Minimal surfaces in the $3$-sphere by stacking Clifford tori

Abstract

Extending work of Kapouleas and Yang, for any integers $N \geq 2, {k , \ell} \geq 1$, and m sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus ${k \ell m}^2 (N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of $O(4)$, where $D_n$ is the dihedral group of order $2n$. Each such surface resembles the union of $N$ nested topological tori, all small perturbations of a single Clifford torus $\mathbb{T}$, that have been connected by ${k \ell m}^2(N-1)$ small catenoidal tunnels, with ${k \ell m}^2$ tunnels joining each pair of neighboring tori. In the large-$m$ limit for fixed $N$, $k$, and $\ell$, the corresponding surfaces converge to $\mathbb{T}$ counted with multiplicity $N$.