One-sided curvature estimates for $H$-disks

In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature, which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [26] to prove a weak chord arc result for these disks. In Section 4 we apply this weak chord arc result to obtain an intrinsic version of the one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature. In a natural sense, these one-sided curvature estimates generalize respectively, the extrinsic and intrinsic one-sided curvature estimates for minimal disks embedded in $\mathbb{R}^3$ given by Colding and Minicozzi in Theorem 0.2 of [8] and in Corollary 0.8 of [9].

Keywords

minimal surface, constant mean curvature, one-sided curvature estimate, curvature estimates, minimal lamination, $H$-surface, chord arc, removable singularity

2010 Mathematics Subject Classification

Primary 53A10. Secondary 49Q05, 53C42.

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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.

The second author was partially supported by EPSRC grant no. EP/M024512/1. 479

Received 8 September 2019

Published 2 October 2020