Existence, characterization and stability of Pansu spheres in sub-Riemannian $3$-space forms

Abstract

Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $\kappa$. For any point $p\in M$ and any number $\lambda \in \mathbb{R}$ with ${\lambda }^2+\kappa >0$, we show existence of a $C^2$ spherical surface ${\mathcal{S}}_{\lambda }(p)$ immersed in $M$ with constant mean curvature $\lambda$. Our construction recovers in particular the description of Pansu spheres in the first Heisenberg group and the sub-Riemannian $3$-sphere. Then, we study variational properties of ${\mathcal{S}}_{\lambda }(p)$ related to the area functional. First, we obtain uniqueness results for the spheres ${\mathcal{S}}_{\lambda }(p)$ as critical points of the area under a volume constraint, thus providing sub-Riemannian counterparts to the theorems of Hopf and Alexandrov for CMC surfaces in Riemannian $3$-space forms. Second, we derive a second variation formula for admissible deformations possibly moving the singular set, and prove that ${\mathcal{S}}_{\lambda }(p)$ is a second order minimum of the area for those preserving volume. We finally give some applications of our results to the isoperimetric problem in sub-Riemannian $3$-space forms.

Final version, to appear in Calculus of Variations and Partial Differential Equations

References in corpus (3)

• Variation of Perimeter Measure in sub-Riemannian geometry
• Umbilicity and characterization of Pansu spheres in the Heisenberg group
• Stable H-minimal hypersurfaces