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

Abstract

We consider the sub-Riemannian 3- sphere (S-3, gh) 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), no. 3, 675-708] that (S-3, g(h)) contains a family of spherical surfaces {S-lambda}(lambda >= 0) with constant mean curvature.. In this work, we first prove that the two closed half-spheres of S-0 with boundary C-0 = {0} x S-1 minimize the sub-Riemannian area among compact C-1 surfaces with the same boundary. We also see that the only 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-lambda with lambda > 0 uniquely solves the isoperimetric problem in (S-3, g(h)) for C-1 sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.