Area-stationary surfaces inside the sub-Riemannian three-sphere

Abstract

We consider the sub-Riemannian metric $g_{h}$ on $\mathbb{S}^3$ 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 $(\mathbb{S}^3,g_{h})$. By following the ideas and techniques in [RR] 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 $(\mathbb{S}^3,g_{h})$ with empty singular set and constant mean curvature $H$ such that $H/\sqrt{1+H^2}$ is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in $(\mathbb{S}^3,g_{h})$.