Determinant of the Laplacian on Tori of Constant Positive Curvature with one Conical Point

Abstract

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extensions) of the Laplacians on a compact Riemann surface of genus one with conformal metric of curvature 1 having a single conical singularity of angle 4π.

Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities

“…Here, for instance, results in [2,3] can be interpreted as a generalization of Polyakov-Alvarez formula to the case of flat conical metrics on a disk and on a sphere, the main result in [21] is a simple consequence of an analog of Polyakov formula for two conformally equivalent flat conical metrics and the results in [22]. Some results were also obtained for determinants of Laplacians in constant positive curvature (spherical) [10,36,23,18,19] and other conical metrics [17], but no Polyakov-Alvarez type formulas for metrics other than smooth or conical flat were available until now.…”

“…Genus one examples will be considered elsewhere. We only note that explicit formulas for determinants of Laplacians on genus one surfaces can be obtained by using results of this paper together with known explicit formulas for the determinant of Laplacian on the (smooth) flat tori [30,29] and the flat annulus [42]; in particular, one can expect to recover the variational formula in [19] and to find the corresponding undetermined constant. Let us also mention that conical metrics with cylindrical and conical ends can be included into consideration by pairing results of this paper with the BFK-type decomposition formulas in [15, Theorem 1] and [16, Theorem 1], however we do not discuss this here.…”

On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions

“…written as the derivative of some function with respect to z k ) before it can be used in explicit formulas for the determinant Det ′ ∆ F . This can be not an easy task even if an explicit expression for b(−∞) was found by separation of variables [11] (e.g. separation of variables can be used to find b(−∞) if m is a constant curvature singular metric) and hence one may want to use the general integrated explicit expression given in Lemma 4 below.…”