Lower bounds for the index of compact constant mean curvature surfaces in $\mathbb R^{3}$ and $\mathbb S^{3}$

TL;DR

This paper establishes a lower bound on the stability index of compact constant mean curvature surfaces in three-dimensional space forms, linking it to the genus and comparing spectral properties of the Jacobi operator with Hodge Laplacian.

Contribution

It provides the first linear lower bound on the stability index in terms of genus for such surfaces and introduces a spectral comparison theorem.

Findings

Stability index is bounded below by a linear function of genus.

Spectral comparison between Jacobi operator and Hodge Laplacian established.

Results apply to surfaces in both $ ext{S}^3$ and $ ext{R}^3$.

Abstract

Let $M$ be a compact constant mean curvature surface either in $\mathbb{S}^3$ or $\mathbb{R}^3$. In this paper we prove that the stability index of $M$ is bounded below by a linear function of the genus. As a by product we obtain a comparison theorem between the spectrum of the Jacobi operator of $M$ and those of Hodge Laplacian of $1$-forms on $M$.