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, $I{I}_0$, for vertically rigid spaces and, as a first application, use $I{I}_0$ to show that minimal surfaces cannot have points of horizontal positive curvature and, that minimal surfaces in Carnot groups cannot be locally horizontally geometrically convex. We note that the convexity condition is distinct from others currently in the literature.

21 pages

References in corpus (5)

• Minimal surfaces in the Heisenberg group
• Exterior Differential Systems and Euler-Lagrange Partial Differential Equations
• The Bernstein Problem in the Heisenberg Group
• Minimal surfaces in pseudohermitian geometry
• Properly embedded and immersed minimal surfaces in the Heisenberg group

Cited by in corpus (7)

• Sub-Riemannian Calculus on Hypersurfaces in Carnot Groups
• A partial solution of the isoperimetric problem for the Heisenberg group
• Generalized mean curvature flow in Carnot groups
• A Natural Connection on (2,3) Sub-Riemannian Manifolds
• Horizontal Gauss Curvature Flow of Graphs in Carnot Groups
• Geometric second derivative estimates in Carnot groups and convexity
• Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips