Volume 15, Number 4, October-December 2009
|Page(s)||839 - 862|
|Published online||19 July 2008|
Minimal surfaces in sub-Riemannian manifolds and structure of their singular sets in the (2,3) case
SISSA/ISAS, via Beirut 2-4, 34100, Trieste, Italy. email@example.com
Revised: 20 April 2008
We study minimal surfaces in sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called horizontal area functional associated with the canonical horizontal area form. We derive the intrinsic equation in the general case and then consider in greater detail 2-dimensional surfaces in contact manifolds of dimension 3. We show that in this case minimal surfaces are projections of a special class of 2-dimensional surfaces in the horizontal spherical bundle over the base manifold. The singularities of minimal surfaces turn out to be the singularities of this projection, and we give a complete local classification of them. We illustrate our results by examples in the Heisenberg group and the group of roto-translations.
Mathematics Subject Classification: 53C17 / 32S25
Key words: Sub-Riemannian geometry / minimal surfaces / singular sets
© EDP Sciences, SMAI, 2008
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