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 geometry, minimal surfaces, singular sets

@article{COCV_2009__15_4_839_0, author = {Shcherbakova, Nataliya}, 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}, pages = {839--862}, publisher = {EDP-Sciences}, volume = {15}, number = {4}, year = {2009}, doi = {10.1051/cocv:2008051}, mrnumber = {2567248}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008051/} }

TY - JOUR AU - Shcherbakova, Nataliya TI - Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the $(2,3)$ case JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 839 EP - 862 VL - 15 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008051/ DO - 10.1051/cocv:2008051 LA - en ID - COCV_2009__15_4_839_0 ER -

%0 Journal Article %A Shcherbakova, Nataliya %T Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the $(2,3)$ case %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 839-862 %V 15 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008051/ %R 10.1051/cocv:2008051 %G en %F COCV_2009__15_4_839_0

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, Volume 15 (2009) no. 4, pp. 839-862. doi : 10.1051/cocv:2008051. http://archive.numdam.org/articles/10.1051/cocv:2008051/

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