On some properties of stable and unstable surfaces with prescribed mean curvature

We investigate the properties of stable (and unstable) hypersurfaces with prescribed mean curvature in Euclidean space and establish some necessary and sufficient tests for stability stated in terms of the external geometric structure of the surface. We prove an analogue of a well-known theorem of A. D. Aleksandrov that generalizes the variational property of the sphere and find an exact estimate for the extent of a stable tubular surface of constant mean curvature. Our method is based on an analysis of the first and second variations of area-type functionals for the surfaces under consideration.