Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the $\left(2,3\right)$ 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 : https://doi.org/10.1051/cocv:2008051
Classification : 53C17,  32S25
Mots clés : 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://www.numdam.org/item/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. http://www.numdam.org/item/COCV_2009__15_4_839_0/

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