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arXiv · 1401.1855

The Dynamics Theorem for properly embedded minimal surfaces

William H. Meeks III·Joaquín Pérez·Antonio Ros
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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 (previously proven by the same authors in [14]) 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 $\mathbb{R}^3$.

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William H. Meeks III, Joaquín Pérez, Antonio Ros. 2014-01-08. The Dynamics Theorem for properly embedded minimal surfaces. https://arxiv.org/abs/1401.1855

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