New minimal surfaces in the sphere from capillary minimal cones

Abstract

For every $p,q\ge 1$, we construct minimal embeddings of ${\mathbb{S}}^p\times {\mathbb{S}}^q\times {\mathbb{S}}^1$ in ${\mathbb{S}}^{p+q+2}$ by doubling the links of free-boundary minimal cones in ${\mathbb{R}}^{p+q+3}$ with bi-orthogonal symmetry. This solves problems posed by Hsiang-Lawson and Hsiang-Hsiang. The equivariance reduces the minimal surface equation to an ODE, and we prove the existence of capillary minimal cones for every contact angle. We obtain free-boundary solutions as limits of capillary surfaces via a singular shooting problem with infinite initial slope. As the contact angle degenerates to $0$, rescalings of the capillary cones converge to a homogeneous solution of the one-phase Bernoulli problem, further illustrating the connection between one-phase free boundaries and minimal surfaces through the capillary functional.