Asymptotic expansion of the heat kernel for orbifolds

TL;DR

This paper extends the asymptotic expansion of the heat kernel to general compact orbifolds, linking spectral data to geometric features, and explicitly computes terms for two-dimensional cases to distinguish orbifold classes.

Contribution

It generalizes Donnelly's heat kernel expansion from good orbifolds to all compact orbifolds and clarifies the role of singularities in spectral geometry.

Findings

Explicit heat invariant formulas for two-dimensional orbifolds

Spectral data distinguishes different classes of two-dimensional orbifolds

Extended asymptotic expansion to general compact 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 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.