## Journal of Differential Geometry

- J. Differential Geom.
- Volume 79, Number 1 (2008), 111-139.

### Constant mean curvature surfaces in sub-Riemannian geometry

#### Abstract

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations for critical points for the associated variational problems. Using the Euler-Lagrange equations, we show that minimal and isoperimetric surfaces satisfy a constant horizontal mean curvature conditions away from characteristic points. Moreover, we use the formalism to construct a horizontal second fundamental form, II_{0}, for vertically rigid spaces and, as a first application, use II_{0} to show that minimal surfaces cannot have points of horizontal positive curvature and that minimal surfaces in Carnot groups cannot be locally strictly horizontally geometrically convex. We note that the convexity condition is distinct from others currently in the literature.

#### Article information

**Source**

J. Differential Geom., Volume 79, Number 1 (2008), 111-139.

**Dates**

First available in Project Euclid: 10 April 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1207834659

**Digital Object Identifier**

doi:10.4310/jdg/1207834659

**Mathematical Reviews number (MathSciNet)**

MR2401420

**Zentralblatt MATH identifier**

1156.53038

#### Citation

Hladky, R. K.; Pauls, S. D. Constant mean curvature surfaces in sub-Riemannian geometry. J. Differential Geom. 79 (2008), no. 1, 111--139. doi:10.4310/jdg/1207834659. https://projecteuclid.org/euclid.jdg/1207834659