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

Aldana, C. L. (2013). Asymptotics of relative heat traces and determinants on open surfaces of finite area. Annals of global analysis and geometry, 44(2), 169-216. doi:10.1007/s10455-012-9362-9.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-000E-74B4-4 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0015-148F-0
Genre: Journal Article

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1001.2033 (Preprint), 450KB
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 Creators:
Aldana, Clara Lucia1, Author              
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1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, escidoc:24012              

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Free keywords: Mathematics, Spectral Theory, math.SP,
 Abstract: The goal of this paper 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.

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 Dates: 2010-01-122012-12-122013
 Publication Status: Published in print
 Pages: This is the final version of the article before it gets published. 51 pages
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 Rev. Method: -
 Identifiers: arXiv: 1001.2033
DOI: 10.1007/s10455-012-9362-9
URI: http://arxiv.org/abs/1001.2033
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Title: Annals of global analysis and geometry
Source Genre: Journal
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Publ. Info: Dordrecht : Kluwer Academic Publishers
Pages: - Volume / Issue: 44 (2) Sequence Number: - Start / End Page: 169 - 216 Identifier: ISSN: 0232-704X
CoNE: http://pubman.mpdl.mpg.de/cone/journals/resource/954926967974