Minimal surfaces in the 3-sphere by doubling the Clifford torus over rectangular lattices

TL;DR

This paper constructs new sequences of minimal surfaces in the 3-sphere that approximate the Clifford torus with increasing genus, using catenoidal bridges arranged over rectangular lattices, extending previous symmetric constructions.

Contribution

It introduces a method to generate minimal surfaces with arbitrary rectangular lattice ratios, breaking previous symmetry constraints and increasing genus while converging to the Clifford torus.

Findings

Sequences converge to the Clifford torus with multiplicity two.

Genus of surfaces tends to infinity.

Constructed surfaces are not symmetric unless lattice is square.

Abstract

Building on work of Kapouleas and Yang, we construct sequences of minimal surfaces embedded in the round 3-sphere which converge to the Clifford torus counted with multiplicity two and have second fundamental form blowing up at every point of the torus and genus tending to infinity. Each surface in a given sequence resembles a pair of tori close to the limit torus and joined by many catenoidal bridges arranged over a rectangular lattice on the limit. The collection of sequences is indexed by the ratio of the lengths of the lattice edges, which may be any prescribed positive rational. Unlike the surfaces of Kapouleas and Yang, these new embeddings are not symmetric with respect to any isometries of the 3-sphere exchanging the two sides of the limit Clifford torus, except when the corresponding lattice is square.