Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator

TL;DR

This paper establishes bounds and classifications for the second eigenvalue of the Jacobi operator on immersed surfaces with nonpositive Euler characteristic, revealing extremal surfaces in specific geometric contexts.

Contribution

It provides a sharp upper bound for the second eigenvalue and classifies surfaces attaining this bound, extending spectral geometry results for nonpositive Euler characteristic surfaces.

Findings

Sharp upper bound for the second eigenvalue in Euclidean sphere

Classification of surfaces attaining the upper bound

Totally geodesic tori maximize the eigenvalue in product spaces

Abstract

In this paper, we investigate the spectral properties of the Jacobi operator for immersed surfaces with nonpositive Euler characteristic, extending previous results in the field. We first prove a sharp upper bound for the second eigenvalue of the Jacobi operator for compact surfaces with nonpositive Euler characteristic that are fully immersed in the Euclidean sphere, and then we classify all such surfaces attaining this upper bound. Furthermore, we demonstrate that totally geodesic tori maximize the second eigenvalue among all compact orientable surfaces with positive genus in the product space $\mathbb{S}^1(r) \times \mathbb{S}^2(s)$.