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

Abstract

We consider the sub-Riemannian metric g _h on ${\mathbb{S}}^{3}$ given 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 by Ritoré and Rosales (Area-stationary surfaces in the Heisenberg group ${{\mathbb{H}}}^1$, arXiv:math.DG/0512547) 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 (${{\mathbb{S}}}^{3},g_{h}$).