The Dynamics Theorem for properly embedded minimal surfaces

摘要: 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 homogeneously regular Riemannian three-manifold, any small neighborhood point large almost-maximal curvature. We next apply theorem and Quadratic Curvature Decay Theorem Meeks et al. (J Differ Geom, arXiv:1308.6439) to deduce compactness, descriptive dynamics- type results concerning space D(M) non-flat limits under dilations given properly minimal M R 3 .