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

Let X be a translation surface of genus g > 1 with 2g − 2 conical points of angle 4π and let γ, γ′ be two homologous saddle connections of length s joining two conical points of X and bounding two surfaces S+ and S− with boundaries ∂S+ = γ − γ′ and ∂S− = γ′ − γ. Gluing the opposite sides of the boundary of each surface S+, S− one gets two (closed) translation surfaces X+, X− of genera g+, g−; g+ + g− = g. Let ∆, ∆+ and ∆− be the Friedrichs extensions of the Laplacians corresponding to the (flat conical) metrics on X , X+ and X− respectively. We study the asymptotical behavior of the (modified, i. e. with zero modes excluded) zeta-regularized determinant det∗ ∆ as γ and γ′ shrink. We find the asymptotics