Spectral properties of bipolar surfaces to Otsuki tori

TL;DR

This paper introduces a new countable family of extremal metrics on the torus related to the spectral properties of bipolar surfaces to Otsuki tori, expanding the known examples of extremal Laplace-Beltrami eigenvalue metrics.

Contribution

It provides the first explicit construction of an infinite family of extremal metrics on the torus for Laplace-Beltrami eigenvalues.

Findings

New countable family of extremal metrics on the torus

Explicit examples related to bipolar surfaces to Otsuki tori

Advances understanding of extremal Laplace-Beltrami eigenvalue metrics

Abstract

The $i$-th eigenvalue $\lambda_i$ of the Laplace-Beltrami operator on a surface can be considered as a functional on the space of all Riemannian metrics of unit volume on this surface. Surprisingly only few examples of extremal metrics for these functionals are known. In the present paper a new countable family of extremal metrics on the torus is provided.