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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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
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by A. Kokotov PDF
Trans. Amer. Math. Soc. 364 (2012), 5645-5671 Request permission


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 $\textrm {det}^* \Delta$ as $\gamma$ and $\gamma ’$ shrink. We find the asymptotics \[ \textrm {det}^* \Delta \sim \kappa s^{1/2}\frac {\textrm {Area} (\mathcal {X})}{\textrm {Area} (\mathcal {X}^+)\textrm {Area} (\mathcal {X}^-)} \textrm {det}^* \Delta ^+\textrm {det}^* \Delta ^-\] 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 $\textrm {det}^* \Delta$ 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
  • MR Author ID: 252297
  • Email:
  • Received by editor(s): July 6, 2010
  • Published electronically: June 14, 2012
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 5645-5671
  • MSC (2010): Primary 58J52; Secondary 32G15, 14H15, 30F10
  • DOI:
  • MathSciNet review: 2946925