Deformation and stability of surfaces with constant mean curvature
From the introduction and abstract: An immersion \({\mathcal X}:M\to \mathbb{R}^3\) of a two-dimensional orientable compact connected \(C^\infty\) manifold \(M\) with boundary \(\partial M\) into the Euclidean three-space \(\mathbb{R}^3\) has constant mean curvature if and only if \({\mathcal X}\) is a critical point of the area functional for all volume-preserving variations of \({\mathcal X}\) that fix the boundary. When the mean curvature of \({\mathcal X}\) is constant (we will say that \({\mathcal X}\) is a CMc immersion), \({\mathcal X}\) is said to be stable if the second variation of the area functional is nonnegative for all such variations of \({\mathcal X}\) is above. For a CMC immersion from a two-dimensional compact smooth manifold with boundary into the Euclidean three-space, we give sufficient conditions under which it has a CMC deformation fixing the boundary. Moreover, we give a criterion of the stability for CMC immersions. Both of these are achieved by using the properties of eigenvalues and eigenfunctions of an eigenvalue problem associated to the second variation of the area functional. In a certain special case, by combining these results, we obtain a `visible' way of judging the stability.
scientific article; zbMATH DE number 1775139
scientific article