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

TL;DR

This paper constructs a sequence of embedded minimal cylinders in a 3-manifold with positive scalar curvature, converging to a lamination with helicoid-like singularities near a stable sphere, revealing complex limit behaviors.

Contribution

It demonstrates the existence of non-proper helicoid-like limits of minimal surfaces in a specific 3-manifold, extending understanding of minimal surface limits.

Findings

Existence of a metric with positive scalar curvature on S2xS1.

Construction of a sequence of minimal cylinders converging to a lamination.

Identification of helicoid-like singularities in the limit.

Abstract

We show that there exists a metric with positive scalar curvature on S2xS1 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.