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March 2020 Heat kernel asymptotics on sequences of elliptically degenerating Riemann surfaces
Daniel Garbin, Jay Jorgenson
Kodai Math. J. 43(1): 84-128 (March 2020). DOI: 10.2996/kmj/1584345689

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 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 Garbin et al. (2018) [8], 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 (Huntley et al. (1995) [14] and (1997) [15], Jorgenson et al. (1997) [20] and (1997) [17]) and on degenerating families of finite volume hyperbolic three manifolds (Dodziuk et al. (1998) [4].) Although the types of results developed here and in Garbin et al. (2018) [8], 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.

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Daniel Garbin. Jay Jorgenson. "Heat kernel asymptotics on sequences of elliptically degenerating Riemann surfaces." Kodai Math. J. 43 (1) 84 - 128, March 2020. https://doi.org/10.2996/kmj/1584345689

Information

Published: March 2020
First available in Project Euclid: 16 March 2020

zbMATH: 07196511
MathSciNet: MR4077206
Digital Object Identifier: 10.2996/kmj/1584345689

Rights: Copyright © 2020 Tokyo Institute of Technology, Department of Mathematics

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Vol.43 • No. 1 • March 2020
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