Minimal surfaces with planar boundary curves

TL;DR

This paper investigates the geometry of minimal surfaces with planar boundary curves, establishing bounds on their height relative to boundary diameters, and extends results to constant mean curvature surfaces.

Contribution

It provides new bounds on the height of minimal and constant mean curvature surfaces with planar boundaries, removing previous restrictions and generalizing to broader classes of surfaces.

Findings

Height of minimal surfaces with planar boundaries is bounded by the maximum boundary diameter.

The height bound is improved from 1.5 times to equal to the maximum diameter.

Results extend to nonminimal constant mean curvature surfaces.

Abstract

We consider compact connected minimal surfaces, with a pair of boundary curves (not necessarily convex) in distinct planes, that have least-area amongst all orientable surfaces with the same boundary. When the planes containing these two boundary curves are either parallel or sufficiently close to parallel, and when the boundary curves themselves are sufficiently close to each other, we draw specific conclusions about the geometry and topology of the surfaces. We also strength the following result: Let $M$ be any compact minimal annulus with two planar boundary curves of diameters $d_1$ and $d_2$ in parallel planes $P_1$ and $P_2$; if the distance between $P_1$ and $P_2$ is $h$, then the inequality $h \leq {3/2}\max\{d_1,d_2\}$ is satisfied. We strength it by removing the assumption that $M$ is an annulus and by showing that the stronger conclusion $h \leq \max\{d_1,d_2\}$ holds. We also include a similar result for nonminimal constant mean curvature surfaces.