Heat coefficients of surfaces with curved conic singularities

TL;DR

This paper derives an explicit formula for the heat trace coefficient at conical singularities on 2D surfaces with rotationally invariant metrics, revealing irrational variation under rescaling when curvature diverges.

Contribution

It provides a new explicit formula for the heat coefficient at conical singularities with non-flat, rotationally invariant metrics, and analyzes its behavior under rescaling.

Findings

Explicit formula for $b_{1/2}(C)$ in heat trace expansion.

Demonstrates irrational variation of $b_{1/2}(C)$ when curvature diverges.

Contrasts behavior with orbifold cone points and polygon corners.

Abstract

Let $(M,g)$ be a two-dimensional Riemannian manifold of finite diameter with a conical singularity. Under the assumption that the metric near the cone point $C$ is rotationally invariant, but not necessarily flat, we give an explicit formula for the coefficient $b_{1/2}(C)$ in the heat trace expansion $\operatorname{tr}(\operatorname{exp}(-t\Delta_g))\sim_{t\searrow0} (4\pi t)^{-1}\sum_{j=0}^\infty a_j(M) t^j+\sum_{j=0}^\infty b_{j/2}(C)t^{j/2}+\sum_{j=0}^\infty c_{j/2}(C) t^{j/2} \log t$. In the case that the Gaussian curvature $K$ of $(M,g)$ satisfies $|K(p)|\to\infty$ as $p\to C$, we show that $b_{1/2}(C)$ varies irrationally under constant rescalings of the distance circles near the cone point. This is a sharp contrast to the behavior of $b_0(C)$ and of those coefficients $b_j(C)$ which appear in certain known formulas in the case of orbifold cone points or corners of geodesic polygons.