Stability of capillary hypersurfaces with constant higher order mean curvature (Q6635795)

Let \(M^{n+1}\) be an oriented Riemannian manifold and \(\varphi:\Sigma^{n} \rightarrow M^{n+1}\) be an oriented hypersurface. The higher order mean curvature of order \(k\) of the hypersurface, \(H_k\), is defined as the normalized \(k\)-th symmetric function of the principal curvatures of \(\varphi\).\N\NGiven a closed domain \(\Omega \subset M^{n+1}\) with smooth boundary, the hypersurface \(\varphi(\Sigma^{n})\) is said to be supported on \(\partial\Omega\) if \(\varphi(\Sigma^{n})\subset \Omega\) and \(\varphi(\partial\Sigma^n)\subset \partial\Omega\). Moreover, \(\varphi\) is capillary if \(\varphi(\partial\Sigma^{n})\subset \varphi(\Sigma^{n})\) meets \(\partial\Omega\) at a constant angle.\N\NIn this paper, the authors introduce and investigate stability for capillary hypersurfaces with constant higher order mean curvature \(H_{k}\). For example, they prove that totally umbilical compact capillary hypersurfaces with constant \(H_{k}\) supported on a totally umbilical hypersurface \(\partial\Omega\) of a space form are stable. In contrast, for \(k > 1\), there does not exist a compact stable hypersurface with \(H_{k}=0\) and free boundary in a geodesic ball of a space form.