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Minimal surfaces in sub-Riemannian manifoldsand structure of their singular setsin the (2,3) case

Published online by Cambridge University Press:  19 July 2008

Nataliya Shcherbakova
Nataliya Shcherbakova*
Affiliation:
SISSA/ISAS, via Beirut 2-4, 34100, Trieste, Italy. chtch@sissa.it
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Abstract

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. 


Keywords

Sub-Riemannian geometryminimal surfacessingular sets
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 4 , October 2009 , pp. 839 - 862
Copyright
© EDP Sciences, SMAI, 2008

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