Limit lamination theorem for H-disks

TL;DR

This paper establishes a limit lamination theorem for sequences of constant mean curvature disks in three-dimensional space, extending previous minimal surface results to non-zero mean curvature cases and deriving a chord arc property.

Contribution

It generalizes the lamination limit theorem and chord arc results from minimal to constant mean curvature disks, broadening understanding of their geometric behavior.

Findings

Proves a lamination limit theorem for H-disks with boundaries tending to infinity.

Establishes a chord arc property for compact H-disks in .

Extends previous minimal surface theorems to non-zero mean curvature cases.

Abstract

In this paper we prove a theorem concerning lamination limits of sequences of compact disks $M_n$ embedded in $\mathbb{R}^3$ with constant mean curvature $H_n$, when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi in [8]. We apply this theorem to prove the existence of a chord arc result for compact disks embedded in $\mathbb{R}^3$ with constant mean curvature; this chord arc result generalizes Theorem 0.5 by Colding and Minicozzi in [9] for minimal disks.