# On the corner contributions to the heat coefficients of geodesic polygons

## TL;DR

This paper derives formulas for how cone points and interior angles in geodesic polygons influence the heat trace asymptotics on Riemannian orbisurfaces, highlighting the role of curvature derivatives at corners.

## Contribution

It provides new explicit formulas for the $t^2$ coefficient in heat trace expansions, including the impact of curvature derivatives at polygon corners.

## Key findings

- Formulas for cone point contributions to heat trace coefficients.
- Analysis of interior angles of the form π/k in geodesic polygons.
- Identification of curvature derivative effects at corners on heat coefficients.

## Abstract

Let $\mathcal O$ be a compact Riemannian orbisurface. We compute formulas for the contribution of cone points of~$\mathcal O$ to the coefficient at $t^2$ of the asymptotic expansion of the heat trace of $\mathcal O$, the contributions at $t^0$ and $t^1$ being known from the literature. As an application, we compute the coefficient at $t^2$ of the contribution of interior angles of the form $\gamma=\pi/k$ in geodesic polygons in surfaces to the asymptotic expansion of the Dirichlet heat kernel of the polygon, under a certain symmetry assumption locally near the corresponding corner. The main novelty here is the determination of the way in which the Laplacian of the Gauss curvature at the corner point enters into the coefficient at $t^2$. We finish with a conjecture concerning the analogous contribution of an arbitrary angle $\gamma$ in a geodesic polygon.