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http://dx.doi.org/10.4134/JKMS.j170757

UNIQUENESS OF FAMILIES OF MINIMAL SURFACES IN ℝ3  

Lee, Eunjoo (Korea Institute for Advanced Study School of Mathematics)
Publication Information
Journal of the Korean Mathematical Society / v.55, no.6, 2018 , pp. 1459-1468 More about this Journal
Abstract
We show that an umbilic-free minimal surface in ${\mathbb{R}}^3$ belongs to the associate family of the catenoid if and only if the geodesic curvatures of its lines of curvature have a constant ratio. As a corollary, the helicoid is shown to be the unique umbilic-free minimal surface whose lines of curvature have the same geodesic curvature. A similar characterization of the deformation family of minimal surfaces with planar lines of curvature is also given.
Keywords
Liouville's equation; geodesic curvature; associate minimal surfaces; helicoid; catenoid; Enneper surface;
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