Finite Index and Do Carmo Question for Constant Mean Curvature Hypersurfaces

TL;DR

This paper proves that finite index constant mean curvature hypersurfaces in Euclidean space are minimal under certain volume or Ricci curvature conditions, and further must be hyperplanes in some cases, with no dimension restrictions.

Contribution

It establishes new conditions under which finite index constant mean curvature hypersurfaces are necessarily minimal or hyperplanes, extending previous results without dimension restrictions.

Findings

Finite index CMC hypersurfaces with sub-exponential volume growth are minimal.

Under Ricci curvature bounds, such hypersurfaces are minimal and must be hyperplanes.

Results are new even for finite index hypersurfaces with zero index.

Abstract

We prove that any finite $\delta$-index hypersurface $M$ in ${\mathbb R}^{n+1}$ with constant mean curvature must be minimal, provided either of the following conditions holds: - the volume growth of $M$ is sub-exponential; - the Ricci curvature of $M$ satisfies $\operatorname{Ric}_M\geq -\frac{3(1-\delta)}{n-1}|A|^2g,$ where $A$ is the second fundamental form and $g$ is the metric on $M.$ In the second case, our result further implies that, in addition to being minimal, such an $M$ must be a hyperplane. We emphasize that no restriction on the dimension is imposed. Moreover, the statement in the second case is new even for finite index hypersurfaces ($\delta=0$).