Frank PACARD
Université Paris 12, Institut Universitaire de France
Constant mean curvature hypersurfaces in Riemannian manifolds

Abstract: Constant mean curvature hypersurfaces appear in the solutions of the isoperimetric problem since they are critical points of the area functional with respect to some volume constraint.
I will concentrate on the study of families of constant mean curvature hypersurfaces which are embedded in some Riemannian manifold, have fixed topology and whose mean curvature may vary from one surface to the other.
Examples of such families of hypersurfaces have been obtained in the earlier 1990 by R.Ye as foliations of a neighborhood of some special points of the manifold by constant mean curvature spheres. More recently new families have also been found in some neighborhood of minimal submanifolds embedded in the ambient Riemannian manifold. The main difficulty in this last construction is due to some resonance phenomena which also appears in many different nonlinear problem (semilinear elliptic partial differential equations related to nonlinear Schroedinger equations, perturbed Hamiltonian systems,...).
It steems from these constructions that the structure of constant mean curvature hypersurfaces should be rather simple when the mean curvature is assumed to be large enough. I will also report some recent work in this direction.