A Characterization of the Critical Catenoid

We show that an embedded minimal annulus Σ² ⊂ B³ which intersects ∂B³ orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique...

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Bibliographic Details
Published in:Indiana University mathematics journal 2018-01, Vol.67 (2), p.889-897
Main Author: McGrath, Peter
Format: Article
Language:English
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Summary:We show that an embedded minimal annulus Σ² ⊂ B³ which intersects ∂B³ orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique free boundary minimal annulus in Bⁿ with lowest Steklov eigenvalue equal to 1. We also give more general criteria which imply that a free boundary minimal surface in B³ invariant under a group of reflections has lowest Steklov eigenvalue 1.
ISSN:0022-2518
1943-5258
DOI:10.1512/iumj.2018.67.7251