Chord arc properties for constant mean curvature disks

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.