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On the asymptotics of determinant of Laplacian at the principal boundary of the principal stratum of the moduli space of Abelian differentials


Author: A. Kokotov
Journal: Trans. Amer. Math. Soc. 364 (2012), 5645-5671
MSC (2010): Primary 58J52; Secondary 32G15, 14H15, 30F10
DOI: https://doi.org/10.1090/S0002-9947-2012-05695-9
Published electronically: June 14, 2012
MathSciNet review: 2946925
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Abstract: Let $ \mathcal {X}$ be a translation surface of genus $ g>1$ with $ 2g-2$ conical points of angle $ 4\pi $ and let $ \gamma $, $ \gamma '$ be two homologous saddle connections of length $ s$ joining two conical points of $ \mathcal {X}$ and bounding two surfaces $ S^+$ and $ S^-$ with boundaries $ \partial S^+=\gamma -\gamma '$ and $ \partial S^-=\gamma '-\gamma $. Gluing the opposite sides of the boundary of each surface $ S^+$, $ S^-$ one gets two (closed) translation surfaces $ \mathcal {X}^+$, $ \mathcal {X}^-$ of genera $ g^+$, $ g^-$; $ g^++g^-=g$. Let $ \Delta $, $ \Delta ^+$ and $ \Delta ^-$ be the Friedrichs extensions of the Laplacians corresponding to the (flat conical) metrics on $ \mathcal {X}$, $ \mathcal {X}^+$ and $ \mathcal {X}^-$ respectively. We study the asymptotical behavior of the (modified, i.e. with zero modes excluded) zeta-regularized determinant $ {\rm det}^*\, \Delta $ as $ \gamma $ and $ \gamma '$ shrink. We find the asymptotics

$\displaystyle {\rm det}^*\,\Delta \sim \kappa s^{1/2}\frac {{\rm Area}\,(\mathc... ...}^+){\rm Area}\,(\mathcal {X}^-)}\,{\rm det}^*\,\Delta ^+{\rm det}^*\,\Delta ^-$

as $ s\to 0$; here $ \kappa $ is a certain absolute constant admitting an explicit expression through spectral characteristics of some model operators. We use the obtained result to fix an undetermined constant in the explicit formula for $ {\rm det}^*\, \Delta $ found in an earlier work by the author and D. Korotkin.

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Additional Information

A. Kokotov
Affiliation: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8
Email: alexey@mathstat.concordia.ca

DOI: https://doi.org/10.1090/S0002-9947-2012-05695-9
Received by editor(s): July 6, 2010
Published electronically: June 14, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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