Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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UNIQUENESS OF FAMILIES OF MINIMAL SURFACES IN ℝ3

  • Lee, Eunjoo (Korea Institute for Advanced Study School of Mathematics)
  • Received : 2017.11.23
  • Accepted : 2018.08.29
  • Published : 2018.11.01
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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.

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References

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