A variational characterization of the catenoid
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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 σ1 and σ2 lying in parallel planes that precludes the existence of a connected minimal surface Σ with ∂Σ=σ1∪σ2.
Date issued
2012-12Department
Massachusetts Institute of Technology. Department of MathematicsJournal
Calculus of Variations and Partial Differential Equations
Publisher
Springer-Verlag
Citation
Bernstein, Jacob, and Christine Breiner. “A Variational Characterization of the Catenoid.” Calculus of Variations and Partial Differential Equations, vol. 49, no. 1–2, December 2012, pp. 215–232.
Version: Author's final manuscript
ISSN
0944-2669
1432-0835