Flat conical metrics on Riemann surfaces and determinants of Laplacians

Regularized determinants of Laplacians in flat singular metrics on compact Riemann surfaces of arbitrary genus are discussed. Analytic surgery theorem and variational formulas for these determinants are proved. This leads to an explicit calculation of the determinants, the resulting expression generalizes the well-known Ray-Singer formula for the determinant of Laplacian in flat smooth metric on an elliptic surface.