DEFORMATION AND STABILITY OF SURFACES WITH CONSTANT MEAN CURVATURE

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1. Let P be a complete surface in the threee-dimensional euclidean space. Each critical point of the area functional, among all immersed surfaces with boundary in P and with a given volume, is called a stationary immersion for supporting surface P.In the special case where P is a plane, we proved that any stable stationary immersion is an embedding onto a hemisphere.2. We studied properties and shapes of nodoids, which are the surfaces of Delaunay (surfaces of revolution with constant mean curvature in the threee-dimensional euclidean space) with self-intersections3. We obtained sufficient conditions for a immersed surface with constant mean curvature (CMC) in the threee-dimensional euclidean space under which it has a CMC-deformation that fixes the boundary. Moreover, we obtained a criterion of the stability for CMC immersions. Both of these are achieved by using the properties of the eigenvalues and the eigenfunctions of the eigenvalue problem associated to the second variation of the area functional. In a certain special case, by combining these results, we obtained a 'visible' way of judging the stability.4. We proved that the well-known second variation formula of the area function for regular minimal surfaces is valid also for generalized minimal surfaces (minimal surfaces with branch points) for 'good' variations.5. We derive sufficient conditions for immersed surfaces with constant mean curvature in three-dimensional space forms to be strongly stable.