A Variational Characterization of the Catenoid

TL;DR

This paper provides a variational characterization of the catenoid, showing it minimizes area among certain minimal annuli, and applies this to derive conditions on boundary curves preventing minimal surfaces.

Contribution

It introduces a novel variational perspective on the catenoid and establishes new geometric conditions for the existence of minimal surfaces bounded by given curves.

Findings

Catenoid minimizes area within a class of minimal annuli

Derived a sharp boundary length condition for minimal surface existence

Connected minimal surfaces are precluded under certain boundary configurations

Abstract

In this note, we use a result of Osserman and Schiffer \cite{OS} to give a variational characterization of the catenoid. Namely, we show that subsets of the catenoid minimize area within a geometrically natural class of minimal annuli. To the best of our knowledge, this fact has gone unremarked upon in the literature. As an application of the techniques, we give a sharp condition on the lengths of a pair of connected, simple closed curves $\sigma_1$ and $\sigma_2$ lying in parallel planes that precludes the existence of a connected minimal surface $\Sigma$ with $\partial \Sigma=\sigma_1\cup\sigma_2$.