Stability of Hypersurfaces of Constant Mean Curvature in Riemannian Manifolds

Abstract

Hypersurfaces \(M^n\)with constant mean curvature in a Riemannian manifold \(\overline{M}^{n+1}\)display many similarities with minimal hypersurfaces of \(\overline{M}^{n+1}\). They are both solutions to the variational problem of minimizing the area function for certain variations. In the first case, however, the admissible variations are only those that leave a certain volume function fixed (for precise definitions, see Sect. 2). This isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces.