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

Kokotov, A.

doi:10.1090/S0002-9947-2012-05695-9

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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Trans. Amer. Math. Soc. 364 (2012), 5645-5671 Request permission

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 $\textrm {det}^* \Delta$ 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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