Spectral geometry of surfaces with curved conic singularities

TL;DR

This paper investigates the spectral geometry of surfaces with curved conic singularities by analyzing the heat trace expansion and relating its terms to geometric features like cone angles and curvature.

Contribution

It provides explicit expressions for the heat trace expansion terms in terms of the geometry of curved conic singularities, linking spectral data to geometric invariants.

Findings

The constant term in the heat trace contains information about the cone angle.

The $bt^{1/2}$ term relates to curvature and cone angle.

Explicit formulas connect spectral expansion to geometric singularity data.

Abstract

Let $(M,g)$ be a surface with Riemannian metric and curved conic singularities. More precisely, a neighbourhood of a singularity is isometric to $(0,1)\times S^1$ with metric $g_{\text{conic}}=dr^2+f(r)^2d\theta^2, r\in(0,1)$. We study the spectral geometry of $(M,g)$ using the heat trace expansion. We express the first few terms in the expansion through the geometry of the singularities. The constant term contains information about the angle at the tip of the cone. The next term, $bt^{1/2}$, is expressed through the curvature and the angle at the tip of the cone.