Compactness of the space of genus-one helicoids

TL;DR

This paper proves that the space of genus-one helicoids is compact when considering sequences of such minimal surfaces, extending previous results and utilizing lamination theory.

Contribution

It generalizes Hoffman and White's result by establishing the compactness of genus-one helicoids using lamination theory.

Findings

The space of genus-one helicoids is compact modulo rigid motions and homotheties.

Lamination theory effectively analyzes sequences of minimal surfaces.

The result extends understanding of the structure of minimal surfaces with genus one.

Abstract

Using the lamination theory developed by Colding and Minicozzi for sequences of embedded, finite genus minimal surfaces with boundaries going to infinity \cite{CM5}, we show that the space of genus-one helicoids is compact (modulo rigid motions and homotheties). This generalizes a result of Hoffman and White \cite{HW}.