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.
CITATION STYLE
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
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