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

  • Kokotov A
2Citations
Citations of this article
5Readers
Mendeley users who have this article in their library.

Abstract

Let X \mathcal {X} be a translation surface of genus g > 1 g>1 with 2 g − 2 2g-2 conical points of angle 4 π 4\pi and let γ \gamma , γ ′ \gamma ’ be two homologous saddle connections of length s s joining two conical points of X \mathcal {X} and bounding two surfaces S + S^+ and S − S^- with boundaries ∂ S + = γ − γ ′ \partial S^+=\gamma -\gamma ’ and ∂ S − = γ ′ − γ \partial S^-=\gamma ’-\gamma . Gluing the opposite sides of the boundary of each surface S + S^+ , S − S^- one gets two (closed) translation surfaces X + \mathcal {X}^+ , X − \mathcal {X}^- of genera g + g^+ , g − g^- ; g + + g − = g g^++g^-=g . Let Δ \Delta , Δ + \Delta ^+ and Δ − \Delta ^- be the Friedrichs extensions of the Laplacians corresponding to the (flat conical) metrics on X \mathcal {X} , X + \mathcal {X}^+ and X − \mathcal {X}^- respectively. We study the asymptotical behavior of the (modified, i.e. with zero modes excluded) zeta-regularized determinant det ∗ Δ \textrm {det}^*\, \Delta as γ \gamma and γ ′ \gamma ’ shrink. We find the asymptotics \[ det ∗ Δ ∼ κ s 1 / 2 Area ( X ) Area ( X + ) Area ( X − ) det ∗ Δ + det ∗ Δ − \textrm {det}^*\,\Delta \sim \kappa s^{1/2}\frac {\textrm {Area}\,(\mathcal {X})}{\textrm {Area}\,(\mathcal {X}^+)\textrm {Area}\,(\mathcal {X}^-)}\,\textrm {det}^*\,\Delta ^+\textrm {det}^*\,\Delta ^- \] as s → 0 s\to 0 ; here κ \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 det ∗ Δ \textrm {det}^*\, \Delta found in an earlier work by the author and D. Korotkin.

Cite

CITATION STYLE

APA

Kokotov, A. (2012). On the asymptotics of determinant of Laplacian at the principal boundary of the principal stratum of the moduli space of Abelian differentials. Transactions of the American Mathematical Society, 364(11), 5645–5671. https://doi.org/10.1090/s0002-9947-2012-05695-9

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free