On stable CMC hypersurfaces with free-boundary in a Euclidean Ball

TL;DR

This paper establishes bounds on the geometric quantities of stable constant mean curvature hypersurfaces with free boundary in a Euclidean ball, leading to classification results in specific dimensions.

Contribution

It improves previous theorems by providing sharper bounds and classification criteria for stable CMC hypersurfaces with free boundary, especially in three dimensions.

Findings

Bounds on length, area, and mean curvature for stable CMC hypersurfaces.

Classification of stable CMC surfaces in 3D as totally geodesic disks or spherical caps.

No use of extended Hersch type balancing argument, relying instead on a Nunes type Stability Lemma.

Abstract

In this note, we observe that if $B$ is a ball in a Euclidean space with dimension $n$, $n\geq3$, then a stable CMC hypersurface $\Sigma$ with free boundary in $B$ satisfies \[ nA\leq L\leq nA\left( \frac{1+\sqrt{1+4(n+1)H^2}}{2} \right)\,, \] where $L$, $A$ and $H$ denote the length of $\partial \Sigma$, the area of $\Sigma$ and the mean curvature of $\Sigma$, respectively. Consequently, if the boundary $\partial \Sigma$ is embedded then $\Sigma$ must be totally geodesic or starshaped with respect to the center of the ball. This result is an improvement of a theorem proved by A. Ros and E. Vergasta \cite{R-V} . In particular, if $n=3$, the only stable CMC surfaces with free boundary in $B$ are the totally geodesic disks or the spherical caps. This last result was proved very recently by I. Nunes \cite{N} using an extended stability result and a modified Hersch type balancing argument to get a better control on the genus. We don't use that modified Hersch type argument. However, we use a Nunes type Stability Lemma and a crucial result due to A. Ros and E. Vergasta.