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≥2\documentclass[12pt]{minimal}
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\begin{document}$$n+1, n\ge 2$$\end{document}, then a CMC free-boundary stable hypersurface Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} in B satisfies InH22∫Σ(1-|x|2)dvolΣ+nA≤L≤nA1+H,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{nH^2}{2}\int _{\Sigma }(1-|x|^2){ dvol}_{\Sigma }+nA\le L\le nA\left( 1+H \right) , \end{aligned}$$\end{document}where L, A and H denote the length of ∂Σ\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Sigma $$\end{document}, the area of Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} and the mean curvature of Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}, respectively, and the orientation of Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} is in a such way that H≥0\documentclass[12pt]{minimal}
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\begin{document}$$H\ge 0$$\end{document}. The left side of (I) is an equality if, and only if, Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} is a totally geodesic disk or a spherical cap. Consequently, if the boundary ∂Σ\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Sigma $$\end{document} of Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} is embedded then Σ\documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$n=2$$\end{document} (in this case its not necessary to assume the boundary ∂Σ\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Sigma $$\end{document} 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.
