Heat Kernel for Simply-Connected Riemann Surfaces

TL;DR

This paper reviews and derives heat kernels for the three types of simply-connected Riemann surfaces, providing a method to generate heat kernels on any such surface from their universal covers.

Contribution

It compiles known heat kernels for the sphere, Euclidean plane, and hyperbolic plane and introduces a method to extend these to general Riemann surfaces via their universal covers.

Findings

Collected and summarized heat kernels for the three simply-connected Riemann surfaces.

Derived differential forms heat kernels for these surfaces.

Proposed a method to generate heat kernels on Riemann surfaces from universal covers.

Abstract

From the uniformization theorem, we know that every Riemann surface has a simply-connected covering space. Moreover, there are only three simply-connected Riemann surfaces: the sphere, the Euclidean plane, and the hyperbolic plane. In this paper, we collect the known heat kernels, or Green's functions, for these three surfaces, and we derive the differential forms heat kernels as well. Then we give a method for using the heat kernels on the universal cover to generate the heat kernel on the underlying surface.