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

Abstract

This paper is the first in a series where we attempt to give a complete description of 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 $\RR^3$ (with the flat metric).