Heat Kernel for Simply-Connected Riemann Surfaces

2010/07/30 by T. Harri Jones, Jones, Trevor H., Dan Kučerovský +1 · 1 citation
Engineering · Mathematics · #35K08 #58J35 #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1007.5467

openalex publication_date 2010/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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