One-sided curvature estimates for H-disks

TL;DR

This paper establishes a new extrinsic one-sided curvature estimate for constant mean curvature disks in three-dimensional space, independent of the mean curvature value, and extends these results to intrinsic settings.

Contribution

It introduces a novel extrinsic one-sided curvature estimate for H-disks that does not depend on the mean curvature, and derives an intrinsic version of this estimate.

Findings

Proved an extrinsic one-sided curvature estimate for H-disks.

Applied the estimate to obtain a weak chord arc result.

Extended the estimates to intrinsic curvature bounds.

Abstract

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 [24] to prove to prove a weak chord arc type 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].