New minimal surfaces in the sphere from capillary minimal cones

TL;DR

This paper constructs new minimal surfaces in spheres by doubling links of minimal cones with symmetry, solving longstanding problems and establishing a connection between capillary surfaces and minimal surfaces.

Contribution

It introduces a method to generate minimal embeddings of product spheres in higher-dimensional spheres using symmetry and solves open problems in the field.

Findings

Existence of minimal embeddings for all p,q ≥ 1.

Construction of capillary minimal cones with arbitrary contact angles.

Limit of capillary cones as contact angle approaches zero converges to a Bernoulli problem solution.

Abstract

For every $p,q\geq 1$, 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.