Area-stationary surfaces in the Heisenberg group H^1
| dc.creator | Ritoré, Manuel | |
| dc.creator | Rosales, César | |
| dc.date | 2005-12-23 | |
| dc.date | 2006-04-08 | |
| dc.date.accessioned | 2026-07-07T06:55:43Z | |
| dc.date.available | 2026-07-07T06:55:43Z | |
| dc.description | We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carathéodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preserving condition is assumed) and such that the characteristic curves meet orthogonally the singular curves. Moreover, a Minkowski type formula relating the area, the mean curvature, and the volume is obtained for volume-preserving area-stationary surfaces enclosing a given region. As a consequence of the characterization of area-stationary surfaces, we refine previous Bernstein type theorems in order to describe entire area-stationary graphs over the xy-plane in H^1. A calibration argument shows that these graphs are globally area-minimizing. Finally, by using the known description of the singular set, the characterization of area-stationary surfaces, and the ruling property of constant mean curvature surfaces, we prove our main results where we classify volume-preserving area-stationary surfaces in H^1 with non-empty singular set. In particular, we deduce the following counterpart to Alexandrov uniqueness theorem in Euclidean space: any compact, connected, C^2 surface in H^1 area-stationary under a volume constraint must be congruent with a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points. As a consequence, we solve the isoperimetric problem in H^1 assuming C^2 smoothness of the solutions. | |
| dc.description | 37 pages, 3 figures; corrected typos, very recent references added | |
| dc.identifier | https://arxiv.org/abs/math/0512547 | |
| dc.identifier | http://arxiv.org/abs/math/0512547 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/106373 | |
| dc.subject | Differential Geometry | |
| dc.subject | Metric Geometry | |
| dc.subject | Primary 53C17, Secondary 49Q20 | |
| dc.title | Area-stationary surfaces in the Heisenberg group H^1 | |
| dc.type | text |