Minimal surfaces in the 3-sphere by stacking Clifford tori

TL;DR

This paper constructs new embedded minimal surfaces in the 3-sphere by stacking and connecting Clifford tori with catenoidal tunnels, extending previous gluing methods to produce highly symmetric, genus-dependent surfaces.

Contribution

It introduces a novel gluing construction for minimal surfaces in the 3-sphere, generalizing prior work to create surfaces with specified genus and symmetry properties.

Findings

Constructed closed embedded minimal surfaces with genus depending on parameters

Surfaces exhibit symmetry under dihedral subgroups of O(4)

As parameter m grows, surfaces converge to a Clifford torus with multiplicity N

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 $N$.