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

2015/01/20 by Hurtado, Ana, Rosales, César
#53C17 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1501.04886

Abstract

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 λ²+κ>0, we show existence of a C² 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 and the sub-Riemannian 3-sphere. 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.

Related

• Area-minimizing properties of Pansu spheres in the sub-riemannian 3-sphere
• An instability criterion for volume-preserving area-stationary surfaces with singular curves in sub-Riemannian 3-space forms
• Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds
• Strongly stable surfaces in sub-Riemannian 3-space forms
• Pansu-Wulff shapes in ℍ¹