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

Abstract

We consider the sub-Riemannian $3$-sphere $({\mathbb{S}}^3,g_h)$ obtained by restriction of the Riemannian metric of constant curvature $1$ to the planar distribution orthogonal to the vertical Hopf vector field. It is known that $({\mathbb{S}}^3,g_h)$ contains a family of spherical surfaces $\{{\mathcal{S}}_{\lambda }{\}}_{\lambda \ge 0}$ with constant mean curvature $\lambda$. In this work we first prove that the two closed half-spheres of ${\mathcal{S}}_0$ with boundary $C_0=\{0\}\times {\mathbb{S}}^1$ 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 ${\mathcal{S}}_{\lambda }$ with $\lambda >0$ uniquely solves the isoperimetric problem in $({\mathbb{S}}^3,g_h)$ for $C^1$ sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.

arXiv comment16 pages, no figures