A Variational Characterization of the Catenoid

Details A Variational Characterization of the Catenoid A Variational Characterization of the Catenoid

2010-12-17

arxiv.org/abs/1012.3941v2

Mathematics

Differential Geometry

21 pages, 1 figure, added Section 4 with additional application

Scientific paper

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 $\partial \Sigma=\sigma_1\cup\sigma_2$.

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