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Zbl 1190.53027
Shcherbakova, Nataliya
Minimal surfaces in sub-Riemannian manifolds and structure of their singular sets in the (2,3) case. (English)
[J] ESAIM, Control Optim. Calc. Var. 15, No. 4, 839-862 (2009). ISSN 1292-8119; ISSN 1262-3377/e

Summary: 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 manifold 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.
[Antonio Masiello (Bari)]

MSC 2000:: *53C17 Sub-Riemannian geometry
32S25 (Hyper-) Surface singularities (analytic spaces)
53C42 Immersions (differential geometry)

Keywords: minimal surfaces; singular sets; horizontal area functional; Heisenberg group

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