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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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참고문헌 (5)
참고문헌 신청
F. Brito / 2003 / Uniqueness and globality of the Liouville formula for entire solutions of @2 logλ/@z@z + λ/2 = 0 / Arch. Math. (Basel) 80(5):501~506
F. Brito / 2004 / Liouville's formula under the viewpoint of minimal surfaces / Commun. Pure Appl. Anal 3(1):41~51
J. Cho / 2017 / Deformation of minimal surfaces with planar curvature lines / J. Geom 108(2):463~479
M. L. Leite / 2015 / Surfaces with planar lines of curvature and orthogonal systems of cycles / J. Math. Anal. Appl 421(2):1254~1273
C. M. C. Riveros / 2017 / Geodesics in minimal surfaces / Math. Notes 101(3-4):497~514
F. Brito / 2004 / Liouville's formula under the viewpoint of minimal surfaces / Commun. Pure Appl. Anal 3(1):41~51
J. Cho / 2017 / Deformation of minimal surfaces with planar curvature lines / J. Geom 108(2):463~479
M. L. Leite / 2015 / Surfaces with planar lines of curvature and orthogonal systems of cycles / J. Math. Anal. Appl 421(2):1254~1273
C. M. C. Riveros / 2017 / Geodesics in minimal surfaces / Math. Notes 101(3-4):497~514
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