Free boundary minimal surfaces in the unit ball with low cohomogeneity

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.