Stability of minimal surfaces in the sub-Riemannian manifold $\widetilde{E(2)}$

Abstract

In the paper we study smooth oriented surfaces in the universal covering space of the group of orientation-preserving Euclidean plane isometries, which has a three-dimensional sub-Riemannian manifold structure. This structure is constructed as a restriction of the Euclidean metric on the group to some completely non-integrable left invariant distribution. The sub-Riemannian area of a surface is then defined as the integral of the length of its unit normal field projected orthogonally onto this distribution. We calculate the first variation formula of the sub-Riemannian surface area and derive the minimality criterion from it. Here we call a surface minimal if it is a critical point of the sub-Riemannian area functional under normal variations with compact support. We show that the minimality in this case is not equivalent to the vanishing of the sub-Riemannian mean curvature. We then prove that a Euclidean plane is minimal if and only if it is parallel or orthogonal to the $z$-axis (where the $z$-coordinate corresponds to the rotation angle of an isometry). Also we obtain the minimality condition for a graph and give examples of minimal graphs. The examples considered in the paper demonstrate, in particular, that the minimality of a surface in the Riemannian (in this case Euclidean) sense does not imply its sub-Riemannian minimality, and vice versa. Next, we consider the stability of minimal surfaces. For this purpose, we derive the second variation formula of the sub-Riemannian area and show with it that minimal Euclidean planes are stable. We introduce a class of surfaces for which the tangent planes are perpendicular to the planes of the sub-Riemannian structure, and call them vertical surfaces. In particular, for such surfaces the second variation formula is simplified significantly. Then we prove that complete connected vertical minimal surfaces are either Euclidean planes or helicoids and that helicoids are unstable. This implies a following Bernstein type result: a complete connected vertical minimal surface is stable if and only if it is a Euclidean plane orthogonal to the $z$-axis.