Minimal Surfaces in Sub-riemannian Manifolds and Structure of Their Singular Sets in the (2, 3) Case

We study minimal surfaces in generic 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 to 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. Generic singularities of minimal surfaces turn out 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. Introduction In the classical Riemannian geometry minimal surfaces realize the critical points of the area functional with respect to variations preserving the boundary of a given domain. In this paper we study the generalization of the notion of minimal surfaces in sub-Riemannian manifolds known also as the Carnot-Carathéodory spaces. This problem was first introduced in the framework of Geometric Measure Theory for the Lie groups. Mainly the obtained results ([8], [9], [10], [11], [4], [5], [13]) concerns the Heisenberg groups, in particular H; in [7] and [12] the authors were studying the group E2 of roto-translations of the plane, in [4] there were also obtained some results for the case of S. In [4], followed by just appeared paper [6], the authors considered the problem in a more general setting and introduced the notion of minimal surfaces associated to CR structures in pseudohermitian manifolds of any dimension. In this paper we develop a different approach using the methods of sub-Riemannian geometry. Though in particular cases of Lie Groups H, E2 and S 3 the surfaces introduced in [4] are minimal also in the sub-Riemannian sense, in general it is not true. The sub-Riemannian point of view on the problem is based on the following construction. Consider an n-dimensional smooth manifold M and a co-rank 1 smooth vector distribution ∆ in it (“horizontal” distribution). It is assumed that the sections of ∆ are endowed with a Euclidean structure, which can be described by fixing an orthonormal basis of vector fields X1, . . . , Xn−1 on ∆ (see [3]). Then ∆ defines a sub-Riemannian structure in M . In this case M is said to be a sub-Riemannian manifold. Given a sub-Riemannian structure there is a canonical way to define a volume form μ ∈ ΛM associated to it. In addition, for any hypersurface W ⊂ M the horizontal unite vector ν such that ∫ Ω iνμ = max X∈∆ ‖X‖∆=1 ∫ Ω iXμ, Ω ⊂ W, plays the role of the Riemannian normal in the classical case, and the n− 1-form iνμ defines the horizontal area form on W . All these notions are direct generalizations of the the classical ones in the Riemannian geometry (i.e., in the case ∆ ≡ TM). Going further in this direction, we define sub-Riemannian minimal surfaces in M as the critical points of the functional associated to the horizontal area form. It turns out that these surfaces 1991 Mathematics Subject Classification. 53C17, 32S25.