Rotationally symmetric critical metrics for Laplace eigenvalues on tori in a conformal class

TL;DR

This paper constructs rotationally symmetric critical metrics for the first Laplace eigenvalue on tori within a conformal class, refining previous results and linking minimal tori in spheres to these metrics through harmonic maps.

Contribution

It constructs explicit $ ext{S}^1$-equivariant harmonic maps from tori to spheres, identifying critical metrics that maximize the eigenvalue in non-rhombic conformal classes, and relates Otsuki tori to these metrics.

Findings

Existence of rotationally symmetric critical metrics with higher eigenvalues than flat metrics.

Explicit parametrization of Otsuki tori using elliptic integrals.

Maximal rotationally invariant metrics are $ ext{S}^1$-equivariant and unique up to scalar.

Abstract

We study the problem of maximizing the first Laplace-Beltrami eigenvalue normalized by area in a conformal class on a torus. By a result of Nadirashvili, El Soufi, and Ilias, critical metrics for the $k$-th normalized Laplace-Beltrami eigenvalue functional $\bar\lambda_k$ in a conformal class correspond to harmonic maps to spheres. In this paper we construct certain $\mathbb S^1$-equivariant harmonic maps $\mathbb T^2\to\mathbb S^3$. For each non-rhombic conformal class on a torus, one of these maps corresponds to a rotationally symmetric critical metric for $\bar\lambda_1$ in this conformal class with the value of $\bar\lambda_1$ being greater than that of the flat metric. This refines a recent result by Karpukhin that answers a question by El Soufi, Ilias, and Ros. Also, we are able to show that if a rotationally invariant metric on a rectangular torus is maximal for $\bar\lambda_1$ in its conformal class, then it is $\mathbb S^1$-equivariant and coincides (up to a scalar factor) with the above metric. Finally, we show that a family of minimal tori in $\mathbb S^3$ called Otsuki tori fits naturally into our family. This gives an explicit parametrization of Otsuki tori in terms of elliptic integrals.