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. Refined logarithmic convexity and concavity results of the classical Jacobi theta functions of one real variable are deeply involved.

Highlights

  • The search for extremal geometries is a popular topic in many branches of mathematics and mathematical physics

  • We pick up a result by Osgood et al [26] on extremals of determinants of Laplace–Beltrami operators on tori and restrict the assumptions, excluding their solution of the following problem

  • Among all 2-dimensional tori of area 1, which torus maximizes the determinant of the Laplace–Beltrami operator?

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Summary

Introduction

The search for extremal geometries is a popular topic in many branches of mathematics and mathematical physics. We pick up a result by Osgood et al [26] on extremals of determinants of Laplace–Beltrami operators on tori and restrict the assumptions, excluding their solution of the following problem. We will prove that this is the case Both problems, the one for general and the one for rectangular lattices, are closely related to the study of extremal values of the heat kernel on the torus [3,5], finding extremal bounds of Gaussian Gabor frames of given density [17–19], as well as the study of certain theta functions [25]. It is worth noting that in all cases the extremal solutions are the same as for the classical sphere packing and covering problem in the plane. The proof of the main result will follow from refined logarithmic convexity and concavity statements related to Jacobi’s theta functions as described by Faulhuber and Steinerberger [19]

The Laplace–Beltrami operator on the torus
The Laplace–Beltrami operator on manifolds and its heat kernel
The determinant on the torus
Rectangular tori
Jacobi’s theta functions
Properties of Theta-2 and Theta-4
Properties of theta-3
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