Strongly stable surfaces in sub-Riemannian $3$-space forms

A surface of constant mean curvature (CMC) equal to $H$ in a sub-Riemannian $3$-manifold is strongly stable if it minimizes the functional $\text{area}+2H\,\text{volume}$ up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian $3$-manifolds. We also produce new examples of $C^1$ complete CMC surfaces with empty singular set in the sub-Riemannian $3$-space forms by studying those ones containing a vertical line. As a consequence, we are able to find complete strongly stable non-vertical surfaces with empty singular set in the sub-Riemannian hyperbolic $3$-space $\mathbb{M}(-1)$. In relation to the Bernstein problem in $\mathbb{M}(-1)$ we discover strongly stable $C^\infty$ entire minimal graphs in $\mathbb{M}(-1)$ different from vertical planes. These examples are in clear contrast with the situation in the first Heisenberg group, where complete strongly stable surfaces with empty singular set are vertical planes. Finally, we analyze the strong stability of CMC surfaces of class $C^2$ and non-empty singular set in the sub-Riemannian $3$-space forms. When these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set.