Chord arc properties for constant mean curvature disks

TL;DR

This paper establishes a chord arc bound for constant mean curvature disks in three-dimensional space, extending minimal surface results and aiding the study of such surfaces with finite topology or positive injectivity radius.

Contribution

It introduces a curvature-independent chord arc bound for CMC disks, generalizing prior minimal surface results to constant mean curvature cases.

Findings

Chord arc bound does not depend on mean curvature value.

Bound is crucial for analyzing complete CMC surfaces with finite topology.

Generalizes minimal surface chord arc properties to CMC surfaces.

Abstract

We prove a chord arc bound for disks embedded in $\mathbb{R}^3$ with constant mean curvature. This bound does not depend on the value of the mean curvature. It is inspired by and generalizes the work of Colding and Minicozzi in [2] 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 $\mathbb{R}^3$ with finite topology or with positive injectivity radius.