Area-stationary surfaces inside the sub-Riemannian three-sphere
| dc.creator | Hurtado, Ana | |
| dc.creator | Rosales, César | |
| dc.date | 2006-08-02 | |
| dc.date.accessioned | 2026-07-07T07:21:18Z | |
| dc.date.available | 2026-07-07T07:21:18Z | |
| dc.description | 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 $(\mathbb{S}^3,g_{h})$. | |
| dc.description | 28 pages, 5 figures | |
| dc.identifier | https://arxiv.org/abs/math/0608067 | |
| dc.identifier | http://arxiv.org/abs/math/0608067 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/115258 | |
| dc.subject | Differential Geometry | |
| dc.subject | Metric Geometry | |
| dc.subject | 53C17; 49Q20 | |
| dc.title | Area-stationary surfaces inside the sub-Riemannian three-sphere | |
| dc.type | text |