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

Shcherbakova, Nataliya

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.

Classification : 53C17, 32S25
Keywords: sub-riemannian geometry, minimal surfaces, singular sets

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