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ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

arXiv:0805.3148v1 [math.DG] 20 May 2008

EMILY B. DRYDEN, CAROLYN S. GORDON, SARAH J. GREENWALD, AND DAVID L. WEBB A BSTRACT. We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the time-zero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbit space of a manifold under the action of a discrete group of isometries), H. Donnelly [10] proved the existence of the heat kernel and constructed the asymptotic expansion for the heat trace. We extend Donnelly’s work to the case of general compact orbifolds. Moreover, in both the good case and the general case, we express the heat invariants in a form that clari?es the asymptotic contribution of each part of the singular set of the orbifold. We calculate several terms in the asymptotic expansion explicitly in the case of two-dimensional orbifolds; we use these terms to prove that the spectrum distinguishes elements within various classes of two-dimensional orbifolds.

C ONTENTS 1. Introduction Acknowledgments 2. Orbifolds and their singular sets 3. Construction of the heat expansion 4. Computation of the heat asymptotics 5. Applications References 1 3 3 9 15 23 34

1. I NTRODUCTION Roughly speaking, a topological orbifold is a space locally homeomorphic to an orbit space of a ?nite group action on Rn . A smooth orbifold consists of a Hausdorff second countable topological space together with an atlas of coordinate charts realizing such local homeomorphisms and satisfying compatibility conditions (see Section 2). Orbifolds were introduced by Satake, then studied by Thurston because of their utility in the investigation of three-manifolds (e.g., a Seifert ?bred three-manifold is naturally a generalized circle bundle over a two-orbifold); today,

Gordon and Webb were supported in part by NSF grants DMS-0072534, DMS-0306752, and DMS-0605247; Greenwald was supported in part by NSF ROA grants 0072533 and 9972304.

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DRYDEN, GORDON, GREENWALD, AND WEBB

orbifolds arise naturally in diverse branches of mathematics and physics, including symplectic geometry, string theory, and vertex operator algebras. We will be interested in orbifolds from a spectral-theoretic point of view. An orbifold endowed with a metric structure is a Riemannian orbifold. As in the manifold case, associated with every Riemannian metric is a Laplace operator acting on smooth functions on the orbifold. In the case of closed orbifolds, the Laplacian has a discrete spectrum. We study the relationship between the geometry and the Laplace spectrum of a closed orbifold via its heat kernel; as in the manifold case, the time-zero asymptotic expansion of the heat kernel furnishes geometric information about the orbifold. Orbifolds began appearing sporadically in the spectral theory literature in the early 1990s, and have received more concentrated attention in the last ?ve years. C. Farsi [15] showed that the spectrum of an orbifold determines its volume by proving that Weyl’s asymptotic formula holds for orbifolds. Dryden and A. Strohmaier [13] showed that for a compact, negatively curved two-dimensional orbifold, the Laplace spectrum determines both the length spectrum and the orders of the singular points and vice versa; on the other hand, P. Doyle and J.P. Rossetti [12] gave (disconnected) examples of isospectral ?at two-dimensional orbifolds with different length spectra and orders of singular points. Further investigations of the relationship between the lengths of closed geodesics and the spectrum were carried out by E. Stanhope and A. Uribe in [29]. It is natural to ask about the singularities that can appear in an isospectral family of orbifolds. Stanhope [28] showed that, in general, there can be at most ?nitely many isotropy types (up to isomorphism) in a set of isospectral Riemannian orbifolds that share a uniform lower bound on Ricci curvature. On the other hand, N. Shams, Stanhope, and Webb [27] constructed arbitrarily large (?nite) isospectral sets of orbifolds satisfying this curvature condition whose isotropy types differ. Rossetti, D. Schueth, and M. Weilandt [25] recently constructed a pair of isospectral Riemannian orbifolds whose isotropy types have different orders. For Riemannian manifolds, the asymptotic expansion of the heat kernel can be used to relate the geometry of the manifold to its spectrum. From the so-called heat invariants appearing in the asymptotic expansion, one can tell the dimension, the volume, and various quantities involving the curvature of the manifold. The heat kernel has been studied in various analogous or more general settings (e.g. [4, 5, 6, 11, 17, 24]). In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbit space of a manifold under the action of a discrete group of isometries), H. Donnelly [10] proved the existence of the heat kernel and constructed the asymptotic expansion for the heat trace. We extend Donnelly’s work to the case of general compact orbifolds. Moreover, in both the good case and the general case, we express the heat invariants in a form that clari?es the asymptotic contribution of each part of the singular set of the orbifold.

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We calculate several terms in the asymptotic expansion explicitly in the case of two-dimensional orbifolds; we use these terms to prove that the spectrum distinguishes elements within various classes of two-dimensional orbifolds. In particular, within the class of all two-dimensional orbifolds with nonnegative Euler characteristic, the spectrum is a complete topological invariant. Additional results are obtained for triangular pillow orbifolds endowed with a hyperbolic structure, and for nonorientable two-dimensional orbifolds. The paper is organized as follows. In Section 2, we give the background necessary for the rest of the paper, recalling several results that clarify the structure of the singular locus of an orbifold. Section 3 is devoted to the construction of the heat kernel on an arbitrary closed Riemannian orbifold by means of the construction of a parametrix. The existence of the heat kernel for closed orbifolds was shown previously by Y.-J. Chiang [8]; existence also follows from more general results for the heat kernel on Riemannian foliations (see [24]). However, we give a different construction in order to express the heat kernel in a convenient form that will allow us, in Section 4, to generalize Donnelly’s asymptotic expansion. Section 5 is devoted to various applications of the heat expansion, including those mentioned in the previous paragraph.

Acknowledgments. The initial inspiration for this project came from Shunhui Zhu, and the authors thank him for sharing his curiosity. We also thank Iosif Polterovich for helpful discussions and encouragement, and Alejandro Uribe and Liz Stanhope for alerting us to the frame bundle approach. The ?rst named author acknowledges the Centre Interfacultaire Bernoulli in Lausanne, Switzerland, and the Centro de An? alise Matem? atica, Geometria e Sistemas Din? amicos, Instituto Superior T? ecnico, in Lisbon, Portugal, for their support during her work on this project.

2. O RBIFOLDS AND THEIR SINGULAR SETS 2.1. D EFINITION . (i) An orbifold chart on a topological space X consists of a connected open subset U of Rn , a ?nite group GU acting on U by diffeomorphisms, and a mapping πU from U onto an open subset U of X inducing a homeomorphism from the orbit space GU \U onto U . We will always assume that the group GU acts effectively on U .

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DRYDEN, GORDON, GREENWALD, AND WEBB

(ii) An embedding λ : (U , GU , πU ) → (U , GU , πU ) between orbifold charts with U ? U is a smooth embedding λ : U → U such that the diagram U /

πU λ

/

U

πU

GU \U

?

G U \U

?

U

/U

commutes. Two charts (U , GU , πU ) and (U , GU , πU ) on X are said to be compatible if, for each point x ∈ U ∩ U , there exists an orbifold chart (U , GU , πU ) with U ? U ∩ U and smooth embeddings (U , GU , πU ) → (U , GU , πU ) and (U , GU , πU ) → (U , GU , πU ). (iii) An n-dimensional orbifold atlas A on X is a compatible family of ndimensional orbifold charts whose images form a covering of X . A re?nement A of an orbifold atlas A is an orbifold atlas each of whose charts embeds into a chart of A. Two orbifold atlases are said to be equivalent if they have a common re?nement. Every orbifold atlas is equivalent to a unique maximal one. An orbifold is a Hausdorff, second countable topological space together with a maximal orbifold atlas. (iv) Let O be an orbifold. A point x of O is said to be singular if for some (hence every) orbifold chart (U , GU , πU ) about x, the points in the inverse image of x in U have nontrivial isotropy in GU . The isomorphism class of the isotropy group, called the abstract isotropy type of x, is independent both of the choice of point in the inverse image of x in U and of the choice of chart (U , GU , πU ) about x. Points that are not singular are called regular. 2.2. R EMARKS . (i) The notion of orbifold generalizes slightly the notion of V -manifold introduced by Satake [26]; V -manifolds are orbifolds for which the singular set has codimension at least two. (ii) Given an embedding λ : (U , GU , πU ) → (U , GU , πU ) as in De?nition 2.1, there exists a homomorphism τ : GU → GU such that λ ? γ = τ (γ ) ? λ for all γ ∈ GU . This was proven by Satake for V -manifolds and was generalized to orbifolds by Moerdijk and Pronk [20]. From our convention that the group actions on each chart are effective, it follows that the homomorphism τ is injective. (iii) There are some subtle differences among the de?nitions of orbifold in the literature; in particular, some authors do not require that the ?nite group actions in the orbifold charts be effective. The de?nition we use is that in [20]. While these distinctions become signi?cant when formulating the

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correct notion of the category of smooth orbifolds, they play no role in our computations. An orbifold is said to be good if it is the orbit space of a manifold under the smooth action of a discrete group; otherwise, it is said to be bad. In particular, every point in an orbifold has a neighborhood that is a good orbifold. Even a bad orbifold can be expressed globally as the quotient of a manifold by a group action, although not a discrete group action. This is done by introducing a Riemannian structure and constructing the “bundle” of orthonormal frames. The orthonormal frame bundle is actually an orbibundle, the appropriate notion of vector bundle over an orbifold as described in [19] or [30]. However, its total space is a smooth manifold with an action of the orthogonal group, and one recovers the original orbifold as the orbit space of this orthogonal action. We will review this construction below after ?rst establishing some notation in the setting of arbitrary group actions. 2.3. D EFINITION . (See [14, Chapter 2].) (i) Consider a smooth proper action of a Lie group H on a smooth manifold M . For x ∈ M , let IsoH (x) denote the subgroup of H that ?xes x. De?ne an equivalence relation on M by x ≡ y if IsoH (x) and IsoH (y ) are conjugate. Each equivalence class is called an H -orbit type. Note that the equivalence classes are invariant under the action of H . We will say that the H -orbit type of x dominates that of y if IsoH (x) is conjugate to a subgroup of IsoH (y ). (ii) Let π : M → H \M be the projection onto the orbit space. Let p ∈ H \M . As p ? ranges over the H -orbit π ?1 (p) in M , the stabilizer IsoH (? p) ranges over a conjugacy class of subgroups of H . We will denote this conjugacy class of subgroups by IsoH (p) and refer to it as the H -isotropy type of p. De?ne an equivalence relation on H \M by p ≡ q if IsoH (p) = IsoH (q ). The equivalence classes will be called H -isotropy equivalence classes. Note that π carries points of the same H -orbit type in M to points of the same H -isotropy equivalence class in H \M . We will say that the H -isotropy equivalence class of p dominates that of q if the groups making up the conjugacy class IsoH (p) are conjugate to subgroups of those in IsoH (q ). By an abuse of notation, we will write |IsoH (p)| to mean the order of each of the groups making up the isotropy type IsoH (p). We will refer to this quantity as the order of the H -isotropy at p. 2.4. D EFINITION . (i) A Riemannian structure on an orbifold O is an assignment to each orbifold chart (U , GU , πU ) of a GU -invariant Riemannian metric gU e on U satisfying the compatibility condition that each embedding λ appearing in De?nition 2.1 is isometric. Every orbifold admits Riemannian structures. (ii) We will say that an orbifold chart (U , GU , πU ) on a Riemannian orbifold O is a distinguished chart of radius r if U is a convex geodesic ball of radius r. In this case, U is a convex geodesic ball in O. The entire group

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DRYDEN, GORDON, GREENWALD, AND WEBB

GU ?xes the center p ? of U , so the abstract isotropy type of p := πU (? p) is represented by GU . 2.5. R EMARK . Recall that for a Riemannian manifold M and a point p ∈ M , the convexity radius at p is the largest positive real number r(p) for which the geodesic ball of radius about p is geodesically convex for all < r(p). If M is compact, the in?mum r of {r(p) : p ∈ M } is positive and is called the convexity radius of M . For a point p in an orbifold O, we may de?ne the convexity radius at p to be the largest real number r(p) such that O admits a distinguished chart of radius centered at p for all < r(p). It is immediate that r(p) is positive. Moreover, if O is compact, then an elementary argument shows that the in?mum r of {r(p) : p ∈ O} is positive; r is called the convexity radius of O. 2.6. O RTHONORMAL FRAME BUNDLE . We give a brief description of the orthonormal frame bundle of a Riemannian orbifold. See [1] for more details. First consider a good Riemannian orbifold O = G\M , where M is a Riemannian manifold and G is a discrete group acting by isometries on M . Let F (M ) → M be the orthonormal frame bundle of M . Each element γ ∈ G, being an isometry of M , induces a diffeomorphism γ? of F (M ) carrying ?bers to ?bers; thus we obtain an action of G on F (M ) covering the action of G on M . The orthonormal frame bundle F (O) of O is de?ned to be G\F (M ) → O. The ?ber of F (O) → O over x ∈ O is the preimage of x in G\F (M ). The right action of O(n) on F (M ) commutes with the left action of G, and hence descends to a right O(n)-action on F (O). For a bad orbifold, the orthonormal frame bundle is de?ned in such a way that its restriction to any good neighborhood U ? = GU \U is the orthonormal frame bundle of the good orbifold U . The orthonormal frame bundle of O is a smooth manifold as well as an orbibundle on which the orthogonal group O(n) acts smoothly on the right, preserving ?bers. In particular, the orbifold O is the orbit space F (O)/O(n) of the right action of O(n) on the manifold F (O). 2.7. N OTATION AND R EMARKS . Let O be an orbifold. Endow O with a Riemannian metric and let F (O) be the associated orthonormal frame bundle as in 2.6. The O(n)-action on the ?ber of F (O) over a point x ∈ O is free if and only if x is a regular point of O. In particular, for each singular point x of O, viewed as an element of F (O)/O(n), the O(n)-isotropy type of x is a non-trivial conjugacy class IsoO(n) (x) of subgroups of O(n). (See De?nition 2.3.) Our realization of the orbifold O as a global quotient of a manifold (namely F (O)) by an action of O(n) depends, of course, on the Riemannian metric. However, it is not dif?cult to show that the conjugacy class IsoO(n) (x) of subgroups of O(n) is actually independent of the choice of Riemannian metric used in the construction. This conjugacy class of subgroups of O(n) will henceforth be denoted Iso(x) (without the subscript O(n) except when needed for clarity) and will be referred to as the isotropy type of the singular point x of O. Its cardinality |Iso(x)| will be called the order of the isotropy at x. Similarly, the equivalence classes of elements of O with the same isotropy type will be called isotropy equivalence classes, without mention of O(n).

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The subgroups of O(n) in the conjugacy class Iso(x) lie in the isomorphism class de?ned by the abstract isotropy type of x given in De?nition 2.1. Indeed, let (U , GU , πU ) be an orbifold chart with x ∈ U . Let x ∈ U with πU (x) = x. With respect to any choice of Riemannian metric on O (and associated Riemannian metric on U ), the group GU acts isometrically on U and thus acts on the left on the orthonormal frame bundle F (U ). The subgroup IsoGU (x) leaves invariant the ?ber of F (U ) over x. For each q in this ?ber, de?ne a homomorphism σq : IsoGU (x) → O(n) by the condition γ (q ) = (q )σq (γ ) where γ (·) and (·)σq (γ ) denote the left action of γ and right action of σq (γ ) ∈ O(n) on the ?ber. By 2.6, the restriction of F (O) to U is given by GU \F (U ). Letting ρ : F (U ) → GU \F (U ) be the projection, then σq maps IsoGU (x) isomorphically to the stabilizer of ρ(q ) in O(n). This stabilizer is a representative of the conjugacy class Iso(x), while IsoGU (x) represents the abstract isotropy type of x. 2.8. D EFINITION . A smooth strati?cation of a manifold or orbifold M is a locally ?nite partition of M into locally closed submanifolds, called the strata, satisfying the following condition: For each stratum N , the closure of N is the union of N with a collection of lower dimensional strata. 2.9. R EMARKS . (i) For any strati?cation of an orbifold (or manifold) O, the strata of maximal dimension are open in O and their union has full measure in O. (ii) The strati?cations that we will discuss below are Whitney strati?cations. As we will not explicitly use the additional properties of Whitney strati?cations here, we omit the de?nition and refer the reader to [14]. The notion of Whitney strati?cation can be de?ned in the more general setting of spaces that can be at least locally embedded in a smooth manifold. As discussed in [14], the orbit space of a proper Lie group action on a smooth manifold has this property. 2.10. P ROPOSITION . [14] Given a smooth action of a Lie group H on a manifold M , then: (i) The connected components of the H -orbit types form a Whitney strati?cation of M . The closure of a stratum N is made up of the union of N with a collection of lower dimensional strata, each lying in an H -orbit type strictly dominated by that of N . (ii) The connected components of the H -isotropy equivalence classes in H \M form a Whitney strati?cation of H \M . The closure of a stratum N is made up of the union of N with a collection of lower dimensional strata, each lying in an H -isotropy equivalence class strictly dominated by that of N . The map π carries each stratum in M onto a stratum in H \M . (iii) For x ∈ H \M and N the stratum through x, there exists a neighborhood U of x in H \M such that the isotropy equivalence class of each element of the complement of N in U strictly dominates that of x. (iv) If M is compact, then the strati?cations of M and of H \M are ?nite.

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DRYDEN, GORDON, GREENWALD, AND WEBB

2.11. C OROLLARY. Let O be an orbifold. Then the action of O(n) on the frame bundle F (O) gives rise to a (Whitney) strati?cation of O. The strata are connected components of the isotropy equivalence classes in O. The set of regular points of O intersects each connected component O0 of O in a single stratum comprising an open dense submanifold of O0 . 2.12. N OTATION . (i) We will refer to the strata of O in Corollary 2.11 as O-strata. (ii) If (U , GU , πU ) is an orbifold chart on O, then the action of GU on U gives rise to strati?cations both of U and of U as in Proposition 2.10. We will refer to these as U -strata and U -strata, respectively. 2.13. P ROPOSITION . Let O be a Riemannian orbifold and (U , GU , πU ) be an orbifold chart. Then: (i) The U -strata are precisely the connected components of the intersections of the O-strata with U . (ii) Any two elements of the same U -stratum have the same stabilizers in GU (not just conjugate stabilizers). (iii) If H is a subgroup of GU , then each connected component W of the ?xed point set Fix(H ) of H in U is a closed submanifold of U . Any U -stratum that intersects W nontrivially lies entirely in W . Thus the strati?cation of U restricts to a strati?cation of W . Proof. (i) A consequence of Proposition 2.10 is that each U -stratum, respectively O-stratum, is a connected component of the set of all points in U , respectively O, having GU -isotropy, respectively O(n)-isotropy, of a given order. Since by 2.7, the order of the GU -isotropy of each x ∈ U is equal to the order of the O(n)-isotropy of x, statement (i) follows. (ii) This follows because GU is discrete and the U -strata are connected. (iii) The ?rst statement is true for the ?xed point set of any smooth proper action by a compact group on a manifold [14]. The second statement follows from (ii). 2.14. N OTATION AND R EMARKS . Let O be a Riemannian orbifold and (U , GU , πU ) be an orbifold chart. Let N be a U -stratum in U . By Proposition 2.13, all the points in N have the same isotropy group in GU ; we will refer to this group as the isotropy group of N , denoted Iso(N ). Given a U -stratum N , denote by Isomax (N ) the set of all γ ∈ Iso(N ) such that N is open in the ?xed point set Fix(γ ) of γ . For γ ∈ GU , Proposition 2.13 tells us that each component W of the ?xed point set Fix(γ ) of γ (equivalently, the ?xed point set of the cyclic group generated by γ ) is a manifold strati?ed by a collection of U -strata. By Remark 2.9(i), the strata in W of maximal dimension are open and their union has full measure in W . In particular, the union of those U -strata N for which γ ∈ Isomax (N ) has full measure in Fix(γ ).

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2.15. E XAMPLE . On R2 , let rx and ry denote the re?ections across the x-axis and y -axis, respectively, and let r0 denote the rotation through angle π about the origin 0. Then G := {rx , ry , r0 , Id} is a Klein four group acting isometrically on R2 . The quotient of R2 by the semi-direct product of G with the lattice Z2 of translations is a closed orbifold O, whose underlying space is a square of side length 1 2 . The points on the boundary of the square are singular points (not boundary points) of the orbifold, comprising eight strata: each corner point forms a single stratum with isotropy of order four, while each open edge forms a stratum with isotropy of order two. The strata of codimension one are called re?ectors or mirrors, and the single point strata are called dihedral points or corner re?ectors. The intersection U of the square with a disk of radius less than 1 2 centered at one of the corners is the image of an orbifold chart (U , G, πU ) where U is a disk in R2 centered at the origin and G is the Klein-four group above. The U -strata of this action consist of the single point 0 and the intersections of the disk U with the positive and negative x-axis and the positive and negative y -axis. If N is the intersection of U with one of the half axes, then Iso(N ) consists of a re?ection and the identity, while Isomax (N ) contains only the re?ection. For N = {0}, we have Iso(N ) = G, but Isomax (N ) = {r0 }. 3. C ONSTRUCTION OF THE HEAT EXPANSION In this section, we address the heat kernel on closed Riemannian orbifolds. 3.1. P ROPOSITION . Let O be a closed Riemannian orbifold. The Laplacian ? of O has a discrete spectrum λ1 ≤ λ2 ≤ . . . , with each eigenvalue having ?nite multiplicity. The normalized eigenfunctions ?j are C ∞ and form an orthonormal basis of L2 (O). This proposition was proved in the case of V -manifolds (as de?ned in Remark 2.2) by Y.-J. Chiang in [8]. For an orbifold O which is not a V -manifold, those strata of the singular set of codimension one are called re?ectors. By doubling along all re?ectors one obtains a V -manifold X that doubly covers O. Thus O is the quotient of a V -manifold X by a Z2 action. Since the eigenfunctions on O are then the Z2 -invariant eigenfunctions on X , Proposition 3.1 follows immediately. 3.2. D EFINITION . Set R+ = [0, ∞) and R? + = (0, ∞). ? We say that K : R+ × O × O → R is a fundamental solution of the heat equation, or heat kernel, if it satis?es: (i) K is C 0 in the three variables, C 1 in the ?rst, and C 2 in the second; ? + ?x K (t, x, y ) = 0 where ?x is the Laplacian with respect to the (ii) ?t second variable; (iii) lim K (t, x, ·) = δx for all x ∈ O.

t→0+

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DRYDEN, GORDON, GREENWALD, AND WEBB

By the same argument as in the manifold case (see [2], III.E.2), Proposition 3.1 implies: 3.3. C OROLLARY. If a heat kernel exists, then it is unique and is given by

∞

K (t, x, y ) =

j =1

e?λj t ?j (x)?j (y ).

Chiang proved the existence of the heat kernel on a compact V -manifold (from which existence on an arbitrary closed orbifold trivially follows) by proving the ?λj t ? (x)? (y ). She also showed that the heat kernel can convergence of ∞ j j j =1 e be approximated on good neighborhoods by the Dirichlet heat kernel on the local manifold covering. The existence also follows from more general results on existence of the heat kernel for the basic Laplacian on Riemannian foliations [22]. However, in order to apply Donnelly’s results on the heat trace for good orbifolds to obtain the terms in the asymptotic expansion of arbitrary orbifolds in an applicable form, we will not assume the earlier existence results for the heat kernel or heat trace. We instead construct a parametrix and then follow the standard construction of the heat kernel from the parametrix as in [2]. Our construction of the parametrix and consequently the heat kernel will use directly the local structure of orbifolds as quotients of manifolds by ?nite group actions. 3.4. D EFINITION . A parametrix for the heat operator on O is a function H : R? +× O × O → R satisfying: (i) H ∈ C ∞ (R? + × O × O ); ? (ii) ?t + ?x H (t, x, y ) extends to a function in C 0 (R+ × O × O); (iii) lim H (t, x, ·) = δx for all x ∈ O.

t→0+

Recall that the heat kernel on a closed n-dimensional Riemannian manifold M has an asymptotic expansion along the diagonal in M × M as t → 0+ of the form (3.5) K (t, x, x) ? (4πt)? 2 (u0 (x, x) + tu1 (x, x) + t2 u2 (x, x) + . . . )

n

where the ui are local Riemannian invariants de?ned in a neighborhood of the diagonal in M × M . Letting ζ be a cut-off function that is identically one near the diagonal, then for m > n 2 , the function (3.6) K (m) (t, x, y ) = ζ (x, y )(4πt)? 2 e?

n d(x,y )2 4t

(u0 (x, y ) + · · · + tm um (x, y ))

is a parametrix for the heat operator on M . 3.7. R EMARK . In what follows, we shall take a local covering of our orbifold O by distinguished charts and piece together a parametrix for the heat operator on O from the expressions in (3.6). The key to piecing together the parametrix on O is to note that while the parametrix K (m) in (3.6) is globally de?ned on M , the three de?ning conditions of a parametrix in De?nition 3.4 are satis?ed locally as well as globally by K (m) . Indeed the ?rst two conditions are trivially local. The third condition is local in the following sense: The expression e?

d(x,y )2 4t

goes

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to zero uniformly as t → 0+ when d(x, y ) is bounded away from zero. Thus for f ∈ C 0 (M ), we have f (x) = lim

t→0+

K (m) (t, x, y )f (y )dy = lim

M

t→0+

K (m) (t, x, y )f (y )dy,

W

where W is any neighborhood of x. ? (m) (t, x, y ) We will also use the fact (see [2]) that if m > n 2 +2l, then ?t + ?x K extends to a function in C l (R+ × M × M ). Moreover, the extension is of class C 2l in the last two variables. 3.8. N OTATION . Let O be an orbifold of dimension n. Fix > 0 so that for each x ∈ O, there exists a distinguished coordinate chart of radius centered at x. Cover O with ?nitely many such charts (Wα , Gα , πα ), α = 1, . . . , s. (Here we write Gα for GWα and πα for πWα .) Let pα be the center of Wα and pα the center of Wα . Let Uα , respectively Vα , be the geodesic ball of radius 4 , respectively 2 , centered at pα , and let Uα and Vα be the corresponding balls centered at pα in Wα . We may assume that the family of balls {Uα }1≤α≤s still covers O. (m) For each α and each nonnegative integer m, we de?ne Hα : R? + × Wα × Wα by

(m) e,y e) Hα (t, x, y ) = (4πt)?n/2 e?d(x

2 /4t

(u0 (x, y ) + · · · + tm um (x, y ))

where the ui are the invariants in (3.5). Since each γ ∈ Gα is an isometry of Wα , we have ui (γ x, γ y ) = ui (x, y ) for all x, y ∈ Wα . It follows that the function (t, x, y ) →

γ ∈Gα (m) Hα (t, x, γ y )

is Gα -invariant in both x and y and thus descends to a well-de?ned function, which (m) we denote by Hα , on R? + × Wα × Wα . Let ψα : O → R be a C ∞ cut-off function, which is identically one on Vα and is supported in Wα . Let {ηα : α = 1, . . . , s} be a partition of unity on O with the support of ηα contained in Uα . De?ne H (m) : R? + × O × O → R by

s

(3.9)

H (m) (t, x, y ) =

α=1

(m) ψα (x)ηα (y )Hα (t, x, y ).

We will show that

H (m)

is a parametrix for the heat kernel on O when m > n 2.

3.10. L EMMA . H (m) ∈ C ∞ (R? + × O × O ). Lemma 3.10 is immediate. 3.11. L EMMA . Let l be a nonnegative integer. Then ? (i) ?t + ?x H (m) (t, x, y ) extends to a function in C l (R+ × O × O) if m > n 2l 2 + 2l. (It is moreover of class C in the last two variables.) (ii) For any given T > 0 and for each m > n 2 , there exists a constant A such m? n ? ( m ) that | ?t + ?x H (t, x, y )| < At 2 when 0 < t < T .

12

DRYDEN, GORDON, GREENWALD, AND WEBB n 2

Proof. (i) Let l ≥ 0 and suppose m >

? ?t

+ 2l. By Remark 3.7, the function

+

(m) ?x e Hα (t, x, y )

on

R? +

× Wα × Wα extends to C l (R+ × Wα × Wα ).

(Here, we are using the notation ? for the Laplacian on Wα for all choices of α.) (m) ? + ?x Thus the same is true for γ ∈Gα ?t e Hα (t, x, γ y ), and hence it follows

? + ?x Hα (t, x, y ) extends to C l (R+ × Wα × Wα ). that ?t Now consider the function ? (m) (t, x, y )). fα (t, x, y ) := + ?x (ψα (x)ηα (y )Hα ?t (m)

Noting that ψα and ηα are compactly supported inside Wα , we may view fα as a function on R? + × O × O which is zero whenever x or y lies outside of Wα . We show fα extends to C l (R+ × O × O). Since ψα ≡ 1 on Vα , it follows immediately from the previous paragraph that fα extends to C l (R+ × Vα × O). Moreover fα ≡ 0 on R? + × O × (O \ Uα ) and so fα also extends smoothly (to zero) on R+ × O × (O \ Uα ). Finally for (x, y ) ∈ (O \ Vα ) × Uα , we have d(x, y ) ≥ 4 . Thus, as t → 0+ , fα (t, x, y ) and all its derivatives converge to zero uniformly for (x, y ) ∈ (O \ Vα ) × Uα . Thus H (m) extends to a function in C l (R+ × O × O). The parenthetical statement in (1) similarly follows from Remark 3.7. (ii) The ui are constructed so that ? (m) ?n/2 ?d(x e)2 /4t m + ?x e e,y t ?x e Hα (t, x, y ) = (4πt) e um (t, x, y ) ?t (see [2]). Since the ui are C ∞ functions, it follows that there exists a constant Bα such that ? (m) m? n 2 + ?x e Hα (t, x, y ) < Bα t ?t on (0, T ] × Wα × Wα . Consequently,

n ? (m) + ?x Hα (t, x, y ) < |Gα |Bα tm? 2 ?t

on (0, T ] × Wα × Wα and

n ? (m) + ?x (ψα (x)ηα (y )Hα (t, x, y )) < |Gα |Bα tm? 2 ?t

on (0, T ]×Vα ×O, since ψα ≡ 1 on Vα . Once again, for x outside of Vα and y in the (m) ? support of ηα , we have d(x, y ) ≥ 4 , and thus ?t + ?x (ψα (x)ηα (y )Hα (t, x, y )) can be bounded in terms of any power of t on (0, T ) × (O \ Vα ) × O. Statement (ii) now follows from (3.9). 3.12. R EMARK . When m > n 2 +2l, one obtains bounds on the partial derivatives of ? order at most l of ?t + ?x H (m) (t, x, y ) by an argument analogous to that used in the proof of Lemma 3.11(ii).

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

13

3.13. L EMMA . Let m be any nonnegative integer. For f ∈ C ∞ (O) and x ∈ O, we have

t→0+

lim

H (m) (t, x, y )f (y )dy = f (x).

O

I.e., limt→0+ = δx (y ). Moreover, if N is a topological space and f is a continuous function on N × O, then the convergence of (m) (t, x, y )f (p, y )dy to f (p, x) is locally uniform on N × O . OH Proof. Let ψα and ηα be the lifts of ψα and ηα to Wα . Since supp(ηα ) ? Uα ? Wα , we have

(m) ψα (x)ηα (y )Hα (t, x, y )f (p, y ) dy

H (m) (t, x, y )

(3.14)

O

=

ψα (x) | Gα |

(m) Hα (t, x, γ y )ηα (y )fα (p, y )dy γ ∈Gα fα W

where fα is a lift of f|N ×Wα to N × Wα and x is an arbitrarily chosen point in the preimage of x under the map Wα → Wα . We change variables in each of the integrals in the right side of (3.14), letting u = γ (y ). Since γ is an isometry and since ηα and fα (p, ·) are γ -invariant, each integral in the summand is equal to (3.15)

fα W (m) Hα (t, x, u)ηα (u)fα (p, u) du.

As t → 0+ , the integral (3.15) above converges to ηα (x)fα (p, x) = ηα (x)f (p, x) (see Remark 3.7). Moreover, this convergence is locally uniform on N × Wα (see [2]). Noting that ψα ≡ 1 on the support of ηα , it follows that both sides of (3.14) converge to ηα (x)f (p, x) as t → 0+ , and the convergence is locally uniform on N × Wα . Since both sides of (3.14) are identically zero when x lies outside of supp(ψα ) ? Wα , we thus have locally uniform convergence to ηα (x)f (p, x) on all of N × O. Finally it follows from (3.9) that

t→0+

lim

H (m) (t, x, y )f (p, y ) dy =

O α

ηα (x)f (p, x) = f (p, x)

locally uniformly. 3.16. P ROPOSITION . H (m) is a parametrix for the heat operator on O if m > n 2. Proof. Immediate from Lemmas 3.10, 3.11(i), and 3.13. The construction of the heat kernel from the parametrix H (m) follows exactly as in [2]. We give only a brief summary.

14

DRYDEN, GORDON, GREENWALD, AND WEBB

3.17. N OTATION . For A, B ∈ C 0 (R+ × O × O), de?ne the convolution A ? B ∈ C 0 (R? + × O × O ) by

t

A ? B (t, x, z ) =

0

dθ

O

A(t ? θ, x, y )B (θ, y, z )dy.

Note that the convolution operator ? is associative. 3.18. L EMMA . Let l be any nonnegative integer, and let m > n 2 + 2l. De?ne ? ( m ) Fm (t, x, y ) = ?t + ?x H (t, x, y ). (See Lemma 3.11 for regularity properties j +1 F ?j (t, x, y ) converges of Fm .) Then for each T > 0, the series ∞ m j =1 (?1) uniformly on [0, T ] × O × O. Let Qm : R+ × O × O → R be the sum of this series. Then Qm ∈ C l (R+ × O × O). Moreover, for any T > 0, there exists a constant C such that n |Qm (t, x, y )| ≤ Ctm? 2 on [0, T ] × O × O. The proof of Lemma 3.18 is identical to that of Lemma E.III.6 of [2] and uses only Lemma 3.11(ii) and Remark 3.12.

0 (m) ? P , 3.19. L EMMA . Let m > n 2 . For P ∈ C (R+ × O × O ), the function H de?ned formally by the expression in Notation 3.17, exists and is in C 0 (R? +×O× n ( m ) l O). Moreover, if m > 2 +l, then H ?P is of class C in the second variable. For ? (m) ?P (t, x, y )) exists and equals (P +H (m) ?P )(t, x, y ). m> n 2 +2, ?t + ?x (H

Again the proof is identical to that of Lemma E.III.7 of [2] and is based on Lemma 3.13. Using Lemmas 3.18 and 3.19, we obtain as in Proposition E.III.8 of [2] that: 3.20. P ROPOSITION . Let m > n 2 + 2 and de?ne Qm as in Lemma 3.18. Then ( m ) ( m ) K := H ?H ? Qm is a fundamental solution of the heat equation on O. Note that uniqueness of the heat kernel implies that H (m) ? H (m) ? Qm is independent of the choice of m > n 2 + 2. 3.21. N OTATION . Let Hα (t, x, y ) =

γ ∈Gα e,γ (y e)) (4πt)?n/2 e?d(x

2 /4t

(u0 (x, γ (y )) + tu1 (x, γ (y )) + . . . ).

Observe that Hα is Gα -invariant in both x and y and thus descends to a wellde?ned function, which we denote by Hα , on R? + × Wα × Wα . 3.22. T HEOREM . In the notation of Proposition 3.1 and 3.21, the trace of the heat kernel has an asymptotic expansion as t → 0+ given by

∞ s

e?λj t ?t→0+

j =1 α=1 O

ηα (x)Hα (t, x, x) dx.

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

15

Proof. By Corollary 3.3 and Proposition 3.20, we have

∞

e?λj t =

j =1 O

(H (m) ? H (m) ? Qm )(t, x, x)dx

n

(m) (t, x, x) is uniformly 2 for m > n 2 +2. By Lemma 3.18 and the fact that (4πt) H bounded for (t, x) ∈ (0, T ] × O for any given T > 0, it follows that ∞

(3.23)

j =1

e?λj t =

O

H (m) (t, x, x)dx + O(tm?n )

on any interval (0, T ]. (Aside: When O is a manifold, then O H (m) (t, x, x)dx = n (4πt)? 2 (a0 + a1 t + · · · + am tm ). For general orbifolds, the arguments in the j n m 2 next section will show that O H (m) (t, x, x)dx is of the form (4πt)? 2 2 j =0 cj t . Thus the error term can be improved to O(tm? 2 ) since O H (m) (t, x, x)dx = 1 (m+n) (t, x, x)dx + O (tm? n? 2 ).) OH Since ψα is identically one on the support of ηα , Notation 3.8 yields (3.24) H (m) (t, x, x) =

α (m) ηα (x)Hα (t, x, x).

n?1

Substituting (3.24) into (3.23), we obtain the theorem. 4. C OMPUTATION OF THE HEAT ASYMPTOTICS 4.1. N OTATION AND R EMARKS . Let γ be an isometry of a Riemannian manifold M and let ?(γ ) denote the set of components of the ?xed point set of γ . Each element of ?(γ ) is a submanifold of M . For each non-negative integer k , Donnelly [10] de?ned a real-valued function, which we temporarily denote bk ((M, γ ), ·), on the ?xed point set of γ . For each W ∈ ?(γ ), the restriction of bk ((M, γ ), ·) to W is smooth. Two key properties of the bk are: ? (Locality) For a ∈ W , bk ((M, γ ), a) depends only on the germs at a of the Riemannian metric of M and of the isometry γ . In particular, if U is a γ -invariant neighborhood of a in M , then bk ((M, γ ), a) = bk ((U, γ ), a). ? (Universality) If M and M are Riemannian manifolds admitting isometries γ and γ , respectively, and if σ : M → M is an isometry satisfying σ ?γ = γ ?σ , then bk ((M, γ ), x) = bk ((M , γ ), σ (x)) for all x ∈ Fix(γ ). In view of the locality property, we will usually delete the explicit reference to M and rewrite these functions as bk (γ, ·), as they are written in [10]. 4.2. C OMPUTATION OF THE bk . [10] In the notation of 4.1, let W ∈ ?(γ ), and let n = dim(M ) and m = dim(W ). For x ∈ W , the orthogonal complement Tx (W )⊥ of Tx (W ) in the tangent space Tx (M ) is invariant under γ? . De?ne Aγ (x) = γ? : Tx (W )⊥ → Tx (W )⊥ , and observe that Aγ (x) is nonsingular. Set Bγ (x) = (I ? Aγ (x))?1 .

16

DRYDEN, GORDON, GREENWALD, AND WEBB

Donnelly showed that bk (γ, x) = | det(Bγ (x))|bk (γ, x), where bk (γ, ·) is an O(m) × O(n ? m) universal invariant polynomial in the components of Bγ and in the curvature tensor R of M and its covariant derivatives. Explicit formulae for b0 and b1 are given in [10, Thm. 5.1] using the following indexing conventions: 1 ≤ α, β ≤ m, m + 1 ≤ i, j, k ≤ n and 1 ≤ a, b, c ≤ n. At each point x ∈ W , choose an orthonormal basis {e1 , . . . , en } of Tx (M ) so that the ?rst m vectors are tangent to W . The sign convention on the curvature tensor R of M is chosen so that Rabab is the sectional curvature of the plane spanned by ea and eb . Set

n

τ=

a,b=1

Rabab

n

and ρab =

Racbc .

c=1

Thus τ is the scalar curvature and ρ the Ricci tensor of M . Then (4.3) b0 (γ, x) = | det(Bγ (x))| 1 1 1 b1 (γ, x) = | det(Bγ (x))|( τ + ρkk + Riksh Bki Bhs 6 6 3 1 + Rikth Bkt Bhi ? Rkaha Bks Bhs ). 3

and, summing over repeated indices, (4.4)

4.5. N OTATION . Let O be an orbifold and let (U , GU , πU ) be an orbifold chart. In the notation of 2.12 and 2.14, let N be a U -stratum and let γ ∈ Isomax (N ). Then N is an open subset of a component of Fix(γ ) and thus by 4.1, bk (γ, ·) (= bk ((U , γ ), ·)) is smooth on N for each nonnegative integer k . De?ne a function bk (N , ·) on N by bk (N , x) =

e) γ ∈Isomax (N

bk (γ, x).

4.6. L EMMA . Let O be a Riemannian orbifold, let N be an O-stratum and let p ∈ N . Let (U , GU , πU ) and (U , GU , πU ) be two orbifold charts with p ∈ U ∩ U . Let p ? ∈ U and p ? ∈ U with πU (? p) = p = πU (? p ), and let N , respectively N , be the U -stratum through p ?, respectively U -stratum through p ? . Then for each k , we have bk (N , p ?) = bk (N , p ? ). Proof. By De?nition 2.1, it suf?ces to consider the case that one chart embeds in the other, say λ : (U , GU , πU ) → (U , GU , πU ) is an isometric embedding p) with λ(? p) = p ? . The associated homomorphism τ : GU → GU carries IsoGU (? max max isomorphically onto IsoGU (? p ) and Iso (? p) to Iso (? p ). The U -stratum N is

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

17

carried to an open subset of the U -stratum N . The lemma is thus an immediate consequence of the universality of the bk , as discussed in 4.1. 4.7. D EFINITION . Let O be a Riemannian orbifold and let N be an O-stratum. (i) For each non-negative integer k , de?ne a real-valued function bk (N, ·) by setting bk (N, p) = bk (N , p ?) where (U , GU , πU ) is any orbifold chart about p, ?1 p ? ∈ πU (p) and N is the U -stratum through p ?. By Lemma 4.6, the function bk (N, ·) is well-de?ned. (ii) The Riemannian metric on O induces a Riemannian metric, and thus a volume element, on the manifold N . Set

∞

IN := (4πt)

? dim(N )/2 k=0

tk

N

bk (N, x)d volN (x)

where d volN is the Riemannian volume element. (iii) Also set

∞

I0 = (4πt)? dim(O)/2

k=0

ak (O)tk

where the ak (O) (which we will usually write simply as ak ) are the familiar heat invariants. More precisely, the invariants ui in (3.5), which are de?ned in terms of the curvature and its covariant derivatives on any Riemannian manifold, also make sense on any Riemannian orbifold. The invariants ak (O) are given by ak = 1 O uk (x, x)d volO (x). In particular, a0 = vol(O ), a1 = 6 O τ (x)d volO (x), etc. Note that if O is ?nitely covered by a Riemannian manifold M , say O = G\M , 1 then ak (O) = |G | ak (M ). 4.8. T HEOREM . Let O be a Riemannian orbifold and let λ1 ≤ λ2 ≤ . . . be the spectrum of the associated Laplacian acting on smooth functions on O. The heat ?λj t of O is asymptotic as t → 0+ to trace ∞ j =1 e I0 +

N ∈S (O )

IN |Iso(N )|

where S (O) is the set of all O-strata and where |Iso(N )| is the order of the isotropy at each p ∈ N as de?ned in Remark 2.7. This asymptotic expansion is of the form

∞

(4.9) for some constants cj .

(4πt)? dim(O)/2

j =0

cj t 2

j

4.10. R EMARK . Suppose O = G\M is a good closed orbifold. Note that M may be noncompact and G may be an in?nite group, although the isotropy group at any point of M must be a ?nite subgroup of G. In this setting, Donnelly [11] proved the existence and uniqueness of the heat kernel K M on M and of an asymptotic expansion for K M . He then obtained an asymptotic expansion for the heat trace on O. Theorem 4.8, in the case of good orbifolds, organizes the information in [11] in a way that clari?es the contribution of each O-stratum to the asymptotics.

18

DRYDEN, GORDON, GREENWALD, AND WEBB

The expression for the heat asymptotics of good orbifolds in [11] differs from that in (4.9) in that the half powers are missing. However, the absence of these powers is apparently a typographical error in transcribing a result of Donnelly’s earlier paper [10], stated below as Proposition 4.11. K. Richardson [24] obtained an asymptotic expansion for the heat trace associated with the basic Laplacian on a Riemannian foliation. Also referring to Donnelly’s work on good orbifolds, he showed that the expansion is of the form given in (4.9). The remainder of this section is devoted to the proof of Theorem 4.8. 4.11. P ROPOSITION . [10]. Let M be a closed Riemannian manifold, let K (t, x, y ) be the heat kernel of M , and let γ be a nontrivial isometry of M . Then, in the notation of 4.1, M K (t, x, γ (x))d volM (x) is asymptotic as t → 0+ to (4πt)

W ∈?(γ ) ?

dim(W ) 2

∞

tk

k=0 W

bk (γ, a)d volW (a)

where d volW is the volume form on W de?ned by the Riemannian metric induced from M . 4.12. P ROOF OF T HEOREM 4.8 IN SPECIAL CASE . We prove Theorem 4.8 for O = G\M a good closed orbifold with G ?nite (and thus M compact). In particular, (M, G, π ) is a global orbifold chart where π : M → O is the projection. In this case, the theorem is an easy consequence of Proposition 4.11. Indeed, letting K denote the heat kernel of M and letting π : M → O be the projection, then the heat kernel K O of O is given by K O (t, x, y ) =

γ ∈G

K (t, x, γ (y ))

where x, respectively y , are any elements of π ?1 (x), respectively π ?1 (y ). Thus K O (t, x, x)d volO (x) =

O

1 | G|

K (t, x, γ (x))d volM (x),

γ ∈G M

so Proposition 4.11 implies that K O (t, x, x)d volO (x) ?t→0+

O

1 |G|

∞

K (t, x, x)d volM (x)

M

(4.13)

1 + |G|

(4πt)

1=γ ∈G W ∈?(γ )

?

dim(W ) 2

tk

k=0 W

bk (γ, a)d volW (a).

The ?rst term on the right-hand-side of (4.13) is given by (4.14) 1 |G|

dim(M ) 1 (4πt)? 2 K (t, x, x)d volM (x) = | G|

∞

ak (M )tk = I0 .

k=0

M

(See the ?nal comment in De?nition 4.7(iii).)

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

19

Next let 1 = γ ∈ G, let W ∈ ?(γ ), and let N be an M -stratum contained in W . Then either N has measure zero in W (in which case γ ∈ / Isomax (N )) or else max N is open in W and γ ∈ Iso (N ). Thus by replacing the integral over W with the integrals over the M -strata that are open in W , reordering the summations in (4.13), and taking note of (4.14), we obtain (4.15)

O

K O (t, x, x)d volO (x) ?t→0+ I0 +

1 |G|

e ∈S e(M ) N

IN e

where S (M ) denotes the set of all M -strata and where IN e = (4πt)

?

e) dim(N 2

∞

tk

k=0 e N

bk (N , a)d volN e (a).

Let N be an O-stratum. Then π ?1 (N ) is a union of ?nitely many mutually isometric M -stratum N1 , . . . , Nk and π : π ?1 (N ) → N is a covering map of |G| degree |Iso( N )| . We have IN e1 + · · · + IN ek = |G| IN |Iso(N )|

and thus Theorem 4.8, in the case of orbifolds ?nitely covered by manifolds, follows from (4.15). The proof in the general case will apply the argument in 4.12 to orbifold charts and then piece the computations together via a partition of unity. We ?rst generalize Proposition 4.11 slightly. The manifolds in the two lemmas below do not have boundaries but could, for example, be bounded domains in a larger manifold. 4.16. L EMMA . We use the notation of 4.1. Let M be an n-dimensional Riemannian manifold (without boundary) of ?nite volume and let γ : M → M be a nontrivial isometry. Assume that the distance d(x, γ (x)) remains bounded away from zero off arbitrarily small tubular neighborhoods of the ?xed point set of γ and that each component of the ?xed point set of γ has ?nite volume. Then as t → 0+ ,

e,γ (x e)) (4πt)?n/2 e?d(x M ∞

2 /4t

(u0 (x, γ (x)) + tu1 (x, γ (x)) + . . . ) dx tk

k=0 W

?

(4πt)? dim(W )/2

W ∈?(γ )

bk (γ, x)d volW (x)

This result is proven in Donnelly [10, Thm. 4.1], in case M is closed. In that case, of course, the hypotheses on the distance function and on the ?xed point set of γ are automatic, and the lemma is a restatement of Proposition 4.11. The proof goes through verbatim in the more general setting of Lemma 4.16. 4.17. L EMMA . With the notation and hypotheses of the previous lemma, let η be a smooth bounded γ -invariant function on M . Then there exists a family of functions

20

DRYDEN, GORDON, GREENWALD, AND WEBB

ck (γ, η, ·), k = 0, 1, 2, . . . , de?ned on the ?xed point set of γ and smooth on each component W ∈ ?(γ ), such that as t → 0+ ,

e,γ (x e)) η (x)(4πt)?n/2 e?d(x M ∞

2 /4t

(u0 (x, γ (x)) + tu1 (x, γ (x)) + . . . ) dx tk

k=0 W

?

W ∈?(γ )

(4πt)? dim(W )/2

ck (γ, η, x)d volW (x).

Moreover, ck (γ, η, ·) satis?es the following: (i) (Locality) ck (γ, η, x) depends only on the germs of γ , η , and the Riemannian metric of M at x ∈ W ; (ii) ck (γ, η, ·) is zero off supp(η ) ∩ W ; (iii) the dependence of ck (γ, η, ·) on η is linear; (iv) ck (γ, 1, ·) = bk (γ, ·) where 1 denotes the constant function η ≡ 1; (v) (Universality) if M is another Riemannian manifold, γ is an isometry of M and σ : M → M is an isometry satisfying σ ? γ = γ ? σ , then ck (γ , η ? σ ?1 , σ (x)) = ck (γ, η, x) for all x in the ?xed point set of γ . The proof requires only minor changes in the proof of Theorem 4.1 of [10]. In the proof of that theorem, the functions bk (γ, ·) are expressed as linear combinations of certain derivatives of explicitly de?ned functions hj , j = 0, . . . , k . To obtain the functions ck (γ, η, ·), one replaces the functions hj by the functions ηhj . 4.18. N OTATION AND R EMARKS . Let O be a closed orbifold and consider the charts Uα , Vα , Wα and partition of unity {ηα } given in Notation 3.8. Let Vα play the role of M in Lemmas 4.16 and 4.17, and let ηα = ηα ? πα play the role of η . Since Vα has compact closure inside the larger Riemannian manifold Wα on which Gα acts by isometries and since the ?xed point set of γ in Wα is connected (in fact, it is the union of a collection of geodesics radiating from the center point p ?α ), one easily veri?es for each γ ∈ Gα that the hypothesis concerning the distance function in the two lemmas holds. (i) For each Vα -stratum N , de?ne a smooth function ck,α (N , ·) on N by ck,α (N , x) =

e) γ ∈Isomax (N

ck (γ, ηα , x).

(ii) Let N be an O-stratum. For each non-negative integer k and each α = 1, . . . , s, de?ne a continuous (in fact, smooth) function ck,α (N, ·) on N as follows: First for x ∈ N ∩ Vα , set ck,α (N, x) = ck (N , x) where ?1 (x) and N is the V -stratum through x. By an x is any element of πα α argument analogous to that of Lemma 4.6. this de?nition is independent ?1 (x). Since η is supported in U , Lemma 4.17(ii) of the choice of x in πα α α implies that ck,α (N, ·) is zero off N ∩ Uα and thus extends to a continuous function on N which is zero off N ∩ Uα .

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

21

(iii) Set

∞

IN,α := (4πt)? dim(N )/2

k=0

tk

N

ck,α (N, x)d volN (x).

4.19. L EMMA . Let O be a closed Riemannian orbifold and let N be an O-stratum. Then for each non-negative integer k , we have

s

ck,α (N, ·) = bk (N, ·)

α=1

and

s

IN,α = IN .

α=1

Proof. Let x ∈ N , and let α1 , · · · , αr be those α ∈ {1, . . . , s} for which x ∈ Vαi . Then we can ?nd a coordinate chart (U , GU , πU ) such that U ? Vα1 ∩ · · · ∩ Vαr and such that the chart (U , GU , πU ) embeds in each of the charts (Vα , Gα , πα ). Let ?1 (x), let N be the U -stratum through λi : U → Vαi be the embedding. Let x ∈ πU x, and let xi = λi (x). As in the proof of Lemma 4.6, λi (N ) is an open subset of the Vαi -stratum Ni through xi . Using the universality property (v) of Lemma 4.17, Notation 4.18, and an argument analogous to that of Lemma 4.6, we see that ck,αi (N, x) := ck,αi (Ni , xi ) =

e) γ ∈Isomax (N

ck (γ, ηαi ? πU , x).

Thus, since ck,α (N, x) = 0 when α is not one of α1 , . . . , αr , we have

s r

(4.20)

α=1

ck,α (N, x) =

e ) i=1 γ ∈Isomax (N

ck (γ, ηαi ? πU , x).

r

From properties (iii) and (iv) of Lemma 4.17 and the fact that

i=1

ηαi ≡ 1 on U ,

we see that

r

(4.21)

i=1

ck (γ, ηαi ? πU , x) = bk (γ, x)

on N . By De?nition 4.7, bk (γ, x) = bk (N, x)

e) γ ∈Isomax (N

and thus the ?rst equation in the lemma follows from (4.20) and (4.21). The second equation is then immediate. We now prove Theorem 4.8.

22

DRYDEN, GORDON, GREENWALD, AND WEBB

Proof. Let n = dim(O). By Theorem 3.22 and the fact that the support of ηα is contained in Vα , we have

∞ s

(4.22)

j =1

e

? λj t

?t→0+

α=1 Vα

ηα (x)Hα (t, x, x)d vol(O)

where Hα (t, x, x) is de?ned in Notation 3.21. By Notation 3.21,

s

ηα (x)Hα (t, x, x) d volO (x) =

α=1 s Vα

(4.23)

α=1

1 |Gα |

s

eα V

ηα (x)(4πt)? 2 (u0 (x, x) + tu1 (x, x) + . . . ) d volV eα (x)

e,γ (x e)) ηα (x)(4πt)?n/2 e?d(x

2 /4t

n

+

α=1

1 |Gα |

(u0 (x, γ (x))

1=γ ∈Gα

eα V

+ tu1 (x, γ (x)) + . . . ) d volV eα (x) where ηα = ηα ? πα . Consider the ?rst sum on the right-hand-side of (4.23). Since ηα is supported in Vα , we have

s α=1 s

n 1 (4πt)? 2 | Gα | n

eα V

ηα (x)(u0 (x, x) + tu1 (x, x) + . . . ) d volV eα (x)

(4.24) =

(4πt)? 2

α=1

ηα (x)(u0 (x, x) + tu1 (x, x) + . . . )d volO (x)

O

= (4πt)? 2

n

(u0 (x, x) + tu1 (x, x) + . . . )d volO (x) = I0 .

O

Next by Lemma 4.17 and the remarks in 4.18, we have for each 1 = γ ∈ Gα , 1 e,γ (x e))2 /4t ηα (x)(4πt)?n/2 e?d(x (u0 (x, γ (x))+ |Gα | eα V (4.25)

1=γ ∈ ga

tu1 (x, γ (x)) + . . . ) d volV eα (x) ? 1 |Gα |

∞

(4πt)? dim(W )/2

1=γ ∈Gα W ∈?(γ ) k=0

tk

W

ck (γ, ηα , x)d volW (x).

By 4.18 and an argument identical to that in 4.12, the right-hand-side of (4.25) is equal to 1 IN,α . |Iso(N )|

N ∈S (O )

Consequently, Lemma 4.19 and (4.25) imply that the second sum in the right-handside of (4.23) is equal to 1 IN . |Iso(N )|

N ∈S (O)

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

23

Thus in view of (4.22), (4.23), and (4.24), the theorem is proved. 5. A PPLICATIONS 5.1. T HEOREM . Let O be a Riemannian orbifold with singularities. If O is even dimensional (respectively, odd dimensional) and some O-stratum of the singular set is odd dimensional (respectively, even dimensional), then O cannot be isospectral to a Riemannian manifold. Proof. It is clear from (4.3) that if N is any O-stratum of the singular set, then the function b0 (γ, ·) is strictly positive on N for each γ ∈ Isomax (N ). Thus, in the two cases, the fact that O is an orbifold can be gleaned from the presence of half-integer powers, respectively integer powers, of t in the asymptotic expansion in Theorem 4.8. 5.2. R EMARK . In [18] this theorem was stated for good orbifolds, but here we also include bad orbifolds. We now restrict our attention to closed two-dimensional orbifolds (2-orbifolds). The singularities which may occur in 2-orbifolds are cone points, dihedral corner re?ectors, and mirror re?ectors. Recall that dihedral corner re?ectors and mirror re?ectors both appeared in Example 2.15. A cone point p of order n is an isolated singularity; an orbifold chart for a neighborhood of p is (D2 , Zn , π ) where D2 is an open 2-disk in R2 and Zn is the cyclic group of order n. The Euler characteristic of a 2-orbifold is 2 minus the sum of the related values: each cone point of order n?1 1 n has value n? n ; each dihedral corner re?ector has value 2n ; each handle has value 2; each cross-cap has value 1; and each mirror re?ector has value 1. Every good 2-orbifold admits a (metrically) spherical, Euclidean or hyperbolic structure depending on whether the Euler characteristic is positive, zero or negative, respectively [30]. In addition, all bad 2-orbifolds have positive Euler characteristic. 5.3. E XAMPLE . Let O be a 2-orbifold and let p be a cone point of order m. If N = {p}, then Iso(N ) is a cyclic group of order m and Isomax (N ) contains all of the nontrivial elements. Letting γ be the generator, then for j = 1, . . . , m ? 1 we have that jπ jπ ) ? sin( 2m ) cos( 2m j , Aγ j = γ? = 2jπ 2jπ sin( m ) cos( m ) where Aγ j is as de?ned in 4.2. Thus b0 (γ j ) = |det((I ? Aγ j )?1 )| = 1 2?

jπ 2 cos( 2m )

=

1 4 sin2 ( jπ m)

.

(We are writing b0 (γ j ) rather than using the function notation b0 (γ j , ·) since N consists of a single point.) 5.4. L EMMA .

m?1 j =1

1 sin2 ( jπ m)

=

m2 ? 1 . 3

24

DRYDEN, GORDON, GREENWALD, AND WEBB

Proof. A well-known formula (see, for example, [23, Example 7.9.1]), proven by the calculus of residues, states that π2 1 . = 2 sin (πz ) k=?∞ (k ? z )2 Thus

m?1 ∞

1

sin2 ( jπ m) j =1 Since

m?1 ∞

=

m2 π2

m?1

∞

j =1 k=?∞ ∞ n=1

1 . (mk ? j )2

∞ n=1

j =1 k=?∞

1 =2 (mk ? j )2

1 ?2 n2

1 m2 n2

and

∞ 1 n=1 n2

=

π2 6 ,

the lemma follows.

5.5. P ROPOSITION . Let O be a 2-orbifold, let p be a cone point in O of order m and let N = {p}. Then in the notation of Theorem 4.8, we have IN = Proof. By 4.7(ii) and 5.3,

m?1

m2 ? 1 + O(t). 12 1 4 sin2 ( jπ m)

IN =

j =1

+ O(t).

Thus Proposition 5.5 follows from Lemma 5.4. 5.6. E XAMPLE . Calculating heat invariants for 2-orbifolds. Degree zero term for orientable 2-orbifolds. An orientable 2-orbifold O can have only isolated singularities, i.e., cone points. Suppose O has k cone points of orders m1 , . . . , mk . In the notation of 4.7(iii), 1 (a0 t?1 + a1 + O(t)). I0 = 4π Thus by Theorem 4.8 and Proposition 5.5, the term of degree zero in the asymptotic expansion in Theorem 4.8 is given by a1 + 4π

k i=1

1 m2 i ?1 . mi 12

By the Gauss-Bonnet Theorem (valid also for orbifolds; see [26, 30]), we have 2π a1 = χ( O ) . 3 Hence the degree zero term is χ(O) + 6

k i=1

(5.7)

m2 i ?1 . 12mi

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

25

Degree zero term for nonorientable 2-orbifolds. For a 2-orbifold O, the dimension zero singular locus is the only portion which contributes to the degree χ(O) a1 zero term of the asymptotic expansion in Theorem 4.8, aside from the 4 π = 6 component. A nonorientable 2-orbifold O can have cone points and/or dihedral corner re?ector points that contribute in the following ways. As in the computation of the degree zero term for orientable 2-orbifolds, here a simple cone point of order 1 m2 ?1 m contibutes m 12 . Let N = {p}, where p is a corner re?ector point created by a rotation of order n and a re?ection. Then |Iso(N )| = 2n. By Notation 2.14(ii), Isomax (N ) contains only the nontrivial elements of the rotation group, since the re?ection ?xes one-dimensional strata of the mirror locus, a higher dimensional stratum of the singular set. Hence the computations in Example 5.3 and Proposition 5.5 remain the same, but the difference in |Iso(N )| is seen as an extra 1 2 factor 1 n2 ?1 in Theorem 4.8. The point contributes 2n 12 to the degree zero term. Thus, for a 2-orbifold O with cone points p1 , . . . , pk of orders m1 , . . . , mk , and dihedral corner re?ector points q1 , . . . , qr of orders n1 , . . . , nr , the term of degree 0 in the asymptotic expansion of the heat trace is (5.8) χ(O) + 6

k i=1

1 m2 i ?1 + mi 12

r j =1

1 n2 j ?1 . 2nj 12

Degree one term for 2-orbifolds. Aside from a2 , only dimension zero strata of the singular set contribute to the t term of the asymptotic expansion in Theorem 4.8. We ?rst note that the last term in (4.4) is zero; this follows from the summation convention and the fact that our singular set is zero-dimensional. Using symmetry properties of the curvature, we can further simplify (4.4) as b1 (γ j ) = 1 4 sin2 ( jπ m) τ ρkk 2 2 2 + + (R1212 (B21 + B12 ? B12 B21 ? B22 B11 )) , 6 6 3

where R1212 is evaluated in the local covering manifold. By the de?nitions of scalar and Ricci curvature, the preceding equation becomes b1 (γ j ) =

2 + B2 ? B B ? B B ) R1212 (1 + B21 12 21 22 11 12

6 sin2 ( jπ m)

1 2

jπ sin( 2m )

.

In general, straightforward calculations show that ? ? Bγ j = (I ? Aγ j )?1 = ? which implies b1 (γ j ) = R1212 8 sin4 ( jπ m) ,

jπ sin( 2m ) jπ 2?2 cos( 2m )

?

? ?

jπ 2?2 cos( 2m )?

1 2

for j = 1, . . . , m ? 1. Thus, for a 2-orbifold O with cone points p1 , . . . , pk of orders m1 , . . . , mk , and dihedral corner re?ector points q1 , . . . , qr of orders n1 , . . . , nr , the coef?cient of

26

DRYDEN, GORDON, GREENWALD, AND WEBB

the term of degree 1 in the asymptotic expansion of the heat trace is ? ? ? ? m ?1 n ?1 k r a2 R1212 ? R1212 ? 1 ? i 1 ? i (5.9) + + . jπ 4π mi 2ni ) 8 sin4 ( m 8 sin4 ( jπ n )

i=1 j =1

i

i=1

j =1

i

1 2 2 2 Recall that a2 (O) = 360 O (2|R| ? 2|ρ| + 5τ )d volO (g ), where R is the curvature, ρ is the Ricci curvature, and τ is the scalar curvature of O (e.g. [2]). We can further simplify (5.9) by making the substitution mi ?1 j =1

1

jπ ) sin4 ( m i

=

2 m4 i + 10mi ? 11 . 45

See [3] and the references therein or [7, 16] for evaluations of this and similar ?nite trigonometric sums. With this substitution, (5.9) becomes (5.10) a2 + 4π

k i=1 2 R1212 (m4 i + 10mi ? 11) + 360mi r i=1 2 R1212 (n4 i + 10ni ? 11) . 720ni

Degree ? 1 2 term for 2-orbifolds. The only O -strata that contribute to the de1 gree √t term are those of codimension one in O. To obtain these O-strata, remove any dihedral points from the mirror locus and then take the connected components of the remaining set. Let x ∈ N , an O-stratum of the mirror locus, and note that γ ∈ Isomax (N ) must act as a re?ection. To compute b0 (γ, x) = |det((I ? Aγ )?1 )|, notice that on the normal bundle to N , γ? = [?1], and so, 1 b0 (γ, x) = |det((I ? Aγ )?1 )| = |[2]?1 | = . 2 Applying 4.7(ii),

∞

IN = (4πt)?1/2

γ ∈Isomax (N ) k=0

tk

N

1 d volN (x) + O(t) 2

=

√ length(N ) 1 √ √ + O( t). 4 π t

1 √ t

We sum over all O-strata of the mirror locus to obtain the coef?cient of the in Theorem 4.8: (5.11)

N

term

IN = |Iso(N )|

N

1 length(N ) length(M irrorLocus(O)) √ √ = . 2 4 π 8 π

1 Degree 2 term for 2-orbifolds. For a 2-orbifold O, the dimension one singular √ locus gives the sole contribution to the t term of the asymptotic expansion in Theorem 4.8.

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

27

We have τ ρkk 1 b1 (γ, x) = |det(Bγ (x))|( + + Riksh Bki Bhs 6 6 3 1 + Rikth Bkt Bhi ? Rkαhα Bks Bhs )(x) 3 where Bij denotes the i, j entry of Bγ (x), τ is the scalar curvature of M at x and ρ is the Ricci curvature. In the case of a 2-orbifold with a point x in its mirror locus (but not a dihedral corner re?ector point), the matrix Bγ (x) is one-dimensional, and thus the third and 1 fourth terms in the sum vanish. Since Bγ (x) = 2 , the last term is ? 1 4 R1212 . We 1 1 also have ρkk = R1212 , while τ = 2R1212 . Thus b1 (γ, x) = 2 ( 4 R1212 )(x) = 1 1 8 R1212 (x) = 16 τ (x). Thus for N a stratum in the mirror locus, the term of degree 1 2 in IN is given by √ √ t 1 t √ τ (x)d volN (x) = √ τ (x)d volN (x). 32 π N 4π N 16 The coef?cient of the degree on O is thus (5.12)

1 2

term in the asymptotic expansion for the heat trace τ

M irrorLocus(O)

1 √ 64 π

where the scalar curvature is the scalar curvature of O computed at points in the mirror locus and the integral is with respect to the induced Riemannian metric on the one-dimensional mirror locus. See Table 1 for the asymptotic expansions of the heat kernel for orbifolds O with χ(O) ≥ 0. Our notation for orbifolds is adapted from Conway’s convention [9], with commas added for readability. Namely O(a, b, ?c, d) denotes an orbifold with simple cone points of orders a and b, and dihedral corner re?ectors of orders 2c and 2d. In addition, O(n×) is a disk with a simple interior cone point and the edge identi?ed via the antipodal map, while O(n?) is a disk with a simple interior cone point and a mirror edge corresponding to a re?ection. A detailed explanation of the orbifold notation can be found in [9] and compared with pictures in [21, pages 80–90]. A proof that these are all of the closed 2-orbifolds with χ(O) ≥ 0 and that the only bad 2-orbifolds are O(m), O(?m), O(m, n) and O(?m, n) (when m > 1 and m = n), can be found in [30]. Let O be an orientable 2-orbifold with k cone points of orders m1 , . . . , mk , denoted O(m1 , . . . , mk ), and consider the quantity c de?ned as 12 times the degree zero term:

k

(5.13)

c = 2χ(O) +

i=1

mi ?

1 mi

.

This quantity is a spectral invariant; note that it depends only on the topology, not on the Riemannian metric. For O(m1 , . . . , mk ), we denote by c(m1 , . . . , mk ) the associated spectral invariant. We now investigate classes of orientable 2-orbifolds for which c is a complete topological invariant. Although Theorem 5.14 is a special

28

DRYDEN, GORDON, GREENWALD, AND WEBB

TABLE 1. 2-orbifold expansions with χ(O) ≥ 0 O with χ(O ) > 0 O(m) O(?m) O(m, n) O(?m, n) O(m×) O(m?) O(2, 2, m) O(?2, 2, m), O(2, ?m) O(2, 3, 3) O(?2, 3, 3), O(3, ?2) O(2, 3, 4) O(?2, 3, 4) O(2, 3, 5) O(?2, 3, 5) O with χ(O ) = 0 torus, Klein bottle *torus, *Klein bottle O(2, 2, 2, 2) O(?2, 2, 2, 2), O(2, ?2, 2), O(2, 2?) O(2, 2×) O(2, 4, 4) O(?2, 4, 4), O(4, ?2) O(3, 3, 3) O(?3, 3, 3), O(3, ?3) O(2, 3, 6) O(?2, 3, 6) Asymptotic Expansion 1 1 V1 t + 12 (2 + m + m ) + O (t) √ 1 1 1 1 ) + O ( t) V t + M L √t + 24 (2 + m + m 1 1 1 V1 t + 12 (m + n + m + n ) + O (t) √ 1 1 1 1 √ V1 t + M L t + 24 (m + n + m + m ) + O ( t) 1 1 V1 t + 12 (m + m ) + O (t) √ 1 1 1 1 (m + m ) + O ( t) V t + M L √t + 12 1 1 V1 t + 12 (3 + m + m ) + O (t) √ 1 1 1 1 V t + M L √t + 24 (3 + m + m ) + O ( t) 43 V1 t + 72 + O (t) √ 1 43 √ V1 t + M L t + 144 + O ( t) 97 V1 t + 144 + O (t) √ 1 1 97 V t + ML√ + 288 + O( t) t 271 V1 t + 360 + O (t) √ 1 271 1 V t + ML√ + + O ( t) 720 t Asymptotic Expansion V1 t + O (t) √ 1 √ V1 t + M L t + O ( t) 1 V1 t + 2 + O (t) V V V V V V V V

1 t 1 t 1 t 1 t 1 t 1 t 1 t 1 t 1 + ML√ + t 1 + 4 + O(t) +3 4 + O (t) 1 + ML√ + t 2 + 3 + O(t) 1 + ML√ + t 5 + 6 + O(t) 1 + ML√ + t 1 4

√ + O ( t) √ + O ( t) √ + O ( t) √ + O ( t)

3 8 1 3

5 12

O) (O )) √ Here m, n ≥ 1, V = vol( and M L = length(M irrorLocus . Note that *torus is an annulus 4π 8 π with two mirror re?ector edges and *Klein bottle is a M¨ obius band with one mirror re?ector edge.

case of Theorem 5.15, we begin with the more restricted class in order to give the reader intuition for the proof techniques used. 5.14. T HEOREM . Within the class of all footballs (good or bad) and all teardrops, the spectral invariant c is a complete topological invariant. I.e., c determines whether the orbifold is a football or teardrop and determines the orders of the cone points.

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

29

TABLE 2. Triangular Pillow Orbifolds O χ(O) 1 O(2, 2, 2) 2 1 O(2, 2, m) m 1 O(2, 3, 3) 6 1 O(2, 3, 4) 12 1 O(2, 3, 5) 30 χ(O) = 0 O(3, 3, 3) 0 O(2, 4, 4) 0 O(2, 3, 6) 0 1 χ(O) < 0 O(3, 3, 4) ? 12 O(3, 4, 4) ?1 6 2 O(3, 3, 5) ? 15 1 O(2, 4, 5) ? 20 . . . ?1 < χ(O) < 0 χ(O) > 0 c(O) 1 52 3+m+ 1 76 1 8 12 1 9 30 8 9 10 8 11 12 5 96 13 9 15 9 19 20

1 m

c(O) > 10

Proof. Denote by O(m) the teardrop with cone point of order m and by O(r, s) the football with cone points of orders r and s. Let c(m) and c(r, s) denote the invariant de?ned in (5.13) in the two cases. Then O(m) has Euler characteristic 1 and thus 1+ m 1 c(m) = 2 + m + . m 1 + The football O(r, s) has Euler characteristic 1 r s , so 1 1 + . r s The invariant is an integer only in the case of O(2, 2), so the football O(2, 2) is spectrally distinguishable from the other footballs and teardrops. Thus for the remainder of the proof, all footballs will be assumed to have at least one cone point of order strictly greater than 2. For teardrops, c(m) trivially determines m. We next claim that footballs are distinguishable from teardrops. Indeed, suppose that c(m) = c(r, s). Then m + 1 1 2 = r + s and m =1 r + s . The latter equation implies that m < min(r, s). Since also r, s ≥ 2, we have 2 + m < r + s, a contradiction, thus proving the claim. It remains only to show that for footballs, c(r, s) determines r and s. From 1 r+s c(r, s), one can read off the quantities r + s and 1 r + s = rs . Hence c(r, s) determines both r + s and rs, thus also |r ? s|, since (r ? s)2 = (r + s)2 ? 4rs. Hence (r, s) is determined up to order, completing the proof. c(r, s) = r + s + 5.15. T HEOREM . Let C be the class consisting of all closed orientable 2-orbifolds with χ(O) ≥ 0. The spectral invariant c is a complete topological invariant within C and moreover, it distinguishes the elements of C from smooth oriented closed surfaces.

30

DRYDEN, GORDON, GREENWALD, AND WEBB

Proof. We ?rst consider the 2-orbifolds for which c is an integer. Note that among teardrops, footballs, and triangular pillows, the only integer values are c(2, 2) = 5, c(2, 3, 6) = 10, c(2, 4, 4) = 9, and c(3, 3, 3) = 8 (cf. Tables 1 and 2). In addition, c(2, 2, 2, 2) = 6, c(S 2 ) = 4, and c(T 2 ) = 0. Let Sg be a Riemann surface of genus g ≥ 2. Then we also have c(Sg ) = 4 ? 4g . It is clear that the values of c are distinct in each case, so that the spectrum distinguishes these 2-orbifolds. For the rest of the proof, it suf?ces to consider orbifolds in C with χ(O) > 0 since these include all orbifolds within C for which c is not an integer. Table 2 lists the values of c for these triangular pillows. Setting c(2, 3, 3) = c(2, 2, m) and √ 25± 481 solving for m gives m = , which contradicts the assumption that m is 12 an integer. Similar calculations for c(2, 3, 4) and c(2, 3, 5) show that the spectrum distinguishes among triangular pillows with χ(O) > 0. By Theorem 5.14, c distinguishes among teardrops and footballs. We next show that c distinguishes both teardrops and footballs from triangular pillows with 1 1 1 , and c(p, q, r) = ?2+ p + q + r + p +1 χ(O) > 0. We have c(m) = 2+ m + m q +r; setting the integer and fractional parts of these equations equal gives 2 + m = ?1 + p + q + r 1 1 1 1 = + + ? 1. m p q r Solving the ?rst equation for m and plugging the result into the second equation yields 0 = pr(p + r ? 3) + pq (p + q ? 3) + qr(q + r ? 3) + pqr(5 ? p ? q ? r). None of the possible triples (p, q, r) satisfy this equation, showing that teardrops are distinguished from triangular pillows with χ(O) > 0. The argument that c distinguishes good footballs from these triangular pillows is analogous. To see that c distinguishes triangular pillows with χ(O) > 0 from bad footballs, we compare the respective integer and fractional parts of c in each case. For example, comparing c(2, 3, 3) and c(r, s), r = s gives 7 = r+s 1 1 1 = + , 6 r s

?119 which implies s2 ? 7s + 42 = 0; thus s = 7± 2 , contradicting s being an integer. The calculations are similar for O(2, 3, 4) and O(2, 3, 5), while for O(2, 2, m) we have √

3+m = r+s 1 1 1 = + , m r s which implies r(r ? 3) + s(s ? 3) + rs = 0. We can assume that one of r and s is strictly greater than 2, say r. Thus r ? 3 ≥ 0 and s ? 3 ≥ ?1, which implies r(r ? 3) + s(s ? 3) + rs ≥ s(r ? 1) > 0.

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

31

Hence triangular pillows with χ(O) > 0 are distinguished from bad footballs. 5.16. R EMARK . Notably absent from the class C are triangular pillows with χ(O) < 0. The invariant c does not seem suf?ciently strong to distinguish among these triangular pillows. However, as a special case of a result in [13], the spectrum does determine the orders of the cone points in such a 2-orbifold, provided that it is endowed with a metric of constant curvature ?1. On the other hand, to distinguish, say, triangular pillows with χ(O) < 0 from triangular pillows with χ(O) > 0, we do not need such a metric assumption. If O(p, q, r) is any triangular pillow with χ(O) < 0 and spec(O(p, q, r)) = spec(O(2, 2, m)), then setting the integer and fractional parts of the respective values of c equal, we have m+3 = p+q+r?2 1 1 1 1 = + + . m p q r Solving the ?rst equation for m and plugging the resulting value into the second equation yields (5.17) 2pqr + pr(p + r ? 5) + pq (p + q ? 5) + qr(q + r ? 5) = 0.

1 1 Note that for a triangular pillow with χ(O) < 0, we must have p +1 q + r < 1, which implies that the sum of any two of p, q, r is at least 5. Thus each term on the right-hand side of (5.17) is nonnegative. This contradiction then implies that O(2, 2, m) cannot be isospectral to such a triangular pillow. For the remaining triples where χ(O) > 0, we set the integer and fractional parts of the respective value of c equal to those of a triangular pillow with χ(O) < 0, and note that there are no χ(O) < 0 triples satisfying these equations. Thus c distinguishes between triangular pillows with χ(O) < 0 and χ(O) > 0. A similar argument shows that c distinguishes triangular pillows with χ(O) < 0 from teardrops; it is clear that c also distinguishes triangular pillows with χ(O) < 0 from the smooth surfaces and the remaining elements of C , with the exception of footballs. In this last case, it seems that metric assumptions are again necessary.

5.18. R EMARK . Other dif?culties arise during the consideration of the expanded class which includes nonorientable orbifolds. Metric assumptions are necessary in numerous cases, such as distinguishing nonorientable orbifolds with the same orientable double cover (e.g. O(?2, 3, 3) and O(3, ?2), see Table 1 and Figure 1). We now examine classes that include nonorientable orbifolds. 5.19. P ROPOSITION . Within the class of all closed 2-orbifolds with χ(O) ≥ 0, the spectrum distinguishes whether the orbifold has zero or positive Euler characteristic. Proof. Note that for 2-orbifolds O with χ(O) = 0, c is either an integer or equal to 4.5. Thus c distinguishes all but the following cases: S 2 from the orbifolds O(?3, 3, 3) and O(3, ?3) (with c = 4), the good football O(2, 2) from O(?2, 3, 6) (with c = 5), and the bad teardrop O(2) from the orbifolds O(?2, 4, 4) and O(4, ?2)

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DRYDEN, GORDON, GREENWALD, AND WEBB

3

3 2

3

3

3 2

3 3 3 2 2

F IGURE 1. The uppermost object is a fundamental domain for O(2, 3, 3), with its quotient O(2, 3, 3) below. Here the vertices are cone points labelled with their orders, and the double edges represent re?ector edges. On the bottom left we show O(?2, 3, 3), which is obtained by re?ecting O(2, 3, 3) in the plane of the paper, and on the right is O(3, ?2), obtained by re?ecting O(2, 3, 3) in the plane containing the dashed loop.

(with c = 4.5). The lack of a mirror locus in the χ(O) > 0 cases and the presence of a mirror locus in the corresponding χ(O) = 0 orbifolds can be gleaned from the degree ? 1 2 term, and so they are distinguished.

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR ORBIFOLDS

33

5.20. P ROPOSITION . Within the class of closed 2-orbifolds of constant nonzero curvature R or ?R the spectrum determines the sign of the curvature, i.e. whether the orbifold is spherical or hyperbolic. Proof. Assume that O has cone points p1 , . . . , pk of orders m1 , . . . , mk , and/or dihedral corner re?ector points q1 , . . . , qr of orders n1 , . . . , nr . Now look at the coef?cient of the t term in the expansion, as in (5.10), which reduces to a2 ±R 4π

k i=1 2 (m2 i + 11)(mi ? 1) + 360mi 1 t r i=1 2 (n2 i + 11)(ni ? 1) 720ni

in the presence of constant curvature. The term in the expansion tells us vol(O) and we know the size of the curvature, so we know the a2 component and may subtract it off. Notice that the summands are nonnegative, and hence we can read the sign of the curvature unless there were no cone points and no dihedral corner re?ector points. In this case, examine the degree zero term, which now has χ(O) a1 no point contributions and reduces to 4 π = 6 , and we can read off the Euler characteristic. 5.21. R EMARK . In the case of closed 2-orbifolds with a nontrivial mirror locus, (5.12) enables us to make a stronger statement. In particular, among such orbifolds that are endowed with a metric of strictly positive, strictly negative, or zero curvature, the spectrum determines the sign of the curvature. This class includes the bad orbifolds O(?m) and O(?p, q ) with p = q since they admit a metric of strictly positive (but variable) curvature. 5.22. P ROPOSITION . Within the class of spherical 2-orbifolds of constant curvature R > 0 the spectrum determines the orbifold. Proof. Notice that the metric requirement eliminates teardrops and bad footballs and their quotients from this class. In Table 1, c distinguishes among the remaining spherical orbifolds with the exception that c is unable to distinguish between orbifolds that are nonorientable but have the same orientable double cover: O(?m, m), O(m×), and O(m?), with double cover O(m, m); O(?2, 2, m) and O(2, ?m), with double cover O(2, 2, m); and O(?2, 3, 3) and O(3, ?2), with double cover O(3, 3, 2) (see Figure 1). However, c is able to distinguish each nonorientable class from the remaining orbifolds. Consider such a class of nonorientable spherical orbifolds with a common orientable double cover. The coef?cient of the degree ? 1 2 term, as given in (5.11), distinguishes nonorientable orbifolds with mirror loci from those without, i.e. in this class it distinguishes orbifolds with only crosscaps from those with mirror loci. In the presence of constant curvature, (5.11) also distinguishes among the remaining spherical cases: among O(?m, m) and O(m?), the length of the mirror locus of O(?m, m) is larger in the constant curvature metric; among O(?2, 2, m) and O(2, ?m), the length of the mirror locus of O(?2, 2, m) is larger; and among O(?2, 3, 3) and O(3, ?2), the length of the mirror locus of O(?2, 3, 3) is larger (see Figure 1).

34

DRYDEN, GORDON, GREENWALD, AND WEBB

5.23. R EMARK . Notice that metric assumptions are only needed to distinguish within each nonorientable class. We cannot make a similar statement for ?at 2orbifolds. For example, it is possible to endow O(2, ?2, 2) and O(2, 2?) with a metric of zero curvature so that they have the same area and also have mirror loci of the same length. They cannot be distinguished by the asymptotic expansion of the heat trace.

R EFERENCES

[1] A. Adem and Y. Ruan, Twisted orbifold K-theory, Comm. Math. Phys. 237 (2003), 533–556. [2] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d’une vari? et? e riemannienne, Lecture Notes in Mathematics, vol. 194, Springer-Verlag (1971). [3] B. C. Berndt and B. P. Yeap, Explicit evaluations and reciprocity theorems for ?nite trigonometric sums, Adv. in Appl. Math. 29 (2002), 358–385. [4] T. P. Branson and P. B. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990), 245–272. [5] J. Br¨ uning and M. Lesch, On the spectral geometry of algebraic curves, J. Reine Angew. Math. 474 (1996), 25–66. [6] J. Br¨ uning and R. Seeley, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (1987), 369–429. [7] Hongwei Chen, On some trigonometric power sums, Int. J. Math. Math. Sci. 30 (2002), 185– 191. [8] Y.-J. Chiang, Spectral geometry of V -manifolds and its application to harmonic maps, in Differential geometry: partial differential equations on manifolds, Proc. Sympos. Pure Math. 54, Part 1, 93–99. [9] J. H. Conway, The orbifold notation for surface groups, in Groups, combinatorics and geometry, London Math. Soc. Lecture Note Ser. 165, Cambridge Univ. Press, Cambridge, 1992, 438–447. [10] H. Donnelly, Spectrum and the ?xed point sets of isometries I, Math. Ann. 224 (1976), 161–170. [11] H. Donnelly, Asymptotic expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math. 23 (1979), 485–496. [12] P. Doyle and J.P. Rossetti, Isospectral hyperbolic surfaces have matching geodesics, preprint, arXiv:math/0605765v1 [math.DG]. [13] E. B. Dryden and A. Strohmaier, Huber’s theorem for hyperbolic orbisurfaces, Canad. Math. Bull. (to appear). [14] J. J. Duistermaat and J. A. C. Kolk, Lie groups, Springer-Verlag, Berlin, 2000. [15] C. Farsi, Orbifold spectral theory, Rocky Mountain J. Math. 31 (2001), 215–235. [16] M. E. Fisher, Problem 69-14, SIAM Review 13 (1971), 116–119. [17] J. B. Gil, Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators, Math. Nachr. 250 (2003), 25–57. [18] C. S. Gordon and J. P. Rossetti, Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn’t reveal, Ann. Inst. Fourier (Grenoble) 53 (2003), 2297–2314. [19] I. Moerdijk, Orbifolds as groupoids, an introduction, in Orbifolds in Mathematics and Physics, A. Adem, ed., Contemp. Math. 310 (2002), 205–222. [20] I. Moerdijk and D. Pronk, Orbifolds, sheaves and groupoids, K -Theory 12 (1997), 3–21. [21] J. M. Montesinos, Classical tessellations and three-manifolds, Springer-Verlag, Berlin, 1987. [22] E. Park and K. Richardson, The basic Laplacian of a Riemannian foliation, Amer. J. Math. 118 (1996), 1249–1275. [23] L. L. Pennisi, Elements of complex variables, Holt, Rinehart and Winston, 1966. [24] K. Richardson, Traces of heat operators on Riemannian foliations, preprint.

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[25] J. P. Rossetti, D. Schueth, and M. Weilandt, Isospectral orbifolds with different maximal isotropy orders, Ann. Glob. Anal. Geom. (to appear). [26] I. Satake, The Gauss-Bonnet theorem for V -manifolds, J. Math. Soc. Japan 9 (1957), 464–492. [27] N. Shams, E. A. Stanhope, and D. L. Webb, One cannot hear orbifold isotropy type, Arch. Math. (Basel) 87 (2006), 375-384. [28] E. A. Stanhope, Spectral bounds on orbifold isotropy, Ann. Global Anal. Geom. 27 (2005), 355–375. [29] E. A. Stanhope and A. Uribe, The trace formula for orbifolds, preprint. [30] W. P. Thurston, Geometry and topology of 3-manifolds, electronic edition of 1980 lecture notes, available at http://www.msri.org/publications/books/gt3m/. (Dryden) D EPARTMENT OF M ATHEMATICS , B UCKNELL U NIVERSITY, L EWISBURG , PA 17837 E-mail address: ed012@bucknell.edu (Gordon) D EPARTMENT OF M ATHEMATICS , DARTMOUTH C OLLEGE , H ANOVER , N EW H AMP 03755 E-mail address: csgordon@dartmouth.edu

SHIRE

(Greenwald) D EPARTMENT OF M ATHEMATICS , A PPALACHIAN S TATE U NIVERSITY, B OONE , NC 28608 E-mail address: greenwaldsj@appstate.edu (Webb) D EPARTMENT OF M ATHEMATICS , DARTMOUTH C OLLEGE , H ANOVER , N EW H AMP 03755 E-mail address: david.l.webb@dartmouth.edu

SHIRE

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