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

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- 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 (2008): 839-862. <http://eudml.org/doc/90940>.

@article{Shcherbakova2008,

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},

month = {7},

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/90940},

volume = {15},

year = {2008},

}

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

DA - 2008/7//

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/90940

ER -

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