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

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.