# Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the $(2,3)$ case

ESAIM: Control, Optimisation and Calculus of Variations (2009)

- Volume: 15, Issue: 4, page 839-862
- ISSN: 1292-8119

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topShcherbakova, 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 15.4 (2009): 839-862. <http://eudml.org/doc/245274>.

@article{Shcherbakova2009,

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.},

author = {Shcherbakova, Nataliya},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {sub-riemannian geometry; minimal surfaces; singular sets; horizontal area functional; Heisenberg group},

language = {eng},

number = {4},

pages = {839-862},

publisher = {EDP-Sciences},

title = {Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the $(2,3)$ case},

url = {http://eudml.org/doc/245274},

volume = {15},

year = {2009},

}

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

PB - EDP-Sciences

VL - 15

IS - 4

SP - 839

EP - 862

AB - 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.

LA - eng

KW - sub-riemannian geometry; minimal surfaces; singular sets; horizontal area functional; Heisenberg group

UR - http://eudml.org/doc/245274

ER -

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