Extremal Determinants of Laplace-Beltrami Operators for Rectangular Tori

In this work we study the determinant of the Laplace-Beltrami operator on rectangular tori of unit area. We will see that the square torus gives the extremal determinant within this class of tori. The result is established by studying properties of the Dedekind eta function for special arguments and refined logarithmic convexity and concavity results of the classical Jacobi theta functions of one real variable are deeply involved.