Non-maximality of known extremal metrics on torus and Klein bottle

TL;DR

This paper investigates extremal metrics on the torus and Klein bottle, demonstrating that known explicit examples are not maximal, thus advancing understanding of eigenvalue optimization on these surfaces.

Contribution

It proves that all previously known extremal metrics on the torus and Klein bottle are not maximal, clarifying their limitations in eigenvalue optimization.

Findings

All known extremal metrics on the torus and Klein bottle are not maximal.

The results refine the understanding of eigenvalue extremality on these surfaces.

Provides insights into the structure of extremal metrics for Laplace eigenvalues.

Abstract

El Soufi-Ilias' theorem establishes a connection between minimal submanifolds of spheres and extremal metrics for eigenvalues of the Laplace-Beltrami operator. Recently, this connection was used to provide several explicit examples of extremal metrics. We investigate the maximality of these metrics and prove that all of them are not maximal.