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 S 3 or R 3 . In this paper we prove that the stability index of M is bounded from 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 1forms on M ..

Stability for a Second Type Partitioning Problem

“…There are plenty of works on the index estimate for closed minimal hypersurfaces or minimal hypersurfaces with free boundary, see for example, Ros [20], Savo [25] and Ambrozio-Carlotto-Sharp [2,3]. See also [19,11,12] for index estimate for CMC surfaces with free boundary, which is related to type-I partitioning problem. The technique in [19,11,12] for non-minimal CMC case only applies for two dimension.…”

“…See also [19,11,12] for index estimate for CMC surfaces with free boundary, which is related to type-I partitioning problem. The technique in [19,11,12] for non-minimal CMC case only applies for two dimension.…”

Index Estimates for Surfaces with Constant Mean Curvature in $3$-dimensional Manifolds

“…Let us mention that Cavalcande-de Oliveira in [6,7] have obtained similar estimates independent of the inequality condition on the embedding of the ambient manifold for the particular case of closed CMC surfaces in R 3 , S 3 and free boundary CMC surfaces in mean convex domains of these two cases.…”

“…First we would like to mention that Barbosa-do Carmo-Eschenburg [4] have studied CMC hypersurfaces in simply connected spaces of constant sectional curvature and proved that geodesic spheres are the only stable ones. In [6] Cavalcante-de Oliveira have proved this using similar index estimates. Let us now present some examples that satisfy the conditions of the above results using canonical embeddings.…”