On branched minimal immersions of surfaces by first eigenfunctions

TL;DR

This paper extends the uniqueness of minimal immersions by first eigenfunctions to metrics with conical singularities, revealing differences in properties depending on the sphere dimension, impacting bounds on Laplace eigenvalues.

Contribution

It generalizes Montiel and Ros's theorem to singular metrics from branched minimal immersions, highlighting dimension-dependent properties affecting eigenvalue bounds.

Findings

Metrics from $\

$ ext{S}^2$ differ significantly from those from higher-dimensional spheres.

Provides detailed proofs of sharp upper bounds for the first non-zero Laplace eigenvalue on tori and Klein bottles.

Abstract

It was proved by Montiel and Ros that for each conformal structure on a compact surface there is at most one metric which admits a minimal immersion into some unit sphere by first eigenfunctions. We generalize this theorem to the setting of metrics with conical singularities induced from branched minimal immersions by first eigenfunctions into spheres. Our primary motivation is the fact that metrics realizing maxima of the first non-zero Laplace eigenvalue are induced by minimal branched immersions into spheres. In particular, we show that the properties of such metrics induced from $\mathbb{S}^2$ differ significantly from the properties of those induced from $\mathbb{S}^m$ with $m>2$. This feature appears to be novel and needs to be taken into account in the existing proofs of the sharp upper bounds for the first non-zero eigenvalue of the Laplacian on the $2$-torus and the Klein bottle. In the present paper we address this issue and give a detailed overview of the complete proofs of these upper bounds following the works of Nadirashvili, Jakobson-Nadirashvili-Polterovich, El Soufi-Giacomini-Jazar, Nadirashvili-Sire and Petrides.