Asymptotic expansion of the heat kernel for orbifolds

Abstract

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 clarifies 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.

Traces of heat operators on Riemannian foliations

“…If a second order differential operator has the same principal symbol as the Laplacian, and if it is geometrically defined and thus commutes with the action of G, then the equivariant trace of the corresponding heat kernel satisfies (1.3). After writing this paper, the author was made aware of the recent work [19], where the authors compute the asymptotics of the heat kernel on orbifolds, related to the work in [18], [10], and to Theorem 3.5 in this paper.…”

Hearing the weights of weighted projective planes

“…For good orbifolds (i.e., those arising as global quotients of manifolds), Donnelly [11] proved the existence of the heat kernel and constructed the asymptotic expansion for the heat trace. This work was extended to general orbifolds in [14], where the expressions obtained also clarify the contributions of the various pieces of the singular set. While [11] and [14] treat the heat trace asymptotics for functions, we will also require the asymptotics for 1-forms.…”

“…In §4, we use a certain polytope associated to a given symplectic toric orbifold to give a description of extremal metrics on weighted projective planes, and to calculate the integral of the square of the scalar curvature on a weighted projective plane. In §5, we recall the asymptotic expansion of the heat trace of an orbifold as given in [14], and we calculate the first few heat invariants for CP 2 (N 1 , N 2 , N 3 ). Finally, we bring all of these tools together to prove Theorem 1 and related results in §6.…”

Equivariant isospectrality and Sunada’s method

“…Now, if π : M → O is a finite Riemannian cover of degree d and {ã k } k≥0 denotes the corresponding heat invariants of M , thenã k = d · a k . Therefore, if we let a i,k denote the corresponding terms of Donnelly's asymptotic expansion for the orbifold O i , it follows from the isospectrality of M 1 and M 2 (and the fact that d 1 = d 2 ) that the heat invariants a 1,k and a 2,k are equal for each non-negative integer k. It then follows, since b 0,S is positive for each singular strata S [DGGW,, that O 1 has singular points if and only if O 2 has singular points.…”