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

DOI : https://doi.org/10.1051/cocv:2008051
Classification : 53C17,  32S25
Mots clés : sub-riemannian geometry, minimal surfaces, singular sets
@article{COCV_2009__15_4_839_0,
     author = {Shcherbakova, Nataliya},
     title = {Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the $(2,3)$ case},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {839--862},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {4},
     year = {2009},
     doi = {10.1051/cocv:2008051},
     zbl = {pre05627158},
     mrnumber = {2567248},
     language = {en},
     url = {www.numdam.org/item/COCV_2009__15_4_839_0/}
}
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, Tome 15 (2009) no. 4, pp. 839-862. doi : 10.1051/cocv:2008051. http://www.numdam.org/item/COCV_2009__15_4_839_0/

[1] A.A. Agrachev, Exponential mappings for contact sub-Riemannian structures. J. Dyn. Contr. Syst. 2 (1996) 321-358. | MR 1403262 | Zbl 0941.53022

[2] A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Berlin, Springer-Verlag (2004). | MR 2062547 | Zbl 1062.93001

[3] V.I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equations. Berlin, Springer-Verlag (1988). | MR 947141 | Zbl 0507.34003

[4] V.I. Arnold, Ordinary differential equations. Berlin, Springer-Verlag (1992). | MR 1162307

[5] A. Bellaı ¨che, The tangent space in sub-Riemannian geometry. Progress in Mathematics 144 (1996) 1-78. | MR 1421822 | Zbl 0862.53031

[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 2103983 | Zbl 1062.35046

[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 2165405 | Zbl 1158.53306

[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 2262784 | Zbl 1109.35009

[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 2235475

[10] B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321 (2001) 479-531. | MR 1871966 | Zbl 1057.49032

[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 1404326 | Zbl 0880.35032

[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 1867362 | Zbl 1044.53022

[15] S. Pauls, Minimal surfaces in the Heisenberg group. Geom. Dedicata 104 (2004) 201-231. | MR 2043961 | Zbl 1054.49029

[16] M. Ritoré and C. Rosales, Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group n . J. Geom. Anal. 16 (2006) 703-720. | MR 2271950 | Zbl 1129.53041

[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 7784 | Zbl 0061.37207