Minimal surfaces in the Heisenberg group

Details Minimal surfaces in the Heisenberg group Minimal surfaces in the Heisenberg group

2001-08-07

arxiv.org/abs/math/0108048v2

Geom. Ded. 104:201-231, 2004

Mathematics

Differential Geometry

26 pages, 12 figures

Scientific paper

10.1023/B:GEOM.0000022861.52942.

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot-Caratheodory metric. We generate a large library of examples of minimal surfaces and use these to show that the solution to the Dirichlet problem need not be unique. Moreover, we show that the minimal surfaces we construct are in fact X-minimal surfaces in the sense of Garofalo and Nhieu.

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