Compactness properties of the space of genus-g helicoids

TL;DR

This paper refines the understanding of the compactness and singular behavior of sequences of genus-g helicoids, leading to new insights into their geometric structure and limits.

Contribution

It sharpens the description of singular behavior in sequences of genus-g helicoids with connected boundary, enhancing the compactness theory of these minimal surfaces.

Findings

Improved characterization of singular limits of genus-g helicoids

Enhanced compactness properties of the space of genus-g helicoids

Refined understanding of boundary behavior in minimal surface sequences

Abstract

In \cite{CM5}, Colding and Minicozzi describe a type of compactness property possessed by sequences of embedded minimal surfaces in $\Real^3$ with finite genus and with boundaries going to $\infty$. They show that any such sequence either contains a sub-sequence with uniformly bounded curvature or the sub-sequence has certain prescribed singular behavior. In this paper, we sharpen their description of the singular behavior when the surfaces have connected boundary. Using this, we deduce certain additional compactness properties of the space of genus-$g$ helicoids.