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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 -dimensional surfaces in contact manifolds of dimension . We show that in this case minimal surfaces are projections of a special class of -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.
@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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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, Tome 15 (2009) no. 4, pp. 839-862. doi: 10.1051/cocv:2008051
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