New minimal surfaces in the sphere from capillary minimal cones

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.

Figures (1)

• Figure 1: The figures show the graphs of the functions $f_\theta$ for angles $\theta$ ranging from $\varepsilon$ to $\tfrac{\pi}{2}$, with the free-boundary profile $f_{\frac{\pi}{2}}$ in black. The left figure illustrates $(n,k) = (4,2)$, when each profile function is symmetric under $t \leftrightsquigarrow \sqrt{1-t^2}$, with a peak at $t = \tfrac{1}{\sqrt{2}}$. The right figure illustrates $(n,k) = (5,2)$, which does not have such a symmetry. As $\theta \downarrow 0$, the rescaled functions $\frac{1}{\theta} (\rho f_{\theta})$ converge in $C^{\infty}_{\text{loc}}$ to a homogeneous solution of the one-phase problem.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4