Weingarten spheres in homogeneous three-manifolds

Résumé

A classical theorem by H. Hopf shows that any compact constant mean curvature (CMC) surface of genus zero immersed in the Euclidean 3-space is a round sphere. This was generalized by Abresch and Rosenberg: any compact CMC surface of genus zero immersed in arotationally symmetric homogeneous three-manifold is a rotational sphere. In this talk we willpresent some wide generalizations of these results. In particular, we will show that any compactspecial Weingarten surface of genus zero immersed in the product space H2 × R is a rotationalsphere, and we will extend the Abresch-Rosenberg classification to the case of surfaces withconstant positive extrinsic curvature.