A variational characterization of the catenoid
Author(s)Bernstein, Jacob; Breiner, Christine Elaine
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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.
Calculus of Variations and Partial Differential Equations
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.
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