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Proceedings of the American Mathematical Society

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On a conjecture of Nitsche


Author: Gregory D. Crow
Journal: Proc. Amer. Math. Soc. 114 (1992), 1063-1068
MSC: Primary 53A10
DOI: https://doi.org/10.1090/S0002-9939-1992-1105038-2
MathSciNet review: 1105038
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Abstract: We show that under the hypothesis of bounded Gaussian curvature, certain minimal surfaces are in fact of finite total curvature. We can then answer the following version of a conjecture of Nitsche (J. Math. Mech. 11 (1962), 295) under the hypothesis of bounded Gaussian curvature:

Conjecture. Let $ {M^2} \subset {{\mathbf{R}}^3}$ be a complete minimal surface such that for some height function $ H$, the level sets are (compact) Jordan curves. Then $ M$ is a catenoid.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1105038-2
Keywords: Minimal surfaces, Nitsche's conjecture, catenoid, bounded Gaussian curvature, finite total curvature
Article copyright: © Copyright 1992 American Mathematical Society