The space of embedded minimal surfaces of fixed genus in a 3-manifolds II; Multi-valued graphs in disks

Abstract

This paper is the second in a series where we give a description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R 3 . We show here that if the curvature of such a disk becomes large at some point, then it contains an almost flat multi-valued graph nearby that continues almost all the way to the boundary. This will be proved by showing the existence of small multi-valued graphs near points of large curvature and then using the extension result for multi-valued graphs proved in the first paper in this series. There are two local models for embedded minimal disks (by an embedded disk, we mean a smooth injective map from the closed unit ball in R 2 into R 3 ). One model is the plane (or, more generally, a minimal graph), the other is a piece of a helicoid. In the first four papers of this series, we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase like a helicoid. Recall that a double spiral staircase consists of two spiral staircases that spiral together around a common axis, one inside the other. This will be done by showing that if the curvature is large at some point (and hence the surface is not a graph), then it is a double spiral staircase. To prove that it is a double spiral staircase, we will first prove that it is built out of N-valued graphs where N is a fixed number. These N-valued graphs are like a single spiral staircase connecting N floors. The existence of the N-valued graphs was initiated in the first paper and will be completed here. The third and fourth papers of this series will deal with how the multi-valued graphs fit together and, in particular, prove regularity of the set of points of large curvature – the axis of the double spiral staircase.