Bryant Surfaces

μ

0.4

l

2

b

0.5

d

1

x

6.283

y

3.141

warped

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].

A one-parameter family of these surfaces of revolution is defined for the parameter

μ

; for

-1/2<μ<0

the surface is embedded, and for

μ>0

the surface is not embedded; as

μ

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

l

warps and parameters

b

and

d

[6, 7]. The parameters

x

and

y

control the range over which the surface is plotted.