The Dynamics Theorem for properly embedded minimal surfaces

Abstract

In this paper we prove two theorems. The first one is a structure result that describes the extrinsic geometry of an embedded surface with constant mean curvature (possibly zero) in a homogeneously regular Riemannian three-manifold, in any small neighborhood of a point of large almost-maximal curvature. We next apply this theorem and the Quadratic Curvature Decay Theorem in Meeks et al. (J Differ Geom, arXiv:1308.6439) to deduce compactness, descriptive and dynamics- type results concerning the space D(M) of non-flat limits under dilations of any given properly embedded minimal surface M in R 3 .