Extremal spectral properties of Otsuki tori

TL;DR

This paper explicitly identifies extremal eigenvalues and metrics for Otsuki tori, a family of minimal tori in the 3-sphere, revealing new spectral extremal properties despite their implicit geometric definition.

Contribution

It explicitly determines the extremal eigenvalues and metrics for Otsuki tori, including an extremal metric for the third eigenvalue, advancing understanding of their spectral geometry.

Findings

Explicit eigenvalue numbers for Otsuki tori

An extremal metric for the third eigenvalue

Spectral extremal properties of implicitly defined tori

Abstract

Otsuki tori form a countable family of immersed minimal two-dimensional tori in the unitary three-dimensional sphere. According to El Soufi-Ilias theorem, the metrics on the Otsuki tori are extremal for some unknown eigenvalues of the Laplace-Beltrami operator. Despite the fact that the Otsuki tori are defined in quite an implicit way, we find explicitly the numbers of the corresponding extremal eigenvalues. In particular we provide an extremal metric for the third eigenvalue of the torus.