A ug 2 00 6 AREA - STATIONARY SURFACES INSIDE THE SUB - RIEMANNIAN THREE - SPHERE

We consider the sub-Riemannian metric g h on S 3 provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot-Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus. We study area-stationary surfaces with or without a volume constraint in (S 3 , g h). By following the ideas and techniques in [RR2] we introduce a variational notion of mean curvature , characterize stationary surfaces, and prove classification results for complete volume-preserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot-Carathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only C 2 compact, connected, embedded surfaces in (S 3 , g h) with empty singular set and constant mean curvature H such that H/ √ 1 + H 2 is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in (S 3 , g h).