2021/06/10 by Ana Hurtado, Hurtado, Ana, César Rosales +1
Mathematics · #49Q05 #49Q20 #53C17 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2106.05661
openalex publication_date 2021/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the sub-Riemannian 3-sphere (\mathbbS3,gh) obtained by restriction of the Riemannian metric of constant curvature 1 to the planar distribution orthogonal to the vertical Hopf vector field. It is known that (\mathbbS3,gh) contains a family of spherical surfaces \Sλ\λ≥ 0 with constant mean curvature λ. In this work we first prove that the two closed half-spheres of S0 with boundary C0=\0\×\mathbbS1 minimize the sub-Riemannian area among compact C1 surfaces with the same boundary. We also see that the only C2 solutions to this Plateau problem are vertical translations of such half-spheres. Second, we establish that the closed 3-ball enclosed by a sphere Sλ with λ>0 uniquely solves the isoperimetric problem in (\mathbbS3,gh) for C1 sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.