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A variational characterization of the catenoid

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Abstract

In this note, we use a result of Osserman and Schiffer (Arch. Rational Mech. Anal. 58:285–307, 1975) 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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Acknowledgments

The authors were supported, respectively, by the NSF grants DMS-0902721 and DMS-0902718. We would like to thank David Hoffman for his interest and many helpful comments. We also thank Brian White for suggesting a simplification of the proof of Lemma 5.5.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Building 380, Stanford, CA, 94305, USA

    Jacob Bernstein

  2. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA, 02139, USA

    Christine Breiner

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  1. Jacob Bernstein
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  2. Christine Breiner
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Correspondence to Christine Breiner.

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Communicated by R. Schoen.

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Bernstein, J., Breiner, C. A variational characterization of the catenoid. Calc. Var. 49, 215–232 (2014). https://doi.org/10.1007/s00526-012-0579-z

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  • DOI: https://doi.org/10.1007/s00526-012-0579-z

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