Resolvent Trace Formula and Determinants of n Laplacians on Orbifold Riemann Surfaces

Gluing formulas for determinants of Dolbeault laplacians on Riemann surfaces

We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, or trivialization of the bundle near the boundary. An application to the comp...

Holomorphic Factorization of Determinants of Laplacians on Riemann Surfaces and a Higher Genus Generalization of Kronecker’s First Limit Formula

For a family of compact Riemann surfaces Xt of genus g > 1, parameterized by the Schottky space Sg, we define a natural basis of H(Xt, ω n Xt ) which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n = 1. We introduce a holomorphic function F (n) on Sg which generalizes the classical product ∏ ∞ m=1 (1 − q) for n = 1 and g = 1. W...

Determinants of Laplacians and an Isopolar Compactness Theorem for Riemann Surfaces of Infinite Area

Tau-functions on spaces of holomorphic differentials over Riemann surfaces and determinants of Laplacians in flat metrics with conic singularities over Riemann surfaces

The main goal of this paper is to compute (up to a moduli-independent constant factor) determinants of Laplacians in flat metrics with conic singularities on compact Riemann surfaces. We consider two classes of metrics: the Ströbel metrics and metrics given by moduli square of a holomorphic differential. For the latter case, if all conic angles equal 4π, our formulas essentially coincide with h...

Tau-functions on spaces of holomorphic differentials over Riemann surfaces and determinants of Laplacians in flat metrics with conic singularities

The main goal of this paper is to compute (up to a moduli-independent constant factor) determinants of Laplacians in flat metrics with conic singularities on compact Riemann surfaces. We consider two classes of metrics: the Ströbel metrics and metrics given by modulus square of a holomorphic differential. For the latter case, if all conic angles equal 4π, our formulas essentially coincide with ...