The Jacobi operator and the stability of vertical minimal surfaces in the sub-Riemannian Lie group SL(2,R)

Abstract

We consider oriented immersed minimal surfaces in three-dimensional sub-Riemannian manifolds which are vertical, i.e., perpendicular to the two-dimensional horizontal distribution of the sub-Riemannian structure. We showed earlier that a vertical surface is minimal in the sub-Riemannian sense if and only if it is minimal in the Riemannian sense and that its sub-Riemannian stability implies its Riemannian stability. We introduce the sub-Riemannian version of the Jacobi operator for such surfaces and prove a sufficient condition for the stability of vertical minimal surfaces similar to a theorem of Fischer-Colbrie and Schoen: if a surface allows a positive function with the vanishing Jacobi operator then it is stable.

Next, we use the Jacobi operator technique to investigate vertical minimal surfaces in the Lie group $\widetilde{\mathrm{SL}(2,\mathbb{R})}$ that can be described as the universal covering of the unit tangent bundle of the hyperbolic plane with the standard left-invariant Sasaki metric (that corresponds to one of the Thurston geometries) and with two different types of sub-Riemannian structures. First, we consider a family of non-left-invariant structures defined by some parameters, find the values of parameters for which vertical minimal surfaces exist, and describe such complete connected surfaces. These are Euclidean half-planes and cylinders, and they all are stable in the sub-Riemannian sense and thus in the Riemannian sense. In particular, this gives us examples of structures that do not allow vertical minimal surfaces. Then, we describe complete connected vertical minimal surfaces for another sub-Riemannian structure that is left-invariant. These are half-planes and helicoidal surfaces that also appear to be stable in the sub-Riemannian sense and thus in the Riemannian sense.