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Annals of Global Analysis and Geometry

, Volume 44, Issue 2, pp 169–216 | Cite as

Asymptotics of relative heat traces and determinants on open surfaces of finite area

  • Clara L. Aldana
Article

Abstract

The goal of this article is to prove that on surfaces with asymptotically cusp ends the relative determinant of pairs of Laplace operators is well defined. We consider a surface with cusps \((M,g)\) and a metric \(h\) on the surface that is a conformal transformation of the initial metric \(g\). We prove the existence of the relative determinant of the pair \((\Delta _{h},\Delta _{g})\) under suitable conditions on the conformal factor. The core of the paper is the proof of the existence of an asymptotic expansion of the relative heat trace for small times. We find the decay of the conformal factor at infinity for which this asymptotic expansion exists and the relative determinant is defined. Following the paper by B. Osgood, R. Phillips, and P. Sarnak about extremal of determinants on compact surfaces, we prove Polyakov’s formula for the relative determinant and discuss the extremal problem inside a conformal class. We discuss necessary conditions for the existence of a maximizer.

Keywords

Surfaces with asymptotically cusp ends Heat kernels  Asymptotic expansion of heat traces Relative determinants 

Notes

Acknowledgments

This article is registered at the MPG, AEI-2012-200. This paper expands part of my doctoral thesis. I thank my supervisor Werner Müller for his guidance. I am grateful to Rafe Mazzeo, Eugenie Hunsicker, and Sylvie Paycha for helpful discussions and their interest in this work. The author thanks an anonymous referee for the suggestions and comments. Finally, the author thanks the Mathematical Institute at the University of Bonn for hosting her during her graduate studies.

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Max P lanck Institute for Gravitational Physics(Albert Einstein Institute), MPGPotsdamGermany

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