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The Dynamics Theorem for properly embedded minimal surfaces

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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 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 \(\mathbb {R}^3\).

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Authors and Affiliations

  1. Mathematics Department, University of Massachusetts, Amherst, MA, 01003, USA

    William H. Meeks III

  2. Department of Geometry and Topology, Institute of Mathematics (IEMath-GR), University of Granada, 18071, Granada, Spain

    Joaquín Pérez & Antonio Ros

Authors
  1. William H. Meeks III
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  2. Joaquín Pérez
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  3. Antonio Ros
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Correspondence to Joaquín Pérez.

Additional information

This material is based upon work for the NSF under Award No. DMS-1309236. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF. J. Pérez and A. Ros were supported in part by the MEC/FEDER Grants no. MTM2011-22547 and MTM2014-52368.

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Meeks, W.H., Pérez, J. & Ros, A. The Dynamics Theorem for properly embedded minimal surfaces. Math. Ann. 365, 1069–1089 (2016). https://doi.org/10.1007/s00208-015-1311-z

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  • DOI: https://doi.org/10.1007/s00208-015-1311-z

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