Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

TL;DR

This paper classifies complete, stable constant mean curvature surfaces with no singularities in Sasakian sub-Riemannian 3-manifolds, revealing their geometric structure and stability properties across different space forms.

Contribution

It provides a second derivative formula for area involving geometric terms and classifies stable CMC surfaces with empty singular set in various sub-Riemannian 3-manifolds.

Findings

In the Heisenberg group, such surfaces are vertical planes.

In hyperbolic space, horocylinders are characterized by squared mean curvature 1.

In the 3-sphere, all such surfaces are unstable.

Abstract

For constant mean curvature surfaces of class $C^2$ immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classification results for complete surfaces with empty singular set which are stable, i.e., second order minima of the area under a volume constraint, inside the 3-dimensional sub-Riemannian space forms. In the first Heisenberg group we show that such a surface is a vertical plane. In the sub-Riemannian hyperbolic 3-space we give an upper bound for the mean curvature of such surfaces, and we characterize the horocylinders as the only ones with squared mean curvature 1. Finally we deduce that any complete surface with empty singular set in the sub-Riemannian 3-sphere is unstable.