Free Boundary Minimal Surfaces in the Unit Ball With Low Cohomogeneity

TL;DR

This paper constructs and analyzes free boundary minimal surfaces in high-dimensional unit balls with low cohomogeneity, revealing new examples, their stability properties, and implications for compactness theorems.

Contribution

It introduces new families of free boundary minimal surfaces in high dimensions, including explicit constructions and stability analysis, extending known examples like the critical catenoid.

Findings

Constructed invariant free boundary minimal surfaces for m,n > 1 with m+n ≥ 8.

Found infinite families of unstable free boundary minimal surfaces when m+n < 8.

Proved uniqueness of certain SO(n)-invariant free boundary minimal surfaces in high dimensions.

Abstract

We study free boundary minimal surfaces in the unit ball of low cohomogeneity. For each pair of positive integers $(m,n)$ such that $m, n >1$ and $m+n\geq 8$, we construct a free boundary minimal surface $\Sigma_{m, n} \subset B^{m+n}$(1) invariant under $O(m)\times O(n)$. When $m+n<8$, an instability of the resulting equation allows us to find an infinite family $\{\Sigma_{m,n, k}\}_{k\in \mathbb{N}}$ of such surfaces. In particular, $\{\Sigma_{2, 2, k}\}_{k\in \mathbb{N}}$ is a family of solid tori which converges to the cone over the Clifford Torus as $k$ goes to infinity. These examples indicate that a smooth compactness theorem for Free Boundary Minimal Surfaces due to Fraser and Li does not generally extend to higher dimensions. For each $n\geq 3$, we prove there is a unique nonplanar $SO(n)$-invariant free boundary minimal surface (a "catenoid") $\Sigma_n \subset B^n(1)$. These surfaces generalize the "critical catenoid" in $B^3(1)$ studied by Fraser and Schoen.