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

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 $M$.

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