Heat kernel asymptotics on sequences of elliptically degenerating Riemann surfaces

TL;DR

This paper investigates the asymptotic behavior of heat kernels on elliptically degenerating hyperbolic Riemann surfaces, providing detailed spectral analysis essential for understanding spectral functions in degenerating geometries.

Contribution

It introduces a detailed analysis of heat kernel asymptotics on elliptically degenerating surfaces, extending previous spectral theory methods to this specific degeneration setting.

Findings

Heat kernel traces exhibit specific asymptotic behaviors during elliptic degeneration.

The results set the stage for analyzing spectral functions like zeta functions and eigenvalues in degenerating families.

Methodology adapts existing spectral analysis techniques to the elliptic degeneration context.

Abstract

This is the first of two articles in which we define an elliptically degenerating family of hyperbolic Riemann surfaces and study the asymptotic behavior of the associated spectral theory. Our study is motivated by a result from \cite{He 83}, which Hejhal attributes to Selberg, proving spectral accumulation for the family of Hecke triangle groups. In this article, we prove various results regarding the asymptotic behavior of heat kernels and traces of heat kernels for both real and complex time. In \cite{GJ 16}, we will use the results from this article and study the asymptotic behavior of numerous spectral functions through elliptic degeneration, including spectral counting functions, Selberg's zeta function, Hurwitz-type zeta functions, determinants of the Laplacian, wave kernels, spectral projections, small eigenfunctions, and small eigenvalues. The method of proof we employ follows the template set in previous articles which study spectral theory on degenerating families of finite volume Riemann surfaces (\cite{HJL 95}, \cite{HJL 97}, \cite{JoLu 97a}, and \cite{JoLu 97b}) and on degenerating families of finite volume hyperbolic three manifolds (\cite{DJ 98}). Although the types of results developed here and in \cite{GJ 16} are similar to those in existing articles, it is necessary to thoroughly present all details in the setting of elliptic degeneration in order to uncover all nuances in this setting.