Weierstrass–Enneper parameterization explained

In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry.

Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863.

Let

f

and

g

be functions on either the entire complex plane or the unit disk, where

g

is meromorphic and

f

is analytic, such that wherever

g

has a pole of order

m

,

f

has a zero of order

2m

(or equivalently, such that the product

fg2

is holomorphic), and let

c1,c2,c3

be constants. Then the surface with coordinates

(x1,x2,x3)

is minimal, where the

xk

are defined using the real part of a complex integral, as follows:\begin x_k(\zeta) &= \mathrm \left\ + c_k, \qquad k=1,2,3 \\ \varphi_1 &= f(1-g^2)/2 \\ \varphi_2 &= i f(1+g^2)/2 \\ \varphi_3 &= fg\end

The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.[1]

For example, Enneper's surface has, .

Parametric surface of complex variables

The Weierstrass-Enneper model defines a minimal surface

X

(

\Reals3

) on a complex plane (

\Complex

). Let

\omega=u+vi

(the complex plane as the

uv

space), the Jacobian matrix of the surface can be written as a column of complex entries:\mathbf = \begin\left(1 - g^2(\omega) \right)f(\omega) \\ i\left(1+ g^2(\omega) \right)f(\omega) \\ 2g(\omega) f(\omega) \end where

f(\omega)

and

g(\omega)

are holomorphic functions of

\omega

.

The Jacobian

J

represents the two orthogonal tangent vectors of the surface:[2] \mathbf = \begin\operatorname\mathbf_1 \\\operatorname\mathbf_2 \\\operatorname \mathbf_3 \end \;\;\;\;\mathbf = \begin-\operatorname\mathbf_1 \\-\operatorname\mathbf_2 \\-\operatorname \mathbf_3 \end

The surface normal is given by\mathbf =\frac

=\frac
^2+1
\begin2\operatorname g \\2\operatorname g \\| g|^2-1\end

The Jacobian

J

leads to a number of important properties:
Xu

Xv=0
,
2
Xu

=\operatorname{Re}(J2)

,
2
Xv

=\operatorname{Im}(J2)

,
Xuu

+

Xvv=0
. The proofs can be found in Sharma's essay: The Weierstrass representation always gives a minimal surface.[3] The derivatives can be used to construct the first fundamental form matrix:\begin\mathbf \cdot \mathbf & \;\; \mathbf \cdot \mathbf\\\mathbf \cdot \mathbf & \;\;\mathbf \cdot \mathbf\end=\begin1 & 0 \\0 & 1\end

and the second fundamental form matrix\begin\mathbf \cdot \mathbf & \;\; \mathbf \cdot \mathbf\\\mathbf \cdot \mathbf & \;\; \mathbf \cdot \mathbf\end

Finally, a point

\omegat

on the complex plane maps to a point

X

on the minimal surface in

\R3

by\mathbf= \begin\operatorname \int_^\mathbf_1 d\omega\\\operatorname \int_^ \mathbf_2 d\omega\\\operatorname \int_^ \mathbf_3 d\omega \endwhere

\omega0=0

for all minimal surfaces throughout this paper except for Costa's minimal surface where

\omega0=(1+i)/2

.

Embedded minimal surfaces and examples

The classical examples of embedded complete minimal surfaces in

R3

with finite topology include the plane, the catenoid, the helicoid, and the Costa's minimal surface. Costa's surface involves Weierstrass's elliptic function

\wp

:[4] g(\omega)=\fracf(\omega)= \wp(\omega)where

A

is a constant.[5]

Helicatenoid

Choosing the functions

f(\omega)=e-ie\omega/A

and

g(\omega)=e-\omega/A

, a one parameter family of minimal surfaces is obtained.

\varphi_1 = e^ \sinh\left(\frac\right)\varphi_2 = i e^ \cosh\left(\frac\right)\varphi_3 = e^\mathbf(\omega) = \operatorname \begine^ A \cosh \left(\frac \right) \\i e^ A \sinh \left(\frac \right) \\e^ \omega \\\end=\cos(\alpha) \beginA \cosh \left(\frac \right) \cos \left(\frac \right)\\- A \cosh \left(\frac \right) \sin \left(\frac \right) \\\operatorname(\omega) \\\end +\sin(\alpha) \beginA \sinh \left(\frac \right) \sin \left(\frac \right)\\A \sinh \left(\frac \right) \cos \left(\frac \right) \\\operatorname(\omega) \\\end

Choosing the parameters of the surface as

\omega=s+i(A\phi)

:\mathbf(s,\phi)=\cos(\alpha) \beginA \cosh \left(\frac \right) \cos \left(\phi \right)\\- A \cosh \left(\frac \right) \sin \left(\phi \right) \\s \\\end +\sin(\alpha) \beginA \sinh \left(\frac \right) \sin \left(\phi \right)\\A \sinh \left(\frac \right) \cos \left(\phi \right) \\A \phi \\\end

At the extremes, the surface is a catenoid

(\alpha=0)

or a helicoid

(\alpha=\pi/2)

. Otherwise,

\alpha

represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the

X3

axis in a helical fashion.

Lines of curvature

One can rewrite each element of second fundamental matrix as a function of

f

and

g

, for example\mathbf \cdot \mathbf = \frac
^2+1
\begin\operatorname \left((1- g^2) f' - 2gfg'\right) \\\operatorname \left((1+ g^2) f'i+ 2gfg'i \right) \\\operatorname \left(2gf' +2fg' \right) \\\end\cdot \begin \operatorname \left(2g \right) \\ \operatorname \left(-2gi \right) \\ \operatorname \left(|g|^2-1 \right) \\ \end = -2\operatorname (fg')

And consequently the second fundamental form matrix can be simplified as\begin-\operatorname f g' & \;\; \operatorname f g' \\\operatorname f g' & \;\; \operatorname f g' \end One of its eigenvectors is \overline which represents the principal direction in the complex domain.[6] Therefore, the two principal directions in the

uv

space turn out to be \phi = -\frac \operatorname(f g') \pm k \pi /2

See also

Notes and References

  1. Book: Dierkes . U. . Hildebrandt . S. . Küster . A. . Wohlrab . O. . Minimal surfaces . I . 108 . Springer . 1992 . 3-540-53169-6 .
  2. Andersson . S. . Hyde . S. T. . Larsson . K. . Lidin . S. . Minimal Surfaces and Structures: From Inorganic and Metal Crystals to Cell Membranes and Biopolymers . Chem. Rev. . 88 . 1 . 221–242 . 1988 . 10.1021/cr00083a011 .
  3. Sharma . R. . The Weierstrass Representation always gives a minimal surface . 1208.5689 . 2012 . math.DG .
  4. Book: Lawden, D. F. . Elliptic Functions and Applications . Applied Mathematical Sciences . 80 . Springer . Berlin . 2011 . 978-1-4419-3090-3 .
  5. Book: Abbena . E. . Salamon . S. . Gray . A. . Minimal Surfaces via Complex Variables . Modern Differential Geometry of Curves and Surfaces with Mathematica . CRC Press . Boca Raton . 2006 . 1-58488-448-7 . 719–766 .
  6. Hua . H. . Jia . T. . 2018 . Wire cut of double-sided minimal surfaces . The Visual Computer . 34 . 6–8 . 985–995 . 10.1007/s00371-018-1548-0 . 13681681 .