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Bryant Surfaces

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A Bryant surface (or catenoid cousin) is a two-dimensional surface embedded in three-dimensional hyperbolic space with constant mean curvature equal to 1. Bryant derived a holomorphic parameterization for such surfaces, similar to the Weierstrass–Enneper parameterization for minimal surfaces [1, 2].

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A one-parameter family of these surfaces of revolution is defined for the parameter ; for the surface is embedded, and for the surface is not embedded; as tends to zero, the surfaces converge to two horospheres [3, 4, 5]. Another family that arises consists of warped surfaces that are not surfaces of revolution with two regular embedded ends, with warps and parameters and [6, 7]. The parameters and control the range over which the surface is plotted.

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Contributed by: Enrique Zeleny (August 2014)
Open content licensed under CC BY-NC-SA

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References

[1] Wikipedia. "Hyperbolic Space." (Jul 29, 2014) en.wikipedia.org/wiki/Hyperbolic_space.

[2] R. Bryant, "Surfaces of Mean Curvature One in Hyperbolic Space," Asterisque, 154–155, 1987 pp. 321–347.

[3] W. Rossman, M. Umehara, and K. Yamada. "Mean Curvature 1 Warped Catenoid Cousins in Hyperbolic 3-Space." (Feb 20, 2002) www.eg-models.de/models/Surfaces/Mean_Curvature_Surfaces/2001.01.048/_direct_link.html.

[4] "Binoids: Constant Mean Curvature 1 Surfaces in Hyperbolic 3-Space." GeometrieWerkstatt. (Jul 29, 2014) www.math.uni-tuebingen.de/user/nick/gallery/CMC1Binoid.html.

[5] Wikipedia. "Horosphere." (Jul 29, 2014) en.wikipedia.org/wiki/Horosphere.

[6] W. Rossman, M. Umehara, and K. Yamada. "Mean Curvature 1 Warped Catenoid Cousins in Hyperbolic 3-Space." (Feb 20, 2002) www.eg-models.de/models/Surfaces/Mean_Curvature_Surfaces/2001.01.050/_direct_link.html.

[7] C. Bohle and G. P. Peters, "Bryant Surfaces with Smooth Ends," Communications in Analysis and Geometry, 17(4), 2009 pp. 587–619.

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