Extremals of determinants for Laplacians on discrete surfaces

Abstract

In this note we pursue a discrete analogue of a celebrated theorem by Osgood, Phillips and Sarnak, which states that in a fixed conformal class of Riemannian metrics of fixed volume on a closed Riemann surface, the zeta-determinant of the corresponding Laplace-Beltrami operator is maximized on a metric of constant Gaussian curvature. We study the corresponding question for the discrete cotan Laplace operator on triangulated surfaces. We show for some types of triangulations that the determinant (excluding zero eigenvalues) of the discrete cotan-Laplacian is stationary at discrete metrics of constant discrete Gaussian curvature. We present some numerical calculations, which suggest that these stationary points are in fact minima, strangely opposite to the smooth situation. We suggest an explanation of this discrepancy.