Minimal surfaces in contact Sub-Riemannian manifolds

TL;DR

This paper develops a framework for understanding Sub-Riemannian minimal surfaces in contact manifolds, deriving intrinsic equations, analyzing characteristic points, and providing explicit solutions in specific groups like Heisenberg and roto-translation groups.

Contribution

It introduces a canonical volume and horizontal area form for generic co-rank 1 Sub-Riemannian structures and derives intrinsic equations for minimal surfaces, extending classical concepts to non-holonomic distributions.

Findings

Characterization of minimal surfaces via intrinsic equations

Classification of characteristic points in the Heisenberg group

Explicit solutions in the (2,3) contact case

Abstract

In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ''horizontal'' area forms. As in the classical case, the Sub-Riemannian minimal surfaces can be defined as the critical points of the '`horizontal'' area functional. We derive an intrinsic equation for minimal surfaces associated to a generic Sub-Riemannian structure of co-rank 1 in terms of the canonical volume form and the ``horizontal'' normal. The presented construction permits to describe the Sub-Riemannian minimal surfaces in a generic Sub-Riemannian manifold and can be easily generalized to the case of non-holonomic vector distributions of greater co-rank. The case of contact vector distributions, in particular the $(2,3)$-case, is studied more in detail. In the latter case the geometry of the Sub-Riemannian minimal surfaces is determined by the structure of their characteristic points (i.e., the points where the hyper-surface touches the horizontal distribution) and characteristic curves. It turns out that the known classification of the characteristic points of the Sub-Riemannian minimal surfaces in the Heisenberg group $H^1$ holds true for the minimal surfaces associated to a generic contact $(2,3)$ distribution. Moreover, we show that in the $(2,3)$ case the Sub-Riemannian minimal surfaces are the integral surfaces of a certain system of ODE in the extended state space. In some particular cases the Cauchy problem for this system can be solved explicitly. We illustrate our results considering Sub-Riemannian minimal surfaces in the Heisenberg group and the group of roto-translations.