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.
Highlights
It is expected that the only two types of singular laminations that can occur as limits of sequences of closed embedded minimal surfaces in a 3-manifold with positive scalar curvature are accumulations of catenoids and non-proper helicoid-like limits
Prior to the construction given here, no non-proper helicoid-like limits were known to exist as limits of closed surfaces
For closed Riemannian manifolds with positive Ricci curvature, combining the work of Choi and Wang [4] and Yang and Yau [45] gives an area bound for embedded minimal surfaces that depends only on the lower bound for the Ricci curvature of the manifold and on the genus of the surface
Summary
It is expected that the only two types of singular laminations that can occur as limits of sequences of closed embedded minimal surfaces in a 3-manifold with positive scalar curvature are accumulations of catenoids and non-proper helicoid-like limits. If we instead use the product metric on S2 × S1 (where each S2 × {z} is minimal but not strictly stable), the same construction will give us a sequence of embedded minimal cylinders that converges smoothly away from two circles to the smooth foliation of S2 × S1 by the parallel minimal spheres S2 × {z}. Each surface in this sequence looks like it is obtained by gluing together two oppositely oriented helicoids. These two minimal surfaces will be embedded but not proper
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