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 (2009)

  • 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 (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 -

References

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