Volume 22, issue 1 (2018)

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Chord arc properties for constant mean curvature disks

William H Meeks, III and Giuseppe Tinaglia

Geometry & Topology 22 (2018) 305–322
Abstract

We prove a chord arc type bound for disks embedded in ${ℝ}^{3}$ with constant mean curvature that does not depend on the value of the mean curvature. This bound is inspired by and generalizes the weak chord arc bound of Colding and Minicozzi in Proposition 2.1 of Ann. of Math. 167 (2008) 211–243 for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamental tool for studying complete constant mean curvature surfaces embedded in ${ℝ}^{3}$ with finite topology or with positive injectivity radius.

Keywords
minimal surface, constant mean curvature, minimal lamination, positive injectivity radius, curvature estimates, one-sided curvature estimate, chord arc
Mathematical Subject Classification 2010
Primary: 53A10
Secondary: 49Q05, 53C42
Publication
Received: 4 November 2015
Revised: 12 March 2017
Accepted: 9 April 2017
Published: 31 October 2017
Proposed: Tobias H. Colding
Seconded: Bruce Kleiner, Gang Tian
Authors
 William H Meeks, III Department of Mathematics University of Massachusetts Amherst, MA United States Giuseppe Tinaglia Department of Mathematics King’s College London London United Kingdom