Extremal Determinants of Laplace-Beltrami Operators for Rectangular Tori

TL;DR

This paper investigates the extremal properties of the Laplace-Beltrami operator's determinant on rectangular tori, showing the square torus maximizes the determinant, using advanced functions like Dedekind eta and Jacobi theta functions.

Contribution

It establishes that the square torus uniquely maximizes the Laplace-Beltrami determinant among rectangular tori of unit area, employing novel analysis of special functions.

Findings

The square torus has the maximal determinant among rectangular tori.

Properties of Dedekind eta function are key to the proof.

Logarithmic convexity and concavity of Jacobi theta functions are utilized.

Abstract

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.