A Characterization of the Critical Catenoid
By Peter J. McGrath. 2016
47 citations recorded by the source corpus.
We show that an embedded minimal annulus $\Sigma^2 \subset B^3$ which intersects $\partial B^3$ 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 $B^n$ with lowest Steklov eigenvalue equal to 1. We also give more general criteria which imply that a free boundary minimal surface in $B^3$ invariant under a group of reflections has lowest Steklov eigenvalue 1.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/ba2b355bf044be936b8ea7470b96944c35f9bfaf
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Compactness analysis for free boundary minimal hypersurfaces (2017, background): “Secondly, one has witnessed a few interesting classification results towards conjectural characterisations of the critical catenoid among free boundary minimal hypersurfaces in the Euclidean unit ball [5,12,28,36,38].”
- Free boundary minimal surfaces in the unit ball : recent advances and open questions (2019, background): “On the other hand, a partial result towards Open Question 5 was obtained by McGrath [73] saying that under additional symmetry assumptions, the critical catenoid is the unique embedded free boundary minimal annulus in B(3).”