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

• Published in: Calculus of variations and partial differential equations 2015-11, Vol.54 (3), p.3183-3227
• Main Authors: Hurtado, Ana, Rosales, César
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-015-0898-y

Let M be a complete Sasakian sub-Riemannian 3-manifold of constant Webster scalar curvature κ . For any point p ∈ M and any number λ ∈ R with λ 2 + κ > 0 , we show existence of a C 2 spherical surface S λ ( p ) immersed in M with constant mean curvature λ . Our construction recovers in particular the description of Pansu spheres in the first Heisenberg group (Pansu, Conference on differential geometry on homogeneous spaces (Turin, 1983), pp 159–174, 1984 ) and the sub-Riemannian 3-sphere (Hurtado and Rosales, Math Ann 340(3):675–708, 2008 ). Then, we study variational properties of S λ ( p ) related to the area functional. First, we obtain uniqueness results for the spheres S λ ( 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 S λ ( 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.