The determinant on flat conic surfaces with excision of disks

Let M be a surface with conical singularities, and consider a degenerating family of surfaces obtained from M by removing disks of smaller and smaller radius around a subset of the conical singularities. Such families arise naturally in the study of the moduli space of flat metrics on higher-genus surfaces with boundary. In particular, they have been used by Khuri to prove that the determinant of the Laplacian is not a proper map on this moduli space when the genus of M is positive. Khuri's work is closely related to the isospectral compactness results of Osgood, Phillips, and Sarnak. Our main theorem is an asymptotic formula for the determinant of the Laplacian on the degenerating family of surfaces, up to terms which vanish in the singular limit. We then apply this theorem to extend and sharpen the results of Khuri. The proof uses the determinant gluing formula of Burghelea, Friedlander, and Kappeler along with an observation of Wentworth on the asymptotics of Dirichlet-to-Neumann operators.