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The determinant on flat conic surfaces with excision of disks


Author: David A. Sher
Journal: Proc. Amer. Math. Soc. 143 (2015), 1333-1346
MSC (2010): Primary 58J50, 58J52
DOI: https://doi.org/10.1090/S0002-9939-2014-12319-7
Published electronically: October 28, 2014
MathSciNet review: 3293746
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Abstract: Let $ M$ be a surface with conical singularities, and consider a family of surfaces $ M_{\epsilon }$ obtained from $ M$ by removing disks of radius $ \epsilon $ around a subset of the conical singularities. Such families arise naturally in the study of the moduli space of flat metrics on higher-genus surfaces with boundary. In particular, they have been used by Khuri to prove that the determinant of the Laplacian is not a proper map on this moduli space when the genus $ p\geq 1$. Khuri's work is closely related to the isospectral compactness results of Osgood, Phillips, and Sarnak. Our main theorem is an asymptotic formula for the determinant of $ M_\epsilon $ as $ \epsilon $ approaches zero up to terms which vanish in the limit. The proof uses the determinant gluing formula of Burghelea, Friedlander, and Kappeler along with an observation of Wentworth on the asymptotics of Dirichlet-to-Neumann operators. We then apply this theorem to extend and sharpen the results of Khuri.


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David A. Sher
Affiliation: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montréal, Quebec H3A 2K6, Canada – and – Centre de Recherches Mathématiques, Université de Montréal CP 6128 succ Centre-Ville, Montréal, H3C 3J7, Canada
Address at time of publication: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043
Email: dsher@umich.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12319-7
Received by editor(s): February 25, 2013
Received by editor(s) in revised form: July 8, 2013
Published electronically: October 28, 2014
Additional Notes: The author was partially supported by an ARCS fellowship in 2011-2012 and a CRM-ISM fellowship in 2012-2013.
Communicated by: Michael Hitrik
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.