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.

[1] A.A. Agrachev, Exponential mappings for contact sub-Riemannian structures. J. Dyn. Contr. Syst. 2 (1996) 321-358. | MR | Zbl

[2] A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Berlin, Springer-Verlag (2004). | MR | Zbl

[3] V.I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equations. Berlin, Springer-Verlag (1988). | MR | Zbl

[4] V.I. Arnold, Ordinary differential equations. Berlin, Springer-Verlag (1992). | MR

[5] A. Bella$\ddot{ı}$che, The tangent space in sub-Riemannian geometry. Progress in Mathematics 144 (1996) 1-78. | MR | Zbl

[6] J.-H. Cheng and J.-F. Hwang, Properly embedded and immersed minimal surfaces in the Heisenberg group. Bull. Austral. Math. Soc. 70 (2004) 507-520. | MR | Zbl

[7] J.-H. Cheng, J.-F. Hwang, A. Malchiodi and P. Yang, Minimal surfaces in pseudohermitian geometry. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4 (2005) 129-177. | Numdam | MR | Zbl

[8] J.-H. Cheng, J.-F. Hwang and P. Yang, Existence and uniqueness for $p$-area minimizers in the Heisenberg group. Math. Ann. 337 (2007) 253-293. | MR | Zbl

[9] G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vision 24 (2006) 307-326. | MR

[10] B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321 (2001) 479-531. | MR | Zbl

[11] N. Garofalo and D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math. 49 (1996) 479-531. | MR | Zbl

[12] N. Garofalo and S. Pauls, The Bernstein problem in the Heisenberg group. Preprint (2004) arXiv:math/0209065v2.

[13] 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.

[14] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications. Providence, R.I. American Mathematical Society (2002). | MR | Zbl

[15] S. Pauls, Minimal surfaces in the Heisenberg group. Geom. Dedicata 104 (2004) 201-231. | MR | Zbl

[16] M. Ritoré and C. Rosales, Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ${ℍ}^n$. J. Geom. Anal. 16 (2006) 703-720. | MR | Zbl

[17] H. Whitney, The general type of singularity of a set of $2n-1$ smooth functions of $n$ variables. Duke Math. J. 10 (1943) 161-172. | MR | Zbl