Polyakov-Alvarez Formula for Curvilinear Polygonal Domains with Slits

We consider the $\zeta$-regularized determinant of the Friedrichs extension of the Dirichlet Laplace-Beltrami operator on curvilinear polygonal domains with corners of arbitrary positive angles. In particular, this includes slit domains. We obtain a short time asymptotic expansion of the heat trace using a classical patchwork method. This allows us to define the $\zeta$-regularized determinant of the Laplacian and prove a comparison formula of Polyakov-Alvarez type for a smooth and conformal change of metric.