Totally umbilic surfaces in homogeneous 3-manifolds

We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in $H^2 \times R$, $S^2 \times R$ and the Sol group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms.Comment: 28 pages, v2 : corrected typos, new section added. To appear in Commentarii Math. Helve

Mean curvature rigidity of horospheres, hyperspheres and hyperplanes

We prove that horospheres, hyperspheres and hyperplanes in a hyperbolic space H n , n ≥ 3, admit no perturbations with compact support which increase their mean curvature. is is an extension of the analogous result in the Euclidean spaces, due to M. Gromov, which states that a hyperplane in a Euclidean space R n admits no compactly supported perturbations having mean curvature ≥ 0