MINIMAL SURFACES IN THE 3-SPHERE BY STACKING CLIFFORD TORI

Extending work of Kapouleas and Yang, for any integers N ≥ 2, k, ` ≥ 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`m2 (N − 1) + 1 and is invariant under a Dkm × D`m subgroup of O(4), where Dn 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 T, that have been connected by k`m2 (N−1) small catenoidal tunnels, with k`m2 tunnels joining each pair of neighboring tori. In the large-m limit for fixed N, k, and `, the corresponding surfaces converge to T counted with multiplicity N.