Stability for hypersurfaces of constant mean curvature with free boundary (Q1900050)

The paper treats the following question. Let \(B\) be a smooth convex body in \(\mathbb{R}^{n+1}\) and let \(M^n\) be a compact smooth manifold with boundary \(\partial M\). Let \(\phi: M^n\rightarrow \mathbb{R}^{n+1}\) be an immersion that maps \(\text{int } M\) into \(\text{int } B\) and \(\partial M\) into \(\partial B\). Consider volume preserving variations \(\phi_t\) such that \(\phi_t(\text{int } M)\subset \text{int } B\) and \(\phi_t(\partial M)\subset\partial B\). The paper considers hypersurfaces that are stationary and stable under such variations and presents various interesting results. The results are more precise when \(n=2\) and \(B\) is a ball in \(\mathbb{R}^3\). For instance, the following result holds. Assume that \(B\subset \mathbb{R}^3\) is a ball and that \(\phi:M^2\rightarrow\mathbb{R}^3\) is stationary and stable. Then: (i) the genus \(g\) of \(M\) is zero and \(\phi(M)\) is either a totally geodesic disk or a spherical cap, or (ii) \(g=1\) and the number of boundary components of \(M\) is \(r=1\) or \(r=2\).