary it can be expressed

m A = B (x)( x∂ )k k − x Xk=0 with B ∞((0,x ), Diffm−k(Y )). We see that a cone differential operator of order m k ∈ C 1 is equal to a rescaled b-differential operator of order m. In other words, if A is an order m cone differential operator then xmA is a b-differential operator. In local coordinates (x,y1,...,yn−1) near the boundary of M0, a b-operator may be expressed as

j A = a (x,y)( x∂ ) (∂ α ). j,α − x y j+X|α|≤m The b-symbol of A is b j α σm(A)= aj,α(x,y)λ η . j+X|α|=m b ∗ Here λ and η are linear functions on T M0 deﬁned by the coordinates so that a generic b ∗ element of T M0 is n dx λ + η dy . x i i Xi=1 The b-operator is b-elliptic if the symbol bσ (A) =0 on bT ∗M 0 . m 6 0 − { } The scalar Laplacian on M0 is

x−2 ( x∂ )2 + ( n +1+ xH−1(∂ H)( x∂ ))+∆ = x−2L { − x − x − x h(x)} b

14 where Lb is an elliptic order 2 b-operator and H is a smooth function depending on the metric. Similarly, a geometric Laplacian ∆0 on M0 is also of the form

−2 ∆0 = x Lb, for an elliptic order two b-operator acting on sections of the vector bundle. The Schwartz 2 kernel of Lb is a distribution on the b-double space M0,b. By the b-calculus theory, (see [26]) L has a parametrix G , such that G is a b-operator of order 2 with b b b − G L = I R b b − where I is the identity operator and R is a b-operator with polyhomogeneous Schwartz kernel on the b-double space. Then, for any u 2(xn−1dxdy) with ∆ u = f 2(xn−1dxdy) ∈L 0 ∈L (x2G )(x−2L u) = (x2G )f = u Ru = u = x2G f + Ru = α + β. b b b − ⇒ b The ﬁrst term, α x2H2 x2 2(xn−1dxdy). The second term β 2(xn−1dxdy) has a ∈ b ⊂ L ∈ L polyhomogeneous expansion as x 0, → ∞ Nj β xγj +kϕ (y). ∼ j Xj=0 Xk=0 Above γj is an indicial root for the operator Lb and ϕj is an eigensection for the induced geometric Laplacian on (Y, h). Then,

∞ Nj γj +k u = α + x ϕj(y) (4) Xj=0 Xk=0 where α x2 2(xn−1dxdy). This decomposition plays a key role in the proof of spectral ∈ L convergence.

4.3 Friedrich’s domain of the conic Laplacian A geometric Laplacian ∆ on a conic manifold is an unbounded operator on 2 sections of 0 L the bundle. It can be extended to various domains in 2; the minimal domain is the L Dmin 2 closure of the graph of ∆ over ∞. The largest domain is the 2 closure of the L 0 C0 Dmax L graph of ∆ over 2. Each of these domains are dense in 2(M ), and the extension of the 0 L L 0 Laplacian to either domain is a closed operator. On complete manifolds = by Dmin Dmax the Gaffney-Stokes Theorem [11]. However, M0 is incomplete and so for a general geometric Laplacian these domains will not be equal. The Friedrich’s domain lies between DF Dmin and and is the closure of the graph of ∆ in 2 with respect to the densely defined Dmax 0 L Hermitian form, Q(u, v)= u, v . h∇ ∇ i ZM0 15 The extension of the Laplacian to the Friedrich’s domain, known as the Friedrich’s extension of the Laplacian, preserves the lower bound and is essentially self adjoint. Here, we work exclusively with the Friedrich’s extension of the Laplacian. For elements of , with u 2 and ∆ u = f 2, we have the expansion (4) from Dmax ∈L 0 ∈L the preceding section, ∞ Nj γj +k u = α + x ϕj(y). Xj=0 Xk=0 n−1 The volume form on M0 near the singularity is asymptotic to x dxdy. Therefore, the n exponents γj must all be strictly greater than 2 . For v min max the decomposition − 2 2∈ D ⊂ D (4) and the definition of min imply that min x . The equality of min and max D D2 ⊂ L D D then depends on the indicial roots of Lb = x ∆0. For further discussion of domains of the conic Laplacian, see [13], whose results include:

2−n+δ = f 2 :∆ f 2 and f = (x 2 ) as x 0, for some δ > 0 . DF { ∈L 0 ∈L O → } We will use this characterization of the domain of the (Friedrich’s extension of the) Lapla- cian in the proof of the ﬁrst theorem.

5 Spectral Convergence

We now have all the necessary ingredients to prove spectral convergence.

Theorem 1. Let (M0, g0) be a compact Riemannian n-manifold with isolated conic singularity, and let (Z, g ) be an asymptotically conic space, with n 3. Assume (M, g ) z ≥ ǫ converges asymptotically conically to (M , g ). Let (E , ) and (E , ) be Hermitian 0 0 0 ∇0 z ∇z vector bundles over (M0, g0) and (Z, gz), respectively, so that each of these bundles in a neighborhood of the boundary is the pullback from a bundle over the cross section (Y, h). Let ∆0, ∆z be the corresponding Friedrich’s extensions of geometric Laplacians, and let ∆ǫ be the induced geometric Laplacian on (M, g ). Assume ∆ has no 2 nullspace. Then the ǫ z L accumulation points of the spectrum of ∆ as ǫ 0 are precisely the points of the spectrum ǫ → of ∆0, counting multiplicity. The theorem follows from the following three statements: the inclusion accumulation σ(∆ ) σ(∆ ), the reverse inclusion accumulation σ(∆ ) σ(∆ ), and correct multiplic- ǫ ⊂ 0 ǫ ⊃ 0 ities.

5.1 Accumulation σ(∆ ) σ(∆ ) ǫ ⊂ 0 We extract a smoothly convergent sequence of eigensections corresponding to a converging sequence of eigenvalues as ǫ 0, and show that the limit section of this sequence is → an eigensection for the conic metric and its eigenvalue is the accumulation point. For

16 this argument, we work with sequences of metrics g which we abbreviate g with { ǫj } { j} Laplacians ∆j. Let λ(ǫ ) be an eigenvalue of ∆ , with eigensection f . Assume that λ(ǫ ) λ¯. Over j j j j → any compact set K M 0, the metric g = g converges smoothly to g by Lemma 1, thus ⊂ 0 j ǫj 0 so do the coeﬃcients of ∆j ∆(ǫj). Hence, normalizing fj by supM fj = 1, it follows − | | ∞ using standard elliptic estimates and the Arzela-Ascoli theorem that fj converges in on 0 ¯ C any compact subset of M0 . Furthermore the limit section f satisﬁes the limiting equation

∆0f¯ = λ¯f.¯

However, we do not know yet that f¯ 0, nor, even if this limit is nontrivial, that it lies in 6≡ the domain of the Friedrichs extension of ∆0. This is the content of the arguments below.

5.1.1 Weight Functions Let φ : M M Z Z as in deﬁnition 4. We identify Z with a ﬁxed ǫ 0,ǫ − 0,δ → 1/ǫ − 1/δ 1/δ K U M so that M M = (U K), K = Z . ⊂ ⊂ 0,ǫ − 0,δ ∼ − ∼ 1/δ Let c on M U. − w = ǫ (φ−1)∗ρ on U K. ǫ  ǫ −  cǫ on K.

Above, c is a constant and no generality is lost by assuming c = 1. Let wj = wǫj . For fj δ some δ > 0 to be chosen later, replacing fj by δ we assume the supremum of fjwj ||wj fj ||∞ | | is 1 on M. Since M is compact, f attains a maximum at some point p M, and we | j| j ∈ may assume p converges to somep ¯ M. The argument splits into three cases depending j ∈ on how and where pj accumulates in M.

5.1.2 Case 1: w (p ) c> 0 as j . j j → →∞ In this case, the points p are accumulating in a compact subset of M 0 at some point { j} 0 p¯ = p (the singularity). The maximum of f wδ on M is 1 and occurs at p so 6 | j j | j f w−δ on M for each j = f (p ) c−δ as j . | j|≤ j ⇒ | j j |→ →∞ The locally uniform ∞ convergence of f to f¯ implies that f¯ satisﬁes a similar bound, C j | | f¯ x−δ as x 0, | |≤ → and clearly f¯(¯p) = c−δ = 0. By the dimension assumption n 3 and the characterization | | 6 ≥ of the Friedrich’s Domain of the Laplacian, we may choose δ so that 2 n − < δ < 0. 2 −

17 Then f¯ lies in the Friedrich’s domain of the Laplacian and satisﬁes

∆0f¯ = λ¯f,¯ so λ¯ is an eigenvalue of ∆0.

5.1.3 Case 2: w (p ) c(ǫ ) as j . | j j |≤ j →∞ ˜ −1 Analysis on Z in this case leads to a contradiction. Let φj = φǫj , fj = fj(φj ). Let δ ˜ −δ p˜j = φj(pj). Because fjwj attains its maximum value of 1 at pj, fj(p ˜j) = (wj(pj)) . | | δ | |δ Rescale fj and f˜j, replacing them respectively with (wj(pj)) fj and (wj(pj)) f˜j so that the maximum of f˜ ρδ occurs at the pointp ˜ Z and is equal to 1. Since w (p ) = (ǫ ), | j | j ∈ j j j O j ρ(p ˜ ) = ǫ−1w (p ) stays bounded for all j, and so we assumep ˜ converges top ˜ Z. j j j j j ∈ By Lemma 1, (Z ,ǫ−2φ∗g ) converges smoothly to (Z , g ). This implies the following j j j j|U j Z equation is satisﬁed by f˜j on Zj, ˜ 2 ˜ ∆Z fj = ǫj λ(ǫj)fj + O(ǫj).

Since the λ(ǫ ) are converging to λ¯ and f˜ ρδ 1 on Z , we have j | j |≤ j ∆ f˜ 0 as j , on any compact subset of Z. Z j → →∞ This implies f f¯ on M and correspondingly, f˜ f˜ locally uniformly ∞ on Z and f˜ j → j → C satisﬁes ∆ f˜ = 0, fρ˜ δ 1, Z | |≤ where equality holds in the second equation at the point p. This shows that f˜ is not identically zero on Z and f˜ = O(ρ−δ) as ρ . Since f˜ is smooth on any compact subset →∞ of Z and is therefore in 2 (Z), choosing δ>n 2 contradicts the assumption that Z has Lloc − no 2 nullspace. L

ǫj 5.1.4 Case 3: wj(pj) 0, 0 as j . → wj (pj ) → →∞ In this case, the points φ (p ) in Z, so we rescale and derive a contradiction on the j j → ∞ complete cone over (Y, h). Consider the coordinates (ρ, y) on Z deﬁned for ρ ρ1. In ǫ ≥ these coordinates g = dρ2 + ρ2h(ρ). Let r = j ρ andg ˜ on Z be deﬁned by z j wj (pj ) j j

ǫ 2 g˜ = j g . j w (p ) Z j j Then, ρ1ǫj 1 (Z , g˜ ) = , Y, dr2 + r2h(r ǫ /w (p ) . j j ∼ w (p ) w (p ) × j j j j j j j j j j 18 As j , h(r ǫ /w (p )) converges smoothly to h, and →∞ j j j j g˜ g = dr2 + r2h j → C on the complete cone C over (Y, h). Let f˜ = w (p )δf (φ−1). Since f wδ 1 with equality j j j j j | j j |≤ at pj, f˜ rδ 1 on (Z , g˜ ) with equality atp ˜ = φ (p ). | j j |≤ j j j j j Let ∆˜j on Zj be the Laplacian induced byg ˜j on Zj,

w (p )2 ∆˜ = j j ∆ , j ǫ2 Z so 2 ˜ ˜ 2 wj(pj) ˜ ∆jfj = ǫj 2 λ(ǫj)fj + O(ǫj) on (Zj, g˜j). ǫj Since w (p ) 0 as j , there is a locally uniform ∞ limit f of f˜ on C which j j → → ∞ C c { j} satisﬁes f rδ 1, ∆ f = 0. | c |≤ c c Since the pointsp ˜j stay at a bounded radial distance with respect to the radial variable rj on Z , we may assumep ˜ p for some p C. At this point, f (p )r(p )δ =1 so f is j j → c c ∈ | c c c | c not identically zero. By separation of variables (see, for example, [22]), fc has an expansion in an orthonormal eigenbasis φ of 2(Y, h), { j} L

γj,+ γj,− fc = aj,+r φj(y)+ aj,−r φj(y) Xj≥0 δ where γj,+/− are indicial roots corresponding to φj and aj,+/− C. In order for fcr 1 ∈ −δ | |≤ globally on C, we must have only one term in this expansion, fc = ajr φj(y). Because the indicial roots are discrete, we may choose δ so that δ is not an indicial root. This is − a contradiction.

5.2 σ(∆ ) Accumulation σ(∆ ) 0 ⊂ ǫ th We use the Rayleigh-Ritz characterization of the eigenvalues [2]. Let λl(ǫj) be the l eigenvalue of ∆j and let ▽f, ▽f R (f) := h ij . j f,f h ij The subscript j indicates that the inner product is taken with respect to the 2 norm L on M with the gj metric. The eigenvalues are characterized using Mini-Max by

λl(ǫj) = infdim L = l, L⊂C1(M) supf∈L,f6=0 Rj(f).

19 Similarly this characterization holds for the eigenvalues of the (Friedrich’s extension of the) conic Laplacian which are known to be discrete (see [4], for example). Because ∞(M ) C0 0 is dense in 2(M ) we may restrict to subspaces contained in ∞(M ). Then, the lth L 0 C0 0 eigenvalue of ∆0 is

¯ ∞ λl = infdim L = l, L⊂ C0 (M0) supf∈L,f6=0 R0(f).

Let λ¯ be the lth eigenvalue in the spectrum of ∆ . Fix ǫ> 0. Then there exists L ∞ l 0 ⊂C0 with dim(L)= l and ¯ supf∈L,f6=0 R0(f) < λl + ǫ. Since any f L is also in ∞(M) and because L is ﬁnite dimensional, by the local ∈ C0 convergence of gj to g0, for large j

R (f) R (f) <ǫ for any f L. | j − 0 | ∈

Since λl(ǫj) is the inﬁmum λ (ǫ ) λ¯ + 2ǫ. l j ≤ l This shows λ (ǫ ) is bounded in j, and so we extract a convergent subsequence and a cor- { l j } responding convergent sequence of eigensections which exists by the preceding arguments. For each l we take λ (ǫ ) µ λ¯ , l j → l ≤ l f u , ∆ u = µ u . j,l → l 0 l l l These limit eigensections u are seen to be orthogonal as follows. Fix l, k, with f u l j,k → k and f u . Since ∞(M ) is dense in 2(M ) we may choose a smooth cutoﬀ function j,l → l C0 0 L 0 χ vanishing identically near the singularity in M0 such that

χu u 2 < ǫ, || k − k||L (M0)

χu u 2 < ǫ, || l − l||L (M0) V ol (M supp(χ)) < ǫ. j − Then on the support of χ, g g uniformly so for large j, j → 0 u , u χu , χu < ǫ, |h k li0 − h k li0| χu , χu χu , χu < ǫ, |h k li0 − h k lij| χu , χu χu ,χf < ǫ, |h k lij − h k j,lij| χu ,χf χf ,χf < ǫ. |h k j,lij − h j,k j,lij|

20 Since the eigensections for ∆ were chosen to be orthonormal and the volume of (M support(χ)) j − is small with respect to gj, χf ,χf < 2ǫ. |h j,k j,lij| Thus, u , u can be made arbitrarily small and u , u are orthogonal for l = k. We h k li0 k l 6 complete this basis to form an eigenbasis of 2(M ). Let f¯ be an arbitrary element of L 0 l this eigenbasis, with eigenvalue λ¯l. We wish to show that this f¯l is actually the ul above, deﬁned to be the limit of (a subsequence of) f , and hence the corresponding µ is equal { j,l} l to λ¯l. Again, assume the smooth cut-oﬀ function χ is chosen so that

χf¯ f¯ 2 < ǫ. || l − l||L (M0) For each j we expand χf¯l in eigensections of ∆j, ∞ χf¯ = a f , where a = χf¯ ,f . l j,k j,k j,k h l j,kij Xk=0 Now, ﬁx k and choose χ such that

χu u 2 < ǫ. || k − k||L (M0) Then, χf¯ ,f χf¯ ,f < ǫ, |h l j,ki0 − h l j,kij| χf¯ ,f χf¯ , u < ǫ, |h l j,kij − h l kij| χf¯ , u χf¯ , u < ǫ, |hh l kij − h l ki0| χf¯ , u f¯ , u < ǫ. |h l ki0 − h l ki0| By the orthogonality f¯ , u = 0 if f¯ = u , and otherwise is 1, so for each k, a 0 h l ki0 l 6 k j,k → as j , for all k with u = f¯ . Because f is not identically zero there must be some k →∞ k 6 l l with uk = f¯l. This shows that every eigensection of ∆0 is the limit of (a subsequence of) f and the corresponding eigenvalue λ¯ is the limit of the corresponding eigenvalues. { j,k} k 5.3 Correct Multiplicities

We argue here by contradiction. Let λ be an eigenvalue for ∆0 with k dimensional eigenspace spanned by u1, . . . , uk. Assume λ occurs as an accumulation point of multiplicity less than k, without loss of generality assume multiplicity k 1. However, preceding − arguments imply the existence of a subsequence f of ∆ with f u , which shows { k,j} j k,j → k that λ is achieved as an accumulation point of multiplicity at least k. Conversely, assume λ occurs as an accumulation point of multiplicity k+1. By preceding orthogonality argument, the limit section u of the converging sequence f is orthogonal to u , . . . , u . This k+1 k+1,j { 1 k} is a contradiction.

♥

21 6 Heat Kernels

The heat kernels for each of the geometries in ac convergence are elements of a pseudod- iﬀerential heat operator calculus that is deﬁned on the corresponding heat space. For the details in the construction of these heat calculi, kernels and spaces, see [29].

6.1 b-heat kernel Let (M, g) bea b-manifold with local coordinates z = (x,y) in a neighborhood of ∂M so that near the boundary dx2 g = + h(x,y). x2 Let (z, z′) be coordinates on M M and let ∆ be a geometric Laplacian on M. The b-heat × b kernel H(z, z′,t) is the Schwartz kernel of the fundamental solution of the heat operator ∂t +∆b. The heat kernel is a distributional section that acts on smooth sections of M, and satisﬁes ′ (∂t +∆b)H(z, z ,t) = 0,t> 0, H = δ(z z′), |t=0 − and by self adjointness since we work with the Friedrich’s extension of ∆b,

H(z, z′,t)= H(z′,z,t)∗.

For a smooth section u on M,

u(z,t) := u(z′),H(z, z′,t) dz′ h i ZM satisﬁes (∂t +∆b)u(z,t)=0 for t> 0, u(z, 0) = u(z). Physically, u(z,t) describes the heat on M at time t> 0 where the initial heat applied to M is given by u(z). Recall the Euclidean heat kernel,

′ 2 n z z G(z, z′,t) = (4πt)− 2 exp | − | . − 2t For a compact manifold without boundary, the heat kernel can be constructed locally using the Euclidean heat kernel and geodesic normal coordinates [28]. On the interior of a manifold with boundary (or singularity) the Euclidean heat kernel is also a good model, however, near the boundary (singularity) a diﬀerent construction is required.

22 6.1.1 b-heat space It is convenient to study the heat kernel on a manifold with boundary (or singularity) as an element of a heat operator calculus deﬁned on a corresponding heat space. This space is a manifold with corners constructed from M M R+ by blowing up along submanifolds at × × which the heat kernel may have interesting or singular behavior. For example, the diagonal is always blown up at t = 0, since away from the boundary the heat kernel behaves like the Euclidean heat kernel which is singular along the diagonal at t = 0. For the b-heat space we ﬁrst blow up the codimension 2 corner at the boundary in both copies of M. The b-heat 2 space Mb,h is then,

M 2 = M 2 R+; ∆(M M) t = 0 , dt , b,h b × t × × { } where ∆(M M) is the diagonal in M M and M 2 is the b-double space (2). The b-heat × × b space has five boundary faces, two of which result from blowing up. The remaining three boundary faces are at t = 0 off the diagonal and at the boundary in each copy of M. More precisely, we have the following.6 Face Geometry of face Defining function in local coordinates + + 2 ′ 2 1 F110 N (Y Y ) R ρ110 = (x + (x ) ) 2 × × 1 + ′ 4 2 4 Fd2 PNt (∆(M M)) ρd2 = ( z z + t ) × + | − | F100 Y (M ∂M) R ρ100 = x × − × + ′ F010 Y (M ∂M) R ρ010 = x × − 2 × F001 (M ∂M) ∆(M M) ρ001 = t +− − × + Above PNt denotes the inward pointing t parabolic normal bundle while SN denotes the inward pointing spherical normal bundle. Note that the local coordinates x,x′,t lift from M M R+ to M 2 as follows, × × b,h ∗ ∗ ′ ∗ 2 β (x)= ρ110ρ100, β (x )= ρ110ρ010, β (t)= ρd2ρ001, so these coordinates are only local defining functions.

6.1.2 b-heat calculus

2 The b-heat calculus consists of distributional section half density kernels on M+ = M M R+ 2 × × which are smooth on the interior and lift to be polyhomogeneous on Mb,h with specified leading orders at the boundary faces. By constructing the b-heat kernel as an element of 2 the b-heat calculus, it is polyhomogenous on Mb,h. and we know the leading order terms. 2 Consequently, the b-heat kernel is polyhomogeneous at the boundaries and corners of M+ with specified leading orders. Once the calculus is defined and the composition rule is proven, construction of the heat kernel as an element of the heat calculus is similar to

6 The subscript “d” indicates a face created by blowing up along the diagonal, so for example Fd2 is the face created by blowing up along the diagonal where the scalar variable t vanishes to second order.

23 solving an ordinary diﬀerential equation using Taylor series. The following deﬁnition is from [26].

Deﬁnition 10. For any k R and index set E110, A is an element of the b-heat calculus, E110,k ∈ Ψb,H if the following hold.

1 − +E110 1. A 2 (F ). ∈Aphg 110 2. A vanishes to inﬁnite order at F001, F100, and F010.

− n+3 −k 3. A ρ 2 ∞(F ). ∈ d2 C d2 Because the heat calculus is deﬁned with half densities, the normalizing factors at F110 and Fd2 simplify the composition rule. An element A of the b-heat calculus is the Schwartz kernel of an operator acting on a smooth half density section f of M by

Af(z,t)= A(z, z′,t),f(z′) dz′. h i ZM Furthermore, A acts by convolution in the t variable so for a smooth half density section R+ f of M t , × t Af(z,t)= A(z, z′,t s),f(z′,s) dz′ds. h − i Z0 ZM Two elements of the b-heat calculus compose as follows.

Technical Theorem 1. Let A Ψka,A and let B Ψkb,B. Then the composition, A B ∈ b,H ∈ b,H ◦ ka+kb,A+B is an element of Ψb,H . The proof of this composition rule is in [26].

6.1.3 Construction of the b-heat kernel

First we construct a model heat kernel H1 as an element of the b-heat calculus that solves 2 the heat equation up to an error vanishing to positive order at the boundary faces of Mb,h. 2 ′ On the interior of Mb,h restricting to a coordinate patch with coordinates (z, z ,t), we locally define ′ 2 ′ −n/2 (|z−z |g) /2t H1(z, z ,t) := (4πt) e , where z z′ is the distance from z to z′ with respect to the metric g. As t 0 away | − |g → from the diagonal this construction immediately implies infinite order vanishing at F001. At F we solve exactly: for each p M and for each point z F in the fiber over d2 ∈ ∈ d2 (p,p, 0) the heat kernel at that point is determined by the coefficients of the metric (and its derivatives) at p. The normal operator of ∂t +∆b is the restriction to F110 of the lift 2 of ∂t +∆b to Mb,h.H1 is defined at F110 to be the kernel of a first order parametrix of

24 this normal operator and is smooth at this face. At F100, F010, and F001 the model kernel vanishes to inﬁnite order. As constructed, H1 satisﬁes

(∂ + ∆)H = K ,H Ψ−2,0 t 1 1 1 ∈ b,H 2 where K1 now vanishes to positive order at the boundary faces of Mb,h. Then, deﬁne

H = H H K , 2 1 − 1 ∗ 1 where now the error term K2 = (∂t + ∆)H2 2 vanishes to higher order on the boundary faces of Mb,h by the composition rule. This −2,0 construction is iterated and Borel summation (see [30]) gives H∞ Ψb,H with H∞ HN = n+3 ∈ − O(tN− 2 ), for N > 0, so that (∂t + ∆)H∞ = K, 2 where K vanishes to inﬁnite order on the boundary faces of Mb,h so we may push K forward to M M R+. We solve away the residual error term using the action of elements of the × × b-heat calculus as t-convolution operators. As a t-convolution operator, the heat kernel is the identity. Above, K as a t-convolution operator is of the form K = Id A where A is − a Volterra operator and Id is the identity. An operator of this form has an inverse of the same form so deﬁning H := H (Id A)−1 ∞ − solves away this residual error term. By construction the leading order behavior of the b-heat kernel is that of the model heat kernel and is summarized below. Face Leading order 1 F 0; O(t− 2 ) as t . 110 →∞ F n+3 ( 2) d2 − 2 − − F 100 ∞ F 010 ∞ F 001 ∞ 6.2 Conic heat kernel Let (M , g ) be a compact manifold with isolated conic singularity and let (E , ) be a 0 0 0 ∇0 Hermitian vector bundle over (M0, g0). Let ∆0 be the Friedrich’s extension of a geometric Laplacian on (M , g ) and let Y be the smooth n 1 dimensional cross section of M so 0 0 − 0 that ∂M0 = Y. The conic heat kernel is constructed analogously to the b-heat kernel.

25 6.2.1 The conic heat space

2 This construction comes from [27]. The conic heat space M0,h is a manifold with corners obtained from M M R+ = M 2 by blowing up along two submanifolds, 0 × 0 × 0,+ M 2 := [M M R+; ∂M ∂M t = 0 , dt]; ∆(M 0 M 0) t = 0 , dt . 0,h 0 × 0 × 0 × 0 × { } 0 × 0 × { } The conic heat space has five boundary faces described in the following table in which z = (x,y) and z′ = (x′,y′) are local coordinates in a neighborhood of the singularity in ′ each copy of M0 so that x = 0,x = 0 define the singularity as well as the boundary of M0. Face Geometry of face Defining function in local coordinates 1 + 4 ′ 4 2 4 F112 PNt (Y Y ) ρ112 = (x + (x ) + t ) + × 1 F PN (∆(M 0 M 0)) ρ = ( z z′ 4 + t2) 4 d2 t 0 × 0 d2 | − | F Y R+ ρ = x 100 × 100 F R+ Y ρ = x′ 010 × 010 F M 0 M 0 ∆(M 0 M 0) ρ = t 001 0 × 0 − 0 × 0 001 Note that the coordinates x,x′,t lift from M M R+ to M 2 as follows, 0 × 0 × 0,h ∗ ∗ ′ ∗ 2 2 β (x)= ρ100ρ112, β (x )= ρ010ρ112, β (t)= ρ112ρd2ρ001, so again these are only local defining functions.

6.2.2 The conic heat calculus

2 Let µ be a conic half density on M0,+; we may assume

n−1 ′ ′ µ = (xx ) 2 √dzdz dt = dVcdt.

2 p Fix also a smooth, nonvanishing half density, ν, on M0,h. Elements of the conic heat calculus 2 are distributional section half densities on M0,+ which are smooth on the interior and lift 2 to be polyhomogeneous on M0,h.

Deﬁnition 11. Let k R and E E E be index sets. Then A Ψk,E100,E010,E112 if ∈ 100 010 112 ∈ 0,H the following hold. 1. A E100 at F . ∈Aphg 100 2. A E010 at F . ∈Aphg 010 3. A E112 at F . ∈Aphg 112 4. A vanishes to inﬁnite order at F001.

− n+3 −k 5. A ρ 2 ∞(F ). ∈ d2 C d2

26 With this normalization the conic heat kernel has order k = 2 and the composition − rule is the following.

Technical Theorem 2. Let A ΨA100,A010,A112,ka , and B ΨB100,B010,B112,kb with the ∈ 0,h ∈ 0,H leading index terms satisfying

β + α > 0, α + β > 0, k > 0, k > 0, β + α > 1. 112 010 112 100 − a − b 100 010 − Then, the composition B A is an element of ΨA100,B010,Γ112,k with Γ = A + B ◦ 0,H 112 112 112 and k = (ka + kb). The proof of this theorem is in [29], and is originally due to [27], see also [22], [12]. The conic heat kernel is constructed analogously to the b-heat kernel ﬁrst using a model heat kernel and then using the composition rule to iteratively solve away the error term. Away from F112, F100 and F010 the model heat kernel comes from the standard local construction using the Euclidean heat kernel. At F112 the model heat kernel comes from an explicit construction of the heat kernel for an exact cone transplanted to F112 and extended smoothly to F100 and F010. See, for example [3] and [27] or for the case of the 2 scalar Laplacian, [29]. The boundary behavior of the conic heat kernel on M0,h is that of the model heat kernel and is summarized in the following table. Face Leading Order 3 2n+1 F112 2 + 2 + 2µ0 −−n−1 F100 2 + µ0 −n−1 F010 2 + µ0 F n+3 ( 2) d2 − 2 − − F 001 ∞ Above, µ0 is the leading term in the polyhomogeneous conormal expansion of H0 at F100 and by symmetry at F010; µ0 is determined by the eigenvalues of the induced Laplacian on Y, the dimension, and the rank of the bundle (see [3]).

6.3 Ac scattering heat kernel A summary of the ac scattering heat kernel, space and calculus is given here; for the details, see the appendix. Let Z¯ be a compactiﬁed ac scattering space with boundary deﬁned by x = 0 and local coordinates (x,y) near the boundary. Let ∆ be the Friedrich’s extension { } z of a geometric Laplacian on Z.

6.3.1 The ac scattering heat space First, we construct the ac scattering double space,

Z¯2 := [Z¯ Z¯; ∂Z¯ ∂Z¯]; ∆(Y Y ) F sc × × × ∩ 110

27 where F110 is the face created by the ﬁrst blowup. This construction comes from [14]. Then, the ac scattering heat space is

Z¯2 = Z¯2 R+; ∆(Z Z) t = 0 , dt . sc,h sc × × × { } The ac scattering heat space has six boundary faces described in the following table. Face Geometry of face Deﬁning function in local coordinates + + 2 ′ 2 ′ 2 1 F220 N (∆(Y Y )) R ρ220 = (x + (x ) + y y ) 2 + × × + 2 ′ 2 1 | − | F220 N ((Y Y ) ∆(Y Y )) R ρ110 = (x + (x ) ) 2 × + − × × F100 Z Y R ρ100 = x × × + ′ F010 Y Z R ρ010 = x ×+ × 1 F PN (∆(Z Z)) ρ = ( z z′ 4 + t2) 2 d2 t × d2 | − | F (Z Z) ∆(Z Z) ρ = t 001 × − × 001 6.3.2 Ac scattering heat calculus

2 Elements of the ac scattering heat calculus are distributional section half densities of Z+ ¯2 which are smooth on the interior and lift to be polyhomogeneous on Zsc,h. Let µ be a ¯2 smooth, non-vanishing half density on Z+ and let ν be a smooth, non-vanishing half density ¯2 on Zsc,h.

Deﬁnition 12. For any k R and index sets E , E , A ΨE110,E220,k if the following ∈ 110 220 ∈ sc,H hold.

1 − +E110 1. A 2 at F . ∈Aphg 110 n+2 − +E220 2. A 2 at F . ∈Aphg 220 3. A vanishes to inﬁnite order at F001, F100, and F010.

− n+3 −k 4. A ρ 2 ∞(F ). ∈ d2 C d2 Two elements of the ac scattering heat calculus compose as follows.

Technical Theorem 3. Let A ΨA110,A220,ka , and B ΨB110,B220,kb . ∈ sc,H ∈ sc,H Then, the composition B A is an element of ΨA110+B110,A220+B220,ka+kb . ◦ sc,H The ac scattering heat kernel is constructed analogously to the b and conic heat kernels. ¯2 The model heat kernel in this case is the lift of the Euclidean heat kernel to Zsc,h and by construction the leading orders of the ac scattering heat kernel are that of the model kernel ¯2 at the boundary faces of Zsc,h. This is stated in the following theorem.

28 Technical Theorem 4. Let (Z, gz) be an asymptotically conic manifold with cross section (Y, h) at inﬁnity. Let (E, ) be a Hermitian vector bundle over (Z, g ) which induces a ∇ z compatible bundle over (Y, h). Let ∆ be a geometric Laplacian on (Z, gz) associated to the bundle (E, ). Then there exists H ΨE110,E220,−2 satisfying: ∇ ∈ sc,H ′ (∂t + ∆)H(z, z ,t) = 0,t> 0, H(z, z′, 0) = δ(z z′). − Moreover, H vanishes to inﬁnite order at F110 and is smooth up to F220. The proof of this theorem is in the appendix.

7 Heat Kernel Convergence

The interaction of the heat kernels will be studied on the asymptotically conic convergence (acc) heat space.

7.1 The acc heat space The acc heat space construction is similar to the heat space constructions of section 6 and the double space construction in section 3. First we let,

2 := [ 2 R+; C C 0 , dt]; C C . Sh S × 1 × 1 × { } 1 × 1 Let = p 2 : ǫ(p)= ǫ′(p) , H0 { ∈ Sh } and let ∆∗( 2) be the lift of the diagonal in 2 (C C ) to . An argument analogous S S − 1 × 1 H0 to the smoothness of the acc double space shows that is a smooth submanifold of 2. H0 Sh Then, the acc heat space is H := [ ;∆∗( 2) 0 , dt]. H H0 S × { } The acc heat space has several boundary faces of codimension one, however, we are only interested in those at ǫ = 0. These are the following. Arising from face geometry x = 0,x′ = 0, F [[Z¯ Z¯ R+; Y Y 0 , dt]; Y Y ]; ∆(Z Z) 0 , dt 1010 × × × × { } × × × { } x = 0, r′ = 0 F [Z¯ M R+; Y Y 0 , dt]; Y Y 1001 × 0 × × × { } × r = 0,x′ = 0 F [M Z¯ R+; Y Y 0 , dt]; Y Y 0110 0 × × × × { } × r = 0, r′ = 0 F [[M M R+; Y Y 0 , dt]; Y Y ]; ∆(M 0 M 0) 0 , dt 0101 0 × 0 × × × { } × 0 × 0 × { } 2 ¯ Note that F1010 ∼= Zb,h the b-heat space associated to Z with an additional benign ¯ blowup at the intersection of the boundary in each copy of Z at t = 0. Similarly, F0101 ∼= 2 e M0,h the conic heat space with an additional benign blowup at the intersection of the

f 29 boundary for t > 0. These are the faces in which we are most interested and should be thought of as the “front faces” while the other two faces should be thought of as the “side faces.” The ǫ = 0 boundary faces meet in codimension 2 corners described in the following table. Arising from corner geometry in faces ′ + x = 0, r = 0,x = 0 C1110 [Y Z¯ R ; Y Y 0 , dt] F1010, F0110. ′ × × + × × { } x = 0, r = 0, r = 0 C1101 [Y M0 R ; Y Y 0 , dt] F1001, F0101 ′ ′ × × + × × { } x = 0,x = 0, r = 0 C1011 [Z¯ Y R ; Y Y 0 , dt] F1010, F1001 ′ ′ × × + × × { } r = 0,x = 0, r = 0 C0111 [M0 Y R ; Y Y 0 , dt] F0101, F0110. There are also higher codimension corners× × contained× in× the { } boundary faces

7.2 The acc heat calculus The acc heat calculus is a parameter (ǫ) dependendent operator calculus which incorporates k the smooth, conic, ac scattering and b-heat calculi. Let Ψǫ,H be the heat calculus of order k for the smooth compact manifold (M, gǫ), (see [26] or [29]). The acc heat calculus consists k of kernels that restrict for each ǫ to an element of Ψǫ,H and which have a polyhomogeneous k,E110 k,E112,E100,E010 expansion at ǫ = 0 in terms of elements of Ψb,H and Ψ0,H . Deﬁnition 13. The asymptotically conic convergence heat calculus of order k, written k,E1010,E0101,E1001,E0110 Ψacc,H consists of kernels A such that the following hold.

k 1. For each ǫ> 0, A restricts to an element of Ψǫ,H.

2. In a neighborhood of F1010, A has an asymptotic expansion in ρ1010 with index set E1010 such that the coeﬃcients are elements of the b-heat calculus of order k. Such an expansion is of the form

A (ρ )αj (log ρ )pA , ∼ 1010 1010 j,l Xj≥1 0≤pX0≤p≤pj k,Ej with A Ψ 110 . Above, if for some j, p = 0, then there are no log terms. j,l ∈ b,H j 3. In a neighborhood of F0101, A has an asymptotic expansion in ρ0101 with index set E0101 such that the coeﬃcients are elements of the conic heat calculus of order k. Such an expansion is of the form

A (ρ )αl (log ρ )pB , ∼ 0101 0101 l,p Xl≥1 0≤pX0≤p≤pl k,El ,El ,El with B Ψ 112 100 010 . l,p ∈ 0,H 4. A has asymptotic expansion in ρ1001 at F1001 with index set E1001 and asymptotic expansion in ρ0110 at F0110 with index set E0110.

30 7.2.1 Composition Let A Ψka,A1010,A0101,A1001,A0110 , B Ψkb,B1010,B0101,B1001,B0110 . We expect A and B to ∈ acc,H ∈ acc,H compose as follows. For each ǫ> 0, the restrictions of A and B to the ǫ slice of compose to give H (A B) Ψka+kb . ◦ |ǫ ∈ ǫ,H At F1010, A has expansion

A (ρ )αj (log ρ )pA , ∼ 1010 1010 j,p Xj≥1 p0≤Xp≤pj k ,Aj with A Ψ a 110 and α ,p = A . Similarly, B has expansion j,p ∈ b,H { j j} 1010 B (ρ )βl (log ρ )qB , ∼ 1010 1010 l,q Xl≥1 q0≤Xq≤ql k ,Bl with B Ψ b 110 and β ,q = B , so at F the composition A B has expansion l,q ∈ b,H { l l} 1010 1010 ◦ A B (ρ )αj +βl (log(ρ ))rA B . ◦ ∼ 1010 1010 j,p ◦ l,q mX≥1 j+Xl=m r0≤Xr≤rm k +k ,Aj +Bl By the composition rule for the b-heat calculus, A B Ψ a b 110 110 so A B has j,p ◦ l,q ∈ b,H ◦ expansion A B (ρ )γk (log(ρ ))rC ◦ ∼ 1010 1010 k,r j,lX≥1 r0≤Xr≤rk k +k ,Aj +Bl with index set γ , r = A + B and C Ψ a b 110 110 . { k k} 1010 1010 k,r ∈ b,H At F0101, A has expansion

A (ρ )aj (log(ρ ))rR ∼ 0101 0101 j,r Xj≥1 r0≤Xr≤rj k ,Aj ,Aj ,Aj with index set a , r = A and R Ψ a 112 100 010 . Similarly at F , B has { j j} 0101 j,r ∈ 0,H 0101 expansion B (ρ )bl (log(ρ ))sS ∼ 0101 0101 l,s Xl≥1 s0X≤s≤sl k ,Bl ,Bl ,Bl with index set b ,s = B and S Ψ b 112 100 010 . The composition A B then { l l} 0101 l,s ∈ 0,H ◦ has an expansion at F0101

A B (ρ )aj +bl (log(ρ ))pR S . ◦ ∼ 0101 0101 j,r ◦ l,s mX≥1 j+Xl=m p0≤Xp≤pj 31 k +k ,Aj +Bl ,Aj +Bl ,Aj +Bl By the composition rule for the conic heat calculus R S Ψ a b 112 112 110 100 010 010 . j,r◦ l,s ∈ 0,H This shows that the composition A B has an expansion of the form ◦ A B (ρ )cj,l (log(ρ ))pG ◦ ∼ 0101 0101 j,l j,lX≥1 p0≤Xp≤pj,l k +k ,Aj +Bl ,Aj +Bl ,Aj +Bl with index set c ,p = A + B and G Ψ a b 112 112 110 100 010 010 . { j,l j,l} 0101 0101 j,l ∈ 0,H At F1001 and F0110, A and B have expansions with index sets A1001,B1001 and A0110,B0110, respectively. The composition then has expansions at F1001 and F0110 with index sets A1001 + B1001 and A0110 + B0110, respectively. The composition rule is not required for the proof of our main theorem, but we expect it to follow as above. A sketch of the proof including construction of the acc triple heat space, the key technical tool for proving the composition rule, is included in appendix B.

Theorem 2. Let (M0, g0) be a compact Riemannian n-manifold with isolated conic singularity, and let (Z, g ) be an asymptotically conic space, with n 3. Assume (M, g ) z ≥ ǫ converges asymptotically conically to (M , g ). Let (E , ) and (E , ) be Hermitian 0 0 0 ∇0 z ∇z vector bundles over (M0, g0) and (Z, gz), respectively, so that each of these bundles in a neighborhood of the boundary is the pullback from a bundle over the cross section (Y, h). Let ∆0, ∆z be the corresponding Friedrich’s extensions of geometric Laplacians, and let ∆ǫ be the induced geometric Laplacian on (M, gǫ). Then the associated heat kernels Hǫ have a full polyhomogeneous expansion as ǫ 0 on the asymptotically conic convergence (acc) → heat space with the following leading terms at the ǫ = 0 faces: H (z, z′,t) H (z, z′,t′) at F ǫ → 0 0101 H (z, z′,t) (ρ )2H (z, z′,τ) at F , ǫ → 1010 b 1010 ′ 2 2 Hǫ(z, z ,t)= O (ρ1001) (ρ0110) at F1001, F0110. ′ Above, H0 is the heat kernel for∆0,Hb is the b-heat kernel for ∆z and t and τ are rescaled time variables with ′ t t t = 2 , τ = 2 . (ρ1010) (ρ1001ρ0110) This convergence is uniform in ǫ for all time and moreover, the error term is bounded by C ǫtN , for any N N . N ∈ 0 7.3 Proof This proof is modeled after the parametrix construction of [26]. The proof is in four steps. First, we lift the operator ∂ +∆ to . Second, we construct the acc model heat kernel as t ǫ H an element of the acc heat calculus to be a solution kernel for the lifted operator. Next, we estimate the error term of this solution kernel. Finally, we use the b-heat calculus, introduce the acc conic triple heat space, and use the conic heat calculus to solve away the error term.

32 7.3.1 Lifted heat operator To analyze the behavior of the heat kernels under ac convergence we must lift the heat operator ∂t +∆ǫ depending on the parameter ǫ to the acc heat space. Speciﬁcally, we are interested in the leading orders of ∂t +∆ǫ near the F1010 and F0101 (front) faces. In a ′ neighborhood of F1010, x and x are small and approaching zero. So, we may be initially tempted to use the local cordinates (r, y, r′,y′,t,ǫ). However, these are not good coordinates ′ up to the face F1010 since r, r ,ǫ vanish at various regions of this face. So instead we consider the projective coordinates r ǫ t σ = , σ′ = r′, η = , τ = . r′ σσ′ η2

′ ′ Near F1010 in the coordinates (r, y, r ,y ,t,ǫ) the heat operator has the form

−2 4 2 2 ∂t + ǫ (r ∂r + r ∆y + lot) where lot are lower order terms, in the sense that they contain spatial derivatives of lower order. In projective coordinates this becomes

−2 η [∂τ +∆b,σ], where 2 ∆b,σ = (σ∂σ) +∆y + lot, is an elliptic order 2 b-operator on (0, )σ Y. Note that these projective coordinates are ∞ ′ × valid up to and on F1010 away from the r = 0 face and corners meeting this face. In these projective coordinates η is a deﬁning function for the side face F1001. By symmetry, the ′ behavior in a neighborhood of the σ = 0 corner of F1010 is the same behavior as at the σ = 0 corner of F1010. ′ ′ Near F0101 in the coordinates (x,y,x ,y ,t,ǫ) the heat operator has the form

2 −2 ∂t + ∂x + x ∆y + lot. Here we consider the projective coordinates x t ǫ s = , s′ = x′, t′ = , ξ = . x′ (s′)2 s′ The heat operator in this coordinates is

′ −2 (s ) (∂t′ +∆0,s),

2 −2 where ∆0,s = ∂s + s ∆y + lot is a geometric Laplacian on (0, )s Y. Note that these ∞ ′ × projective coordinates are valid up to and on F0101 away from the x = 0 face and corners ′ meeting this face. Since s is a deﬁning function for the side face F0110 as well as F1010

33 in these coordinates we see that the heat operator lifted to the acc heat space has the following leading order behavior at the boundary faces. Face ∂t +∆ǫ leading term time variable −2 t F (ρ ) (∂ +∆ ) τ = 2 1010 1010 τ b,σ (ρ1001 ρ0110) ′ t F ∂ ′ +∆ t = 2 0101 t 0,s (ρ1010 ) −2 F0110 (ρ0110) −2 F1001 (ρ1001) We can now construct the acc model heat kernel H1 as an element of the acc heat calculus that solves the heat equation at the ǫ = 0 boundary faces (and interior) of to H at least ﬁrst order.

7.3.2 Acc model heat kernel

′ ′ At F0101, let H1 = H0(z, z ,t ), where H0 is the heat kernel for (M0, g0) with rescaled time ′ variable t . Extend H1 smoothly oﬀ F0101, and at F1010 let H1 be asymptotic to

H (ρ )2H (z, z′,τ), 1 ∼ 1010 b where Hb is the b-heat kernel with rescaled time variable τ, and let H1 restrict to Hb at F1010. At F1001 and F0110, let H1 be asymptotic to

H (ρ )2 at F , H (ρ )2 at F . 1 ∼ 1001 1001 1 ∼ 0110 0110

Extend H1 smoothly off the boundary faces so that for ǫ>ǫ0 > 0,H1 is the heat kernel Hǫ for (M, g ). As t 0 away from the diagonal H vanishes to infinite order. At t = 0 along ǫ → 1 the diagonal, a purely local construction away from the boundaries and singularities using the Euclidean heat kernel solves the heat equation up to O(t). The standard parametrix construction [26] using the jet of the metric at the base point of each fiber improves this to an error term vanishing to infinite order in t as t 0 uniformly down to ǫ = 0. Since → the setting is a bit different here, we include this construction.

7.3.3 Acc model heat kernel construction along diagonal at t = 0 In a neighborhood of the faces diﬀeomorphic to PN +(∆) in we carry out a local con- H struction as in [26] chapter 7. Let X be a manifold; since this construction is the same for X = M, X = M0 and X = Z¯ we use X to simplify notation. Let FX denote the PN +(∆(X)) face in , where ∆(X) is the diagonal in X X. Let N(∂ +∆ ) be the H × t X restriction to F of the lift of (∂ +∆ ) to , where ∆ is our geometric Laplacian on X t X H X X. With the heat calculus normalization at FX , an element A of the acc heat calculus of order k restricts to F as follows − X N(A)= t(k+n+2)/2A . |FX

34 As in [26] we observe that FX is naturally diffeomorphic to a radial compactification of the tangent space of X, with each fiber of FX over (x,x, 0) diffeomorphic to the tangent space at x, fiber(x) ∼= TxX. Then, from [26] 7.15, 1 N(t(∂ +∆ )A) = [σ(∆ ) (R + n + k + 2)]N(A), (5) t X X − 2

where R is the radial vector field on the fibers of T X. Note that if G0 satisfies

t(∂ +∆ )G = O(t∞) as t 0, G t =0= δ(x x′), t X 0 → 0| −

then G0 also satisﬁes

(∂ +∆ )G = O(t∞) as t 0, G t =0= δ(x x′). (6) t X 0 → 0| −

So, we may work with t(∂t +∆X ) as in [26]. Our initial parametrix G0 will have order k = 2 at F . Then, from 5 we have the following equation for G − X 0 1 [σ(∆ ) (R + n)]N(G ) = 0. X − 2 0

From [26] 7.13, in order for G0 to satisfy the initial condition it must satisfy,

N(G0) = 1. Zfiber Since these conditions are fiber-by-fiber we introduce local coordinates so that

2 2 σ(∆X )= D1 + . . . + Dn on TxX.

Then we have 1 D2 + . . . + D2 (R + n) N(G ) = 0, (7) 1 n − 2 0 so 2 n X x N(G ) = (2π)− 2 exp | | (8) 0 − 4 ′ is the desired solution, where X is a projective local coordinate on F , X = x−x (see [26] x t1/2 (7.36)), and is the Riemannian norm on T X induced by the metric at x. To see that | ∗ |x this is the desired solution, consider the Fourier transform of (7) with u = N(G ) , 0 |TxX (ξ∂ + 2 ξ 2)ˆu = 0, uˆ(0) = 1. ξ | | Then by standard results in ordinary diﬀerential equations, the expression in (8) is the unique decaying solution.

35 Now we may iterate this to solve up to higher order. Assume we have found G0,...,Gk satisfying t(∂t +∆X )Gj = Rj, where R is of order 3 j at F . To ﬁnd G = G T , we wish to solve j − − X k+1 k − k

t(∂t +∆X )Tk = Rk + Rk+1, where we have already found R of order 3 k, and R will be of order 4 k. Lifting k − − k+1 − − to T X this becomes 1 σ(∆ ) (R + n j 1) N(T )= N(R ), X − 2 − − k k which we may again solve via Fourier transform. Letting u = N(Tk) and f = N(Rk) we find 1 uˆ(ξ)= exp((r 1) ξ 2)fˆ(rξ)rk+1dr − | | Z0 is the desired solution. This completes the inductive construction for all k. Now the suc- cessive T = G G give a formal power series at F which can be summed by Borel’s j j+1 − j X Lemma so that G is order 2 at F and satisfies (6). We then set the acc model heat − X kernel H1 = G in a neighborhood of FX . Since this construction is the same for X = M, X = M and X = Z¯ and for each the error term vanishes to infinite order as t 0, the 0 → error term for H has infinite order vanishing as t 0 for all ǫ 0. 1 → ≥ 7.3.4 Error term approximation For each ǫ > 0, let E(z, z′,t,ǫ) = H (z, z′,t) H (z, z′,t,ǫ). Let K be defined for each ǫ − 1 ǫ> 0 by ′ ′ (∂t +∆ǫ)E(z, z ,t,ǫ)= K(z, z ,t,ǫ). By construction of H , K = O(ǫt∞) as ǫ,t 0, so for anyN N, there is C > 0 such that 1 → ∈ for any (z, z′) M M, ∈ × K(z, z′,t,ǫ) < CǫtN . | | Moreover, K has a polyhomogeneous expansion down to ǫ = 0. For each ǫ > 0, E is smooth on for t > 0 by parabolic regularity applied for each H ǫ > 0 since K is O(t∞). By construction E is smooth down to t = 0, so E(z, z′,t,ǫ) is smooth on the blown down space, M M R+ (0, δ] . The following maximum principle × × × ǫ argument on M [0, T ] shows that E is also O(ǫt∞) as ǫ,t 0 in the same sense as K. × t → Fix ǫ > 0, z′ M. Since K = O(ǫt∞), fix C > 1 and N N >> 1 such that ∈ ∋ K(z, z′,t,ǫ) 2 Cǫ2t2N for all z M. Let u(z,t) = E(z, z′,t,ǫ) 2. Let ∆ be the scalar | | ≤ ∈ | | Laplacian for (M, gǫ). Then u satisfies

(∂ + ∆)u = 2 (∂ + ∗ )E, E E 2 = 2 K E, E E 2 t h t ∇ ∇ i−|∇ | h − R i−|∇ |

36 2 K, E 2 K E K 2 + E 2 = K 2 + u. ≤ h i≤ | || | ≤ | | | | | | Above we have used the positivity of and the compatibility of the bundle connection R with the metric. Now, letu ˜ = e−tu. Thenu ˜ satisﬁes

(∂ + ∆)˜u e−t K 2 Cǫ2t2N . t ≤ | | ≤ Let w =u ˜ Cǫ2t2N+1. Since E and hence u andu ˜ vanish at t = 0, w = 0 and w − |t=0 satisﬁes (∂ + ∆)w Cǫ2t2N C(2N + 1)ǫ2t2N < 0. t ≤ − Fix T > 0 and consider w on M [0, T ] . If w has a local maximum for z M and t (0, T ) × t ∈ ∈ then (∂t + ∆)w> 0, and this is a contradiction. If w has a maximum at t = T then ∂ w 0 and t ≥

(∂t + ∆)w> 0, which is again a contradiction. Therefore, the maximum of w occurs at t =0 and so

w Cǫ2t2N+1. ≤ This implies u eT Cǫ2t2N , for 0

We solve away the error term of the acc model heat kernel H1 and construct the full asymptotic expansion of the acc heat kernel H on . First consider F = Z2 . This acc H 1010 ∼ b,h face has the following geometry. Boundary Face Geometry of Face Arising from e F PN +(∆(Z Z)) parabolic blowup of diagonal at t = 0 bd2 × F PN +(Y Y ) parabolic blowup of Y Y at t = 0 b112 × × F SN +(Y Y ) R+ blowup of Y Y for all time b110 × × × F Y Z R+ boundary in ﬁrst copy of Z b100 × × F Z Y R+ boundary in second copy of Z b010 × × Fb001 (Z Z) ∆(Z Z) t = 0 away from diagonal × − × ′ Since at this face H1 is asymptotic to Hb(z, z ,τ), at the boundary face where τ vanishes away from the diagonal, H1 vanishes to inﬁnite order. Recall t τ = 2 , (ρ1001ρ0110)

37 so τ vanishes at Fb001. Moreover, Hb vanishes to infinite order at the side faces Fb100 and Fb010. At the diagonal face Fbd2 we’ve solved H1 up to error vanishing to infinite order in t. So, we have at this point an approximation H1 whose error vanishes to infinite order on the interior of F1010 and at all boundary faces except Fb112 and Fb110. Since the face Fb112 arises from Fb110 and the blowup of these two faces may be performed in either order, we solve the error term up to infinite order on the blown-down space in which Fb112 has been blown down. Then, lifting the solution to the fully blown up space, the error term will still vanish to infinite order. The indicial operator for ∆b,σ at Fb110 is

2 (σ∂σ) +∆y on R+ Y. We would like to solve σ × (∂ +∆ ) u = K , τ b |Fb110 − |Fb110 2 so that u is polyhomogeneous on Zb,h; then H1 + u would vanish identically at Fb110. Since H1 is polyhomogeneous on F1010 and smooth up to these boundary faces, the error term K is also. Expanding K, where we usee simply ρ for the projective deﬁning function for Fb110,

K (ρ)j k , k ∞, ∼ j j ∈C0 Xj≥0 and expanding the desired solution u,

u (ρ)ju ∼ j Xj≥0 we may use either separation of variables expanding in eigenfunctions of ∆y or the Mellin transform (see [26]) to ﬁnd u0 satisfying

(∂ + I(∆ ))u = k , τ b 0 − 0 with u vanishing to infinite order as σ 0, : at the side faces F , F . Since 0 → ∞ b100 b010 ∆ I(∆ ) = (ρ)(L ), where L is also a b-differential operator, we may now iteratively b − b 1 1 solve for u1, u2,..., to solve the equation to increasingly higher order. Recall that in ′ the projective coordinates near this face, σ defines Fb110 and since the operator does not differentiate with respect to σ′ = r′ the defining function commutes past the operator. Using Borel summation we construct u so that

(∂ +∆ )u = K + K τ b − 2 where K2 vanishes to infinite order at Fb110, Fb112. Using a smooth cutoff function χ sup- ported in a neighborhood of these faces, the second approximation H2 = H1 + χu now satisfies (∂ +∆ )H = K τ b 2|F11 2

38 where K2 vanishes to infinite order on both the interior and all boundary faces of F1010. Since the error now vanishes to infinite order at all boundary faces except those arising 2 from F0101 ∼= M0,h, we restrict attention to this face. It is convenient to use the conic heat calculus composition rule, but this requires the conic triple space. Since our error vanishes to infinite orderf at all boundary faces except F0101, we construct a partial acc triple heat space, the acc conic triple heat space, which contains the conic triple heat space so that we may use the conic heat calculus composition rule.

7.3.6 Acc conic triple heat space The acc conic triple heat space 3 is a submanifold constructed from 3 R+ R+ by Hc S × × eight blowups. Let X, X′, X′′ denote the three copies of the submanifold X in X3. Then the blowups are listed in the following table in the order which the blowups are performed together with the name of the face created. Blowup Face Y Y ′ Y ′′ t = 0,t′ = 0 , dt, dt′ F × × × { } 11122 Y Y ′ t = 0 , dt F × × { } 11020 Y ′ Y ′′ t′ = 0 , dt′ F × × { } 01102 Y Y ′′ t′′ = t t′ = 0 , dt′′ F × × { | − | } 10122 ∆( ′ ′′) t = 0,t′ = 0 , dt, dt′ F S × S × S × { } d3 ∆( ′) t = 0 , dt F S × S × { } d20 ∆( ′ ′′) t′ = 0 , dt′ F S × S × { } d02 ∆( ′′) t′′ = 0 , dt′′ F S × S × { } d22 Let β∗H be the lift of H to 3, and let β∗K be the lift of K to 3. Then, β∗K 2 2 Hc 2 2 Hc 2 vanishes to infinite order at all boundary faces except those arising from the lift of F0101. Now let H := β (β∗H β∗H β∗K ), 3 ∗ 2 − 2 2 where β is the push forward to from 3. Since β∗K vanishes to infinite order at all ∗ H Hc 2 boundary faces except F , the push forward of (β∗H )(β∗K ) to vanishes to infinite 0101 2 2 H order at all boundary faces except F0101, where the result is given by the conic heat calculus composition rule. Consequently, H = H β (β∗H β∗K ) 3 2 − ∗ 2 2 vanishes to higher order at the boundary faces of F0101, by the conic heat calculus composition rule. Continuing this construction and using Borel summation, we arrive at H∞ with expansion asymptotic to H2,H3,... and satisfying

(∂t′ +∆0)H∞ = K∞,

where K∞ now vanishes to infinite order on F0101. Using a smooth cutoff function we now have H defined on all of satisfying ∞ H (∂t +∆ǫ)H∞ = K∞

39 where K vanishes to infinite order at all boundary faces of . ∞ H 7.3.7 Solving away the residual error term To complete this construction we must remove the residual error term which vanishes to infinite order at the boundary faces of . It is now convenient to consider the elements of H the acc heat calculus as t-convolution operators acting on R+. For an element A which S× vanishes to infinite order at the boundary faces of and u a smooth half density section H of R+, the t-action of A on u is S× t Au(t)= Au(t s), u(s) ds, (9) h − i Z0 where the spatial variables have been suppressed. As a function of s′,s 0, [Au(s′)](s) ≥ vanishes to infinite order at s′ = 0. Restricting to s′ = t s, − [Au(t s)](s)= s−k/2−1(t s)ju (t s,s), − − k,j − for any k, j N with u a smooth half density, so for any k 1 this is integrable and − ∈ 0 k,j − ≥ consequently, Au(t) as in (9) is smooth in t and vanishes rapidly as t 0. So, an element → A of the acc heat calculus which vanishes to infinite order at all boundary faces of gives H rise to a Volterra operator. Since as a t-convolution operator we have

(∂ + ∆)H = Id K , t ∞ − ∞ we would like to invert (Id K ). Formally, the inverse should be − ∞ (Id K )−1 = Kj , − ∞ ∞ Xj≥0 j where K∞ is the j-fold composition of K∞. To show that this Neumann series converges, j we estimate the kernel of K∞. Since K∞ vanishes to inﬁnite order at all boundary faces of we may restrict to submanifolds of , estimating as in [26] and then combine these H j H j j + estimates to estimate K∞ on . The kernel k∞ of the restriction of K∞ to M M R ǫ H × × ×{ } is bounded by tj kj (z, z′,t,ǫ) C , t

40 j + Similarly, the kernel of the restriction of K∞ to Z¯ Z¯ R is bounded by × × tj kj (z, z′,t) C , t

These three bounds imply that the constants Cǫ,j stay bounded as ǫ 0 and so we have j → the following global bound for the kernel of K∞ on both and the blown-down space H ǫ = ǫ′ R+ { }⊂S×S× tj kj (z, z′,t,ǫ) C , t

It follows that the Neumann series for (Id K )−1 is summable and has an inverse which − ∞ as a t-convolution operator is also of the form (Id A) where A is an element of the acc − heat calculus that vanishes to inﬁnite order at the boundary faces of . Then, the full acc H heat kernel is H = H (Id K )−1. ∞ − ∞ As a consequence of this construction, H has a fully polyhomogeneous expansion down to ǫ = 0 with leading order terms given by the acc model heat kernel.

♥ Remarks A consequence of this theorem is the convergence

2 H H ′ + ǫ H as ǫ 0, ǫ → 0,t b,τ → with error term bounded by

C ǫtN , for any N N , and for all t. N ∈ 0 A Asymptotically conic scattering heat kernel

Let Z¯ be a compactified ac scattering space with boundary defined by x = 0 and local { } coordinates (x,y) near the boundary. Let ∆z be the Friedrich’s extension of a geometric Laplacian on Z. We motivate the definition of the acc heat space by lifting the Euclidean ¯2 heat kernel to Z+. Recall the Euclidean heat kernel for Rn,

′ 2 ′ − n − |z−z | G(z, z ,t) = (4πt) 2 e 2t .

41 Here the coordinate z = (r, y) has not been compactiﬁed. With the compactiﬁcation of Z 1 ′ ′ ¯2 ¯ given by x = r in the local coordinates (x,y,x ,y ,t) on Z+ near the boundary of Z the Euclidean heat kernel is

1 1 2 ′ 2 n ′ + y y G(x,y,x′,y′,t) = (4πt)− 2 exp | x − x | | − | . − 2t ! This motivates blowing up

S = (x,y,x′,y′,t) : x = 0,x′ = 0 . 110 { } x ′ ′ In the projective coordinates s = x′ , s = x the Euclidean heat kernel is

s−1 2 ′ 2 n ′ + y y G(s,y,s′,y′,t) = (4πt)− 2 exp | ss | | − | . − 2t ! This motivates a second blowup at s = 1, along the submanifold where the diagonal in Z¯ Z¯ meets the ﬁrst blown up face × S = (x,y,x′,y′,t) : x = 0,x′ = 0,y = y′ . 220 { } A.1 The Ac scattering heat space ¯2 As motivated above, the ac scattering heat space is constructed from Z+ by performing three blowups. ¯2 First, the scattering double space, Zsc is constructed:

Z¯2 := [Z¯ Z¯; ∂Z¯ ∂Z¯]; ∆(Y Y ) F sc × × × ∩ 110 where F110 is the face created by the ﬁrst blowup. Including the time variable we perform one more blowup to construct the ac scattering heat space,

42 A.2 Ac scattering heat calculus

2 Elements of the ac scattering heat calculus are distributional section half densities on Z+ ¯2 which are smooth on the interior and lift to be polyhomogeneous on Zsc,h. Let µ be a smooth, non-vanishing half density on Z¯ Z¯ R+ and let ν be a smooth, non-vanishing ¯2 × × half density on Zsc,h. Deﬁnition 14. For any k R and index sets E , E , A inΨE110,E220,k if the following ∈ 110 220 sc,H hold.

1 − +E110 1. A 2 at F . ∈Aphg 110 n+2 − +E220 2. A 2 at F . ∈Aphg 220 3. A vanishes to inﬁnite order at F001, F100, and F010.

− n+3 −k 4. A ρ 2 ∞(F ). ∈ d2 C d2 Elements of the ac scattering heat calculus are Schwartz kernels of operators acting on R+ sections of Z in the usual way and on sections of Z t by t-convolution. The composition × ¯3 rule is proven using the ac scattering triple heat space, Zsc,h. This space has partial blow down/projection maps to three identical copies of the ac scattering heat space as well as ¯2 full blow down/projection maps to three identical copies of Z+; these are called the left, right, and center. Formally, two elements of the ac heat calculus are composed by lifting ¯2 ¯3 from the left and right copies of Zsc,h to Zsc,h, multiplying and blowing down/projecting ¯2 to the center copy of Zsc,h. It is key that the triple space be constructed so that these lifts and push-forward maps are b-ﬁbrations in order that polyhomogeneity be preserved.

A.2.1 The ac scattering triple heat space ¯3 We first construct the ac scattering triple space Zsc and later include the time variables. In a neighborhood of the boundary in each copy of Z¯ we have the local coordinates (x,y), which provide the local coordinates (x,y,x′,y′,x′′,y′′) on Z¯3. First we blow up the codimension three corner defined by x = 0,x′ = 0,x′′ = 0 . We call this face F with defining { } 11100 function locally given by 2 ′ 2 ′′ 2 1 ρ11100 = (x + (x ) + (x ) ) 2 .

Next, we blow up the three codimension two corners corresponding to the F110 faces in ¯2 each of the three copies of Zsc,h. These faces are as follows. Face Submanifold to be blown up Defining Function ′ 2 ′ 2 1 F11000 S11000 = x = 0,x = 0 F11100 ρ11000 = (x + (x ) ) 2 { ′ ′′ }− ′ 2 ′′ 2 1 F01100 S01100 = x = 0,x = 0 F11100 ρ01100 = ((x ) + (x ) ) 2 { }− 1 F S = x = 0,x′′ = 0 F ρ = ((x)2 + (x′′)2) 2 10100 10100 { }− 11100 10100 43 Next we blow up the codimension 2n + 1 corner where the diagonals meet F11100. After ′ ′′ ′ ′′ the F11100 blowup, we have coordinates (θ,θ ,θ ,y,y ,y , ρ11100), with ′ ′ ′′ ′′ 2 ′ 2 ′′ 2 x = (ρ11100)θ, x = (ρ11100)θ , x = (ρ11100)θ , (θ) + (θ ) + (θ ) = 1. Using these coordinates, we next blow up S = θ = θ′ = θ′′, y = y′ = y′′, r = 0 . 22200 { 0 } The face created by this blowup is called F22200 with defining function 1 ρ = ((θ θ′)2 + (θ′ θ′′)2 + y y′ 2 + y′ y′′ 2 + r2) 2 . 22200 − − | − | | − | 0 After this we blow up the three codimension n corners corresponding to the F220 faces in the three copies of the double heat space. These are as follows. Face Submanifold to be blown up Defining Function ′ ′ 2 ′ 2 ′ 2 1 F22000 S22000 = θ = 0,θ = 0,y = y ρ22000 = (θ + (θ ) + y y ) 2 { ′ ′′ ′ }′′ ′ 2 ′′ 2| − ′ | ′′ 2 1 F02200 S02200 = θ = 0,θ = 0,y = y ρ02200 = ((θ ) + (θ ) + y y ) 2 { ′′ ′′ } 2 ′′ 2 | − ′′ 2| 1 F20200 S20200 = θ = 0,θ = 0,y = y ρ20200 = ((θ) + (θ ) + y y ) 2 { } ¯3 | − | We have now constructed the ac scattering triple space, Zsc. We next introduce the time variables and perform the parabolic temporal diagonal blowups. We must first blow up the codimension 2 corner of R+ R+ to preserve symmetry. Let × 2 = [R+ R+; t = t′ = 0]. T0 × The defining function for the blowup of t = t′ = 0 is ρ , which we call t′′ because it { } 00011 plays the role of the third time variable. We now take Z3 2 and blow up the temporal sc ×T0 diagonal faces. First, we blow up the codimension 2n + 3 triple diagonal, Sd3, defined by z = z′ = z′′, t′′ = 0 . { } The defining function of this face is ρd3, 1 ρ = ( z z′ 4 + z z′′ 4 + (t′′)2) 4 . d3 | − | | − | Next, we blow up the three temporal diagonals corresponding to the diagonal faces in the three copies of the double heat space. These are as follows. Face Submanifold to be blown up Defining Function ′ ′ 4 2 1 Fd20 Sd20 = z = z ρd20 = ( z z + t ) 4 { ′ ′′} | ′− |′′ 4 ′ 2 1 Fd02 Sd02 = z = z ρd02 = ( z z + (t ) ) 4 { ′′ } | − ′′ |4 ′′ 2 1 Fd22 Sd22 = z = z ρd22 = ( z z + (t ) ) 4 We have now constructed{ } the ac scattering triple| − heat| space and proceed with the composition rule. Technical Theorem 5. Let A ΨA110,A220,ka , and B ΨB110,B220,kb . ∈ sc,H ∈ sc,H Then, the composition B A is an element of ΨA110+B110,A220+B220,ka+kb . ◦ sc,H

44 A.2.2 Proof Formally we have, ∗ ∗ κB◦Aν = (βC )∗ ((βR) (κAν)(βL) (κBν)) . (10) ∗ Multiplying both sides of (10) by ν and using the fact that (βc)∗(βc) (ν)= ν

2 ∗ ∗ ∗ κB◦Aν = (βC )∗ ((βR) (κAν)(βL) (κBν)(βc) (ν)) . (11) ¯2 Next we calculate the lifts of the deﬁning functions and half densities from Zsc,h to ¯3 Zsc,h. A calculation gives the half density on the heat space ν in terms of the half density ¯2 µ on Z+

∗ − 1 − n − n+1 ν = (βh) (ρ110) 2 (ρ220) 2 (ρd2) 2 µ . The ac scattering triple heat space has partial blow down/projection maps βL, βR, and ¯2 ¯ ¯ ¯′ ¯′′ βC to three identical copies of Zsc,h. If we denote the three copies of Z by Z, Z , Z , and the three time variables (t,t′,t′′) where t′′ is from the blowup of R+ R+ then the three ¯2 × copies of Zsc,h are as follows. ¯2 ¯3 Copy of Zsc,h Associated to in Zsc,h ¯ ¯′ R+ Left Z Z t ¯′× ¯′′× R+ Right Z Z t′ ¯ × ¯′′ ×R+ Center Z Z t′′ Next, we compute the× lifts× of the deﬁning functions for the boundary faces of the heat space to the triple heat space.

45 ¯2 ¯3 Lifting map Deﬁning function on Zsc,h Lift to Zsc,h ∗ (βL) ρ100 ρ10000ρ10100 ∗ (βL) ρ010 ρ01000ρ01100 ∗ (βL) ρ110 ρ11100ρ11000 ∗ (βL) ρ220 ρ22200ρ22000 ∗ (βL) ρd2 ρd3ρd20 ∗ (βL) ρ001 ρ00010ρ00011ρd22 ∗ (βR) ρ100 ρ01000ρ01100 ∗ (βR) ρ010 ρ00100ρ10100 ∗ (βR) ρ110 ρ11100ρ01100 ∗ (βR) ρ220 ρ22200ρ02200 ∗ (βR) ρd2 ρd3ρd02 ∗ (βR) ρ001 ρ00001ρ00011ρd22 ∗ (βC ) ρ100 ρ10000ρ11000 ∗ (βC ) ρ010 ρ001000ρ01100 ∗ (βC ) ρ110 ρ11100ρ10100 ∗ (βC ) ρ220 ρ22200ρ20200 ∗ (βC ) ρd2 ρd3ρd22 ∗ (βC ) ρ001 ρ00022ρ00011ρd22 Then, ∗ ∗ − 1 − n − n+1 (βL) (ν) = (βL) ((ρ110) 2 (ρ220) 2 (ρd2) 2 µ). Next, we use the fact that

∗ ∗ ∗ 2 (βL) (µ)(βR) (µ)(βC ) (µ)= µ3.

Here, µ2 is a smooth density on Z¯ Z¯ Z¯ R+ R+, so we may assume 3 × × × × 2 ′ ′′ ′ µ3 =dzdz dz dtdt .

2 A Jacobian calculation gives the lift of µ3 to the triple heat space. First note

∗ (β3) (x) = (ρ11100)(ρ11000)(ρ10100)(ρ10000),

∗ ′ (β3) (x ) = (ρ11100)(ρ11000)(ρ01100)(ρ01000), ∗ ′′ (β3) (x ) = (ρ11100)(ρ01100)(ρ10100)(ρ00100). This implies

∗ 2 2 n (β3) (µ3) = (ρ11100) (ρ11000ρ01100ρ10100)(ρ22000ρ02200ρ20200)

2n+1 n+1 2n+3 ′′ 2 (ρ22200) (ρd20ρd02ρd22) ρd3 (t )ν3 .

46 2 Here, ν3 is a smooth, nonvanishing density on the triple heat space. Combining this with the above lifts, we arrive at the following formula

∗ ∗ ∗ 1 1 (βL) (ν)(βR) (ν)(βC ) (ν) = (ρ11100) 2 (ρ10100ρ01100ρ10100) 2 n n+1 n+3 n+1 2 2 2 2 ′′ 2 (ρ22000ρ02200ρ20200) (ρ22200) (ρd3) (ρd20ρd02ρd22) (t )ν3 . To use the push forward theorem of [23], we need to write each of these in terms of ¯2 b-densities. First, we have on the center copy of Zsc,h

b 2 −1 2 ν = (ρ100ρ010ρ110ρ220ρ001ρd2) ν . Then, we have b 2 ∗ −1 2 ν = (βc)∗(βc) ((ρ100ρ010ρ110ρ220ρ001ρd2) ν ). We observe ∗ −1 (βc) (ρ100ρ010ρ110ρ220ρ001ρd2) = −1 (ρ10000ρ00100ρ11000ρ01100 ρ10100ρ11100ρ22200ρ02200ρ20200 ρd3ρd22ρ00011) . ∗ −1 So now we multiply both sides of (11) by (βc)∗(βc) (ρ100ρ010ρ110ρ220ρ001ρd2) ) and inside the right side of (11) we have

− 1 n n−2 (ρ11100ρ11000ρ01100ρ10100) 2 (ρ22000ρ02200ρ22200) 2 (ρ20200) 2

n+1 n+1 n 2 2 2 −1 2 (ρd3) (ρd20ρd02) (ρd22) (ρ10000ρ00100) ν3 . 2 To use the push forward theorem, we must change the density ν3 to a b-density. We observe b 2 ν3 = (ρ11100ρ11000ρ01100ρ10100ρ22200ρ22000ρ02200ρ20200 −1 2 ρ10000ρ01000ρ00100ρd3ρd20ρd02ρd22ρ00011ρ00010ρ00001) ν3 . So, we now have for the composition formula

1 n+2 (βc)∗(κ Ãκ˜B(ρ11100ρ11000ρ01100ρ10100) 2 (ρ22200ρ22000ρ02200) 2 n n+3 n+1 2 2 2 b 2 (ρ20200) (ρd3ρd20ρd02) (ρd22) ρ01000ρ00011ρ00010ρ00001( ν3 )). ¯3 We observe the following orders ofκ Ã on Zsc,h. Face κÃ Index Set/Leading Order F 1 + A 11100 − 2 110 F 1 + A 11000 − 2 220 F , F , F , F , F 01100 10100 02200 20200 d22 ∞ F , F n+2 + A 22200 22000 − 2 220 F , F n+3 k d3 d20 − 2 − a F , F , F , F , F 10000 01000 00100 00010 00011 ∞ 47 Similarly, forκ ˜B we have orders as follows. Face κ˜B Index Set/Leading Order F 1 + B 11100 − 2 110 F 1 + B 01100 − 2 220 F , F F , F , F 11000 10100, 22000 20200 d22 ∞ F , F n+2 + B 22200 02200 − 2 220 F , F n+3 k d3 d02 − 2 − b F10000, F01000, F00100, F00010, F00011 Now, recalling the formula: ∞

1 n+2 (βc)∗(κ ˜Aκ˜B(ρ11100ρ11000ρ01100ρ10100) 2 (ρ22200ρ22000ρ02200) 2

n n+3 n+1 2 2 2 b 2 (ρ20200) (ρd3ρd20ρd02) (ρd22) ρ01000ρ00011ρ00010ρ00001( ν3 ))

We see that the quantity on the right hand side to be pushed forward by (βc)∗ has the following indices on the boundary faces. Face Index Set/Leading Order F 1 + A + B 11100 − 2 110 110 F , F , F , F , F , F 11000 01100 10100 22000 02200 20200 ∞ F n+2 + A + B 22200 − 2 220 220 F n+3 (k + k ) d3 − 2 − a b F , F , F d20 d02 d22 ∞ F10000, F01000, F00100, F00010, F00001, F00011 ∗ ∞ ¯3 ¯2 The push forward under (βc) sends the boundary faces of Zsc,h to Zsc,h as follows. ¯3 ¯2 Zh Face Boundary face of Zsc,h or Interior F11100 F110 F10100 F110 F22200, F20200 F220 Fd3, Fd22 Fd2 F10000 F100 F00100 F010 F00011 F001 F11000, F01100 F22000, F02200, Interior Fd20, Fd02, F01000, F00010, F00001 Interior b 2 The quantity to be pushed forward is integrable with respect to ν3 at the faces that are mapped to the interior, so we may apply the push forward theorem (see [23]) to arrive at the result of the composition rule. The kernel, κB◦A will have the following polyhomogeneous ¯2 index sets and leading orders on Zh.

48 ¯2 Face of Zh Index Set/Leading Order F 1 + A + B 110 − 2 110 110 F n+2 + A + B 220 − 2 220 220 F n+3 (k + k ) d2 − 2 − a b F 100 ∞ F 010 ∞ F 001 ∞ This concludes the proof of the composition rule: B A is an element of ΨA110+B110,A220+B220,ka+kb . ◦ ac,H

♥ Technical Theorem 6. Let (Z, gz) be an asymptotically conic scattering space with cross section (Y, h) at infinity. Let (E, ) be a Hermitian vector bundle over (Z, g ) so that near ∇ z the boundary E is the pullback of a bundle over (Y, h). Let ∆ be a geometric Laplacian on (Z, g ) associated to the bundle (E, ). Then there exists H ΨE110,E220,−2 satisfying: z ∇ ∈ ac,H ′ (∂t + ∆)H(z, z ,t) = 0,t> 0, H(z, z′, 0) = δ(z z′). − Moreover, H vanishes to infinite order at F110 and is smooth up to F220. ¯2 On the interior of Zac,h the ac scattering model heat kernel is locally defined by the Euclidean heat kernel and a partition of unity. At Fd2 we construct the model heat kernel explicitly using the jet of the metric at the base point of each fiber. At F001 the model heat kernel vanishes to infinite order. At F110 and F220 the model heat kernel is the lift of the Euclidean heat kernel. Then the ac scattering model heat kernel H1 satisfies

(∂t + ∆)H1 = K1,

¯2 where K1 vanishes to positive order on the boundary faces of Zsc,h. We now deﬁne

H = H H K , 2 1 − 1 1 with (∂t + ∆)H2 = K2 ¯2 where K2 vanishes to one order higher on the boundary faces of Zsc,h. Similarly,

H := H H K . 3 2 − 2 2

Using Borel summation we construct H∞ with expansion asymptotic to H1,H2,H3,... and satisfying (∂t + ∆)H∞ = K,

49 ¯2 where now K vanishes to inﬁnite order on the boundary faces of Zsc,h. As a t-convolution operator we wish to have H∞ = Id, however, we currently have H∞ = Id + K, but this is not a problem since (Id + K) is invertible with inverse of the same form. Then the ac scattering heat kernel H = H (Id K)−1 ∞ − is an element of the ac scattering heat calculus with leading orders on the boundary faces ¯2 of Zsc,h determined by those of the model kernel.

♥ B Acc triple heat space

Let T := [[[[ ; Y Y ′ Y ′′]; Y Y ′]; Y ′ Y ′′]; Y Y ′′], S × S × S × × × × × where we have used Y,Y ′,Y ′′ to denote the three copies of Y in 3. Let S := p T : f (p) = 0, i = 1, 2 , f (p)= x(p)r(p) x′(p)r′(p), f (p)= x′(p)r′(p) x′′(p)r′′(p). T { ∈ i } 1 − 2 − Like the acc double and heat space, is a smooth manifold with corners. Let T R+ := [R+ R+; 0 0 ]. 2,b t × s { } × { } Then the acc triple heat space is constructed from R+ by blowing up along twelve T × 2,b submanifolds creating the following twelve boundary faces. Below, let tD be the lift to ′ ′′ ′ T of the diagonal in , let D110 be the lift of the diagonal in , D011 be the S × S′ × S′′ S × S′′ lift of the diagonal in and D101 be the lift of the diagonal in . Submanifold blownS × up S Face created S × S Y Y ′ Y ′′ 0 0 , dt, ds S × × × { } × { } 11122 Y Y ′ t = 0 , dt S , × × { } 11020 Y ′ Y ′′ s = 0 , ds S × × { } 01102 Y Y ′′ t = s = 0 , ds, dt S × × { } 10122 Y Y ′ Y ′′ S × × 111 Y Y ′ S × 110 Y ′ Y ′′ S × 011 Y Y ′′ S × 101 tD t,s = 0 , ds, dt S × { } td D t = 0 , dt S 110 × { } d20 D s = 0 , ds S 011 × { } d02 D s = t = 0 , ds, dt S 101 × { } d22 50 As constructed, the acc triple heat space has full and partial projection/blow down maps to three identical copies, left, right and center, of the acc heat space and to three corresponding copies of the blown down space ǫ = ǫ′ 2 R+. To compose two elements { } ⊂ S × A and B we view the element A as acting from the left to the right while B acts from the right to the center. Formally, the composition B A is the pushforward from the acc triple ◦ heat space of the product of the lifts of A and B. Compatibility assumptions on the leading orders of A and B at boundary faces of the acc heat space are required so that we can push forward. With these assumptions and with the possible inclusion of normalizing factors at boundary faces of the acc heat space, two elements compose as one would expect. The technical details in the proof of this composition rule are expected to be analogous to the technical details in the proof of the ac scattering heat calculus composition rule (appendix A).

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