Embeddedness of the solutions to the H-Plateau Problem

TL;DR

This paper extends the embeddedness results of minimal surfaces to constant mean curvature disks, showing that such disks are embedded within convex domains for a range of mean curvature values, including the unit ball.

Contribution

It generalizes previous embeddedness results to constant mean curvature disks, broadening understanding of their geometric properties in convex domains.

Findings

Any minimizing H-disk in an H_0-convex domain is embedded for H in [0,H_0).

In the unit ball, any Jordan curve on the sphere bounds an embedded H-disk for H in [0,1].

The results extend classical minimal surface embeddedness to constant mean curvature cases.

Abstract

We generalize Meeks and Yau's embeddedness result on the solutions of the Plateau problem to the constant mean curvature disks. We show that any minimizing H-disk in an H_0-convex domain is embedded for any H in [0,H_0). In particular, for the unit ball B in R^3, this implies that for any H in [0,1], any Jordan curve in the unit sphere bounds an embedded H-disk in B.