Finite index constant mean curvature hypersurfaces in low dimensions

TL;DR

This paper classifies complete finite index constant mean curvature hypersurfaces in certain low-dimensional ambient spaces, showing they are either minimal or compact under specific curvature conditions.

Contribution

It extends known results by proving that such hypersurfaces are minimal or compact in six-dimensional product manifolds and completes the classification in positive curvature space forms.

Findings

Complete finite index CMC hypersurfaces are minimal or compact in certain 6D manifolds.

Hypersurfaces in hyperbolic 6-space with mean curvature vector length >7 are necessarily compact.

The paper extends classification results to broader classes of ambient manifolds.

Abstract

We prove that every complete finite index immersed CMC hypersurface is either minimal or compact, provided that the ambient six-dimensional manifold is a Riemannian product of a closed manifold with non-negative sectional curvature and a Euclidean factor. This answers affirmatively a question posed by do Carmo, for this class of ambient spaces, and extends known lower dimensional results. As a consequence, we complete the classification of two-sided, complete weakly stable CMC hypersurfaces immersed in the space forms of positive curvature in dimension six. More generally, we study the class of Riemannian manifolds with bounded curvature and obtain several partial results. In particular, we show that a complete, finite index CMC hypersurface immersed in the hyperbolic six-space with mean curvature vector of length greater than seven is necessarily compact.