Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere

Abstract

Abstract We consider the sub-Riemannian 3-sphere ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} obtained by restriction of the Riemannian metric of constant curvature 1 to the planar distribution orthogonal to the vertical Hopf vector field. It was shown in [A. Hurtado and C. Rosales, Area-stationary surfaces inside the sub-Riemannian three-sphere, Math. Ann. 340 2008, 3, 675–708] that ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} contains a family of spherical surfaces { 𝒮 λ } λ ⩾ 0 {\{\mathcal{S}_{\lambda}\}_{\lambda\geqslant 0}} with constant mean curvature λ. In this work, we first prove that the two closed half-spheres of 𝒮 0 {\mathcal{S}_{0}} with boundary C 0 = { 0 } × 𝕊 1 {C_{0}=\{0\}\times\mathbb{S}^{1}} minimize the sub-Riemannian area among compact C 1 {C^{1}} surfaces with the same boundary. We also see that the only C 2 {C^{2}} solutions to this Plateau problem are vertical translations of such half-spheres. Second, we establish that the closed 3-ball enclosed by a sphere 𝒮 λ {\mathcal{S}_{\lambda}} with λ > 0 {\lambda>0} uniquely solves the isoperimetric problem in ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} for C 1 {C^{1}} sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.