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arXiv · 1012.3941

A Variational Characterization of the Catenoid

Jacob Bernstein·Christine Breiner
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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 $σ_1$ and $σ_2$ lying in parallel planes that precludes the existence of a connected minimal surface $Σ$ with $\partial Σ=σ_1\cupσ_2$.

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Jacob Bernstein, Christine Breiner. 2011-02-24. A Variational Characterization of the Catenoid. https://arxiv.org/abs/1012.3941

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