Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-24T22:38:49.133Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  19 July 2008

Nataliya Shcherbakova
Nataliya Shcherbakova*
Affiliation:
SISSA/ISAS, via Beirut 2-4, 34100, Trieste, Italy. chtch@sissa.it
Get access

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. 


Keywords

Sub-Riemannian geometryminimal surfacessingular sets
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 4 , October 2009 , pp. 839 - 862
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agrachev, A.A., Exponential mappings for contact sub-Riemannian structures. J. Dyn. Contr. Syst. 2 (1996) 321–358. CrossRef
A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Berlin, Springer-Verlag (2004).
V.I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equations. Berlin, Springer-Verlag (1988).
V.I. Arnold, Ordinary differential equations. Berlin, Springer-Verlag (1992).
Bella, A. $\ddot \i$ che, The tangent space in sub-Riemannian geometry. Progress in Mathematics 144 (1996) 1–78.
Cheng, J.-H. and Hwang, J.-F., Properly embedded and immersed minimal surfaces in the Heisenberg group. Bull. Austral. Math. Soc. 70 (2004) 507–520. CrossRef
Cheng, J.-H., Hwang, J.-F., Malchiodi, A. and Yang, P., Minimal surfaces in pseudohermitian geometry. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4 (2005) 129–177.
Cheng, J.-H., Hwang, J.-F. and Yang, P., Existence and uniqueness for p-area minimizers in the Heisenberg group. Math. Ann. 337 (2007) 253–293. CrossRef
Citti, G. and Sarti, A., A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vision 24 (2006) 307–326. CrossRef
Franchi, B., Serapioni, R. and Serra Cassano, F., Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321 (2001) 479–531.
Garofalo, N. and Nhieu, D.-M., Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math. 49 (1996) 479–531. 3.0.CO;2-A>CrossRef
N. Garofalo and S. Pauls, The Bernstein problem in the Heisenberg group. Preprint (2004) arXiv:math/0209065v2.
R. Hladky and S. Pauls, Minimal surfaces in the roto-translational group with applications to a neuro-biological image completion model. Preprint (2005) arXiv:math/0509636v1.
R. Montgomery, A tour of subriemannian geometries, their geodesics and applications. Providence, R.I. American Mathematical Society (2002).
Pauls, S., Minimal surfaces in the Heisenberg group. Geom. Dedicata 104 (2004) 201–231. CrossRef
Ritoré, M. and Rosales, C., Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ${\mathbb H}^n$ . J. Geom. Anal. 16 (2006) 703–720. CrossRef
Whitney, H., The general type of singularity of a set of $2n-1$ smooth functions of n variables. Duke Math. J. 10 (1943) 161–172. CrossRef