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

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.

arXiv commentThis paper has been withdrawn by the author now that its results have been subsumed under arXiv:1502.07420. Readers wishing to review the withdrawn document should beware erroneous decay factors appearing in Lemma 3.8 (where the superscripts of $\gamma$ should be removed) and in Lemma 3.30 and Proposition 3.36 (where the exponent $2$ in the decay factors should be replaced by $\gamma \in (0,1)$)