Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the (2,3) case
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 839-862.

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.

DOI : 10.1051/cocv:2008051
Classification : 53C17, 32S25
Mots clés : sub-riemannian geometry, minimal surfaces, singular sets
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     title = {Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the $(2,3)$ case},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
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Shcherbakova, Nataliya. Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the $(2,3)$ case. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 839-862. doi : 10.1051/cocv:2008051. http://www.numdam.org/articles/10.1051/cocv:2008051/

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