On CMC free-boundary stable hypersurfaces in a Euclidean ball

Abstract

In this note, we prove that if B is the unit ball centred in the origin in the Euclidean space with dimension $n+1, n\ge 2$, then a CMC free-boundary stable hypersurface $\Sigma$ in B satisfies

$$\begin{aligned} \frac{nH^2}{2}\int _{\Sigma }(1-|x|^2){ dvol}_{\Sigma }+nA\le L\le nA\left( 1+H \right) , \end{aligned}$$

(I)

where L, A and H denote the length of $\partial \Sigma$, the area of $\Sigma$ and the mean curvature of $\Sigma$, respectively, and the orientation of $\Sigma$ is in a such way that $H\ge 0$. The left side of (I) is an equality if, and only if, $\Sigma$ is a totally geodesic disk or a spherical cap. Consequently, if the boundary $\partial \Sigma$ of $\Sigma$ is embedded then $\Sigma$ must be either totally geodesic or starshaped with respect to the center of the ball. This result is a slightly improvement of a theorem proved by Ros and Vergasta. In particular, if $n=2$ (in this case its not necessary to assume the boundary $\partial \Sigma$ is embedded), the only CMC free-boundary stable surfaces in B are the totally geodesic disks or the spherical caps. This classification result was proved very recently by Nunes using an extended stability result and a modified Hersch type balancing argument to get a better control on the genus and on the number of connected components of the boundary of the surfaces. We don’t use that modified Hersch type argument. However, we use a Nunes stability type lemma and a crucial result due to Ros and Vergasta.Our technique, considering a Nunes stability type lemma, can be applied to study sets which are stable for the volume-constrained least area problem within the unit ball, and provide a proof for the Sternberg–Zumbrun’s conjecture.