[Paper Review] Minimal surfaces in contact Sub-Riemannian manifolds

Nataliya Shcherbakova|ArXiv.org|Apr 23, 2006

Topological and Geometric Data Analysis5 references3 citations

TL;DR

This paper introduces a coordinate-free, intrinsic definition of minimal surfaces in contact Sub-Riemannian manifolds of co-rank 1 by formulating the minimal surface condition as the vanishing of the exterior derivative of the horizontal area form restricted to the surface. It establishes that such surfaces are integral manifolds of a system of ODEs in the extended space $M \times S^1$, and proves that in the $(2,3)$-contact case, the classification of characteristic points and the condition for characteristic curves to be geodesics generalize beyond the Heisenberg group to all generic contact structures.

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.

Motivation & Objective

• To provide a coordinate-free, intrinsic definition of minimal surfaces in co-rank 1 Sub-Riemannian manifolds using canonical volume and horizontal area forms.
• To derive an intrinsic equation for minimal surfaces in terms of the canonical volume form and horizontal normal, valid for generic Sub-Riemannian structures.
• To generalize the known classification of characteristic points in the Heisenberg group to all generic $(2,3)$-contact Sub-Riemannian structures.
• To show that Sub-Riemannian minimal surfaces are integral surfaces of a system of ODEs in the extended space $M \times S^1$.
• To characterize when characteristic curves coincide with Sub-Riemannian geodesics, identifying necessary and sufficient conditions on structural constants.

Proposed method

• Define the canonical volume form $\mu$ and horizontal area form $i_\nu \mu$ on a co-rank 1 Sub-Riemannian manifold using an orthonormal frame on the horizontal distribution $\Delta$.
• Define the Sub-Riemannian normal $\nu$ as the unit vector field maximizing the integral of $i_X \mu$ over unit horizontal vector fields $X$.
• Formulate the minimal surface condition as $(d \circ i_\nu \mu)|_W = 0$, an intrinsic PDE on the surface $W$.
• Apply the method of characteristics to the minimal surface equation, leading to a system of ODEs in the extended space $M \times S^1$ with a $S^1$-valued parameter $\psi$.
• Use the Pontryagin Maximum Principle to relate the extremal curves in $T^*M$ to Sub-Riemannian geodesics, and identify conditions under which characteristic curves are geodesics.
• Derive explicit conditions on the structural constants $c_{ij}^k$ for characteristic curves to be geodesics, distinguishing cases based on vanishing of $c_{12}^1$, $c_{12}^2$, and $c_{32}^1 + c_{31}^2$.

Experimental results

Research questions

• RQ1 Can minimal surfaces in generic co-rank 1 Sub-Riemannian manifolds be defined intrinsically without relying on global structures like CR-geometry?
• RQ2 What is the intrinsic PDE governing minimal surfaces in terms of the canonical volume and horizontal normal?
• RQ3 Does the classification of characteristic points in the Heisenberg group extend to all $(2,3)$-contact Sub-Riemannian structures?
• RQ4 Under what conditions are the characteristic curves of a minimal surface also Sub-Riemannian geodesics?
• RQ5 Can the geometry of minimal surfaces be fully described by a system of ODEs in an extended state space?

Key findings

• The intrinsic minimal surface equation $(d \circ i_\nu \mu)|_W = 0$ provides a coordinate-free characterization valid for all co-rank 1 Sub-Riemannian structures.
• Sub-Riemannian minimal surfaces in the $(2,3)$-contact case are integral surfaces of a system of ODEs in $M \times S^1$, with the $S^1$-coordinate corresponding to the horizontal direction.
• The classification of characteristic points—elliptic, parabolic, hyperbolic—holds for all generic $(2,3)$-contact structures, not only in the Heisenberg group.
• Characteristic curves are Sub-Riemannian geodesics if and only if the structural constants satisfy $c_{31}^1 \cos 2\phi + \frac{c_{32}^1 + c_{31}^2}{2} \sin 2\phi = 0$, with explicit solutions for $\phi^*$ depending on the constants.
• In the Heisenberg group, all characteristic curves are geodesics and form totally geodesic planes with one characteristic point.
• In the group of roto-translations, only two families of characteristic curves are geodesics: straight lines in the $xy$-plane and vertical lines, corresponding to $\phi^* = \pi/2, 3\pi/2$.