Minimal surfaces in the 3-sphere by stacking Clifford tori

Abstract

Extending work of Kapouleas and Yang, for any integers $N\ge 2$, $k,\ell \ge 1$, 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 $N$.

The final version will appear in JDG

Cited by in corpus (2)

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