Non-proper helicoid-like limits of closed minimal surfaces in 3-manifolds

Abstract

We show that there exists a metric with positive scalar curvature on S 2 × S 1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two helicoid-like singularities on the 2-sphere. The construction is inspired by a recent example by D. Hoffman and B. White.

Topological type of limit laminations of embedded minimal disks

“…Proof. If γ is such a geodesic, then Ψ(γ ∩ B ǫ ) is a proper geodesic in B 3 4 π that contains p. Hence Ψ(γ ∩ B ǫ ), has length at least 3 2 π > π and so is unstable.…”

“…Two of the limit leaves are the tori L 2 = T 2 × {−1} and L 3 = T 2 × {1}, the other two L 4 and L 5 are non-proper annuli with L 4 = L 5 = L 2 ∪ L 3 . The original idea for this construction is due to D. Hoffman; we refer also to [3] for a related construction.…”