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

Nataliya Shcherbakova

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

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

Abstract

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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 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. 


How to cite

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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 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 -

References

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