Asymptotic expansion of the heat kernel for orbifolds

Abstract

We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifoldO 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. CONTENTS