An Enneper-Weierstrass Minimal Surface

outer radius

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1.

inner radius

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

periods

π

2π

root r

1

2

3

4

5

6

7

8

9

part

real

imaginary

A minimal surface has zero mean curvature. An Enneper-Weierstrass parametrization for such a surface is based on two suitably defined holomorphic functions

f(z)

and

g(z)

. The functions chosen here are

f(z)=

1/r

z

+

z



z

and

g(z)=z

. Embedding in

3



is given by the indefinite integrals

x

k

=∫

ϕ

k

(z)z

, where

ϕ

1

=f(z)1-

2

g(z)



,

ϕ

2

=f(z)1+

2

g(z)



and

ϕ

3

=2f(z)g(z)

. The surface shown is the parametric plot of the real and imaginary parts of the

x

k

as

z

ranges over an annulus.