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