On the spectral geometry of manifolds with conic singularities

TL;DR

This paper explores how the heat trace expansion on manifolds with conic singularities reflects geometric properties, linking the vanishing of specific terms to smoothness and the shape of the cross sections.

Contribution

It provides a detailed analysis of the heat trace expansion's terms, relating them to the geometry and smoothness of manifolds with conic singularities across various dimensions.

Findings

Logarithmic term vanishes iff the manifold is smooth.

In 2D, the cross section length equals 2π for smoothness.

In 4D, the cross section must be a spherical space form for the logarithmic term to vanish.

Abstract

In the previous article we derived a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities. In this article we investigate how the terms in the expansion reflect the geometry of the manifold. Since the general expansion contains a logarithmic term, its vanishing is a necessary condition for smoothness of the manifold. In the two-dimensional case this implies that the constant term of the expansion contains a non-local term that determines the length of the (circular) cross section and vanishes precisely if this length equals $2\pi$, that is, in the smooth case. We proceed to the study of higher dimensions. In the four-dimensional case, the logarithmic term in the expansion vanishes precisely when the cross section is a spherical space form, and we expect that the vanishing of a further singular term will imply again smoothness, but this is not yet clear beyond the case of cyclic space forms. In higher dimensions the situation is naturally more difficult. We illustrate this in the case of cross sections with constant curvature. Then the logarithmic term becomes a polynomial in the curvature with roots that are different from 1, which necessitates more vanishing of other terms, not isolated so far.