The space of embedded minimal surfaces of fixed genus in a 3-manifold I; Estimates off the axis for disks

This paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed Riemannian 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 R3 (with the flat metric). This study is undertaken here and completed in [CM6]. These local results are then applied in [CM7] where we describe the general structure of fixed genus surfaces in 3-manifolds. 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 R2 into R3). 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 where each staircase is a multi-valued graph. 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 such a disk is a double spiral staircase, we will first prove that it is built out of N -valued graphs where N is a fixed number. This is initiated here and will be completed in the second paper. 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. The reader may find it useful to also look at the survey [CM8] and the expository article [CM9] for an outline of our results, and their proofs, and how these results fit together. The article [CM9] is the best to start with.