Global Extrema of the Zeta Regularized Determinant on Orthogonal Flat Tori

Abstract

The search for extremal geometries is a central theme in several areas of mathematics. Here, we address the following question: among all n-dimensional orthogonal tori of unit volume, which one maximizes the zeta regularized determinant of the Laplacian? We prove that the equilateral torus is the unique maximizer in each dimension n, for all n greater than or equal to 2, validating Sarnak's conjecture in this context. We also investigate the analogous question for the Laplacian on Euclidean boxes with the Neumann and Dirichlet boundary conditions. For orthogonal flat tori of unit volume and dimension n, we show further that the determinant is strictly decreasing with the dimension and tends to zero as the dimension n tends to infinity.