View source for Weierstrass–Enneper parameterization

{{Short description|Construction for minimal surfaces}}
In [[mathematics]], the '''Weierstrass–Enneper parameterization''' of [[minimal surface]]s is a classical piece of [[differential geometry]].
[[Alfred Enneper]] and [[Karl Weierstrass]] studied minimal surfaces as far back as 1863.
[[File:Weierstrass parameterization facilities fabrication of periodic minimal surfaces.jpg|thumb|Weierstrass parameterization facilities fabrication of periodic minimal surfaces]]
Let <math>f</math> and <math>g</math> be functions on either the entire complex plane or the unit disk, where <math>g</math> is [[meromorphic function|meromorphic]] and <math>f</math> is [[analytic function|analytic]], such that wherever <math>g</math> has a pole of order <math>m</math>, <math>f</math> has a zero of order <math>2m</math> (or equivalently, such that the product <math>f g^2</math> is [[Holomorphic function|holomorphic]]), and let <math>c_1,c_2,c_3</math> be constants.  Then the surface with coordinates <math>(x_1, x_2, x_3)</math> is minimal, where the <math>x_k</math> are defined using the real part of a complex integral, as follows:
<math display="block">\begin{align}
x_k(\zeta) &{}= \mathrm{Re} \left\{ \int_{0}^{\zeta} \varphi_{k}(z) \, dz \right\} + 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{align}</math>
The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.<ref name="DHWK">{{cite book |last1=Dierkes |first1=U. |last2=Hildebrandt |first2=S. |last3=Küster |first3=A. |last4=Wohlrab |first4=O. |title=Minimal surfaces |volume=I |page=108 |publisher=Springer |year=1992 |isbn=3-540-53169-6 }}</ref>
For example, [[Enneper's surface]] has {{math|1=''f''(''z'') = 1}}, {{math|1=''g''(''z'') = ''z<sup>m</sup>''}}.
==Parametric surface of complex variables==
The Weierstrass-Enneper model defines  a minimal surface <math>X</math> (<math>\Reals^3</math>) on a complex plane (<math>\Complex</math>). Let <math>\omega=u+v i</math> (the complex plane as the <math>uv</math> space), the [[Jacobian matrix and determinant|Jacobian matrix]] of the surface can be written as a column of complex entries:
<math display="block">\mathbf{J} = \begin{bmatrix}
\left( 1 - g^2(\omega) \right)f(\omega)  \\
i\left( 1+ g^2(\omega) \right)f(\omega)  \\
2g(\omega) f(\omega)
\end{bmatrix} </math>
where <math>f(\omega)</math> and <math>g(\omega)</math> are holomorphic functions of <math>\omega</math>.
The Jacobian <math>\mathbf{J}</math> represents the two orthogonal tangent vectors of the surface:<ref>{{cite journal |last1=Andersson |first1=S. |last2=Hyde |first2=S. T. |last3=Larsson |first3=K. |last4=Lidin |first4=S. |title=Minimal Surfaces and Structures: From Inorganic and Metal Crystals to Cell Membranes and Biopolymers |journal=Chem. Rev. |volume=88 |issue=1 |pages=221–242 |year=1988 |doi=10.1021/cr00083a011 }}</ref>
<math display="block">
\mathbf{X_u} = \begin{bmatrix}
\operatorname{Re}\mathbf{J}_1 \\
\operatorname{Re}\mathbf{J}_2  \\
\operatorname{Re} \mathbf{J}_3
\end{bmatrix}   \;\;\;\;
\mathbf{X_v} = \begin{bmatrix}
-\operatorname{Im}\mathbf{J}_1 \\
-\operatorname{Im}\mathbf{J}_2  \\
-\operatorname{Im} \mathbf{J}_3
\end{bmatrix}
</math>
The surface normal is given by
<math display="block">
\mathbf{\hat{n}} =
\frac{\mathbf{X_u}\times \mathbf{X_v}}{|\mathbf{X_u}\times \mathbf{X_v}|} =
\frac{1}{| g|^2+1}
\begin{bmatrix}
2\operatorname{Re} g \\
2\operatorname{Im} g  \\
| g|^2-1
\end{bmatrix}
</math>
The Jacobian <math>\mathbf{J}</math> leads to a number of important properties: <math>\mathbf{X_u} \cdot \mathbf{X_v}=0</math>, <math>\mathbf{X_u}^2 = \operatorname{Re}(\mathbf{J}^2)</math>, <math>\mathbf{X_v}^2 = \operatorname{Im}(\mathbf{J}^2)</math>, <math>\mathbf{X_{uu}} + \mathbf{X_{vv}}=0</math>. The proofs can be found in Sharma's essay: The Weierstrass representation always gives a minimal surface.<ref>{{cite arXiv|last=Sharma |first=R. |title=The Weierstrass Representation always gives a minimal surface  |eprint=1208.5689 |date=2012 |class=math.DG }}</ref>  The  derivatives can be used to construct the [[first fundamental form]] matrix:
<math display="block">
\begin{bmatrix}
\mathbf{X_u} \cdot \mathbf{X_u} & \;\; \mathbf{X_u} \cdot \mathbf{X_v}\\
\mathbf{X_v} \cdot \mathbf{X_u} &  \;\;\mathbf{X_v} \cdot \mathbf{X_v}
\end{bmatrix}=
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
</math>
and the [[second fundamental form]] matrix
<math display="block">\begin{bmatrix}
\mathbf{X_{uu}} \cdot \mathbf{\hat{n}} & \;\; \mathbf{X_{uv}} \cdot \mathbf{\hat{n}}\\
\mathbf{X_{vu}} \cdot \mathbf{\hat{n}} & \;\; \mathbf{X_{vv}} \cdot \mathbf{\hat{n}}
\end{bmatrix}</math>
Finally, a point <math>\omega_t</math> on the complex plane maps to a point <math>\mathbf{X}</math> on the minimal surface in <math>\R^3</math> by
<math display="block">\mathbf{X}= \begin{bmatrix}
\operatorname{Re} \int_{\omega_0}^{\omega_ t}\mathbf{J}_1 d\omega\\
\operatorname{Re} \int_{\omega_0}^{\omega_ t} \mathbf{J}_2 d\omega\\
\operatorname{Re} \int_{\omega_0}^{\omega_ t} \mathbf{J}_3 d\omega
\end{bmatrix}</math>
where <math>\omega_0 = 0</math> for all minimal surfaces throughout this paper except for [[Costa's minimal surface]] where <math>\omega_0=(1+i)/2</math>.
==Embedded minimal surfaces and examples==
The classical examples of embedded  complete minimal surfaces in <math>\mathbb{R}^3</math> with finite topology include the plane, the [[catenoid]], the [[helicoid]], and the [[Costa's minimal surface]]. Costa's surface involves [[Weierstrass's elliptic function]] <math>\wp </math>:<ref>{{cite book |last=Lawden |first=D. F. |title=Elliptic Functions and Applications |series=Applied Mathematical Sciences |volume=80 |publisher=Springer |location=Berlin |year=2011 |isbn=978-1-4419-3090-3 }}</ref>
<math display="block">g(\omega)=\frac{A}{\wp' (\omega)}</math>
<math display="block">f(\omega)= \wp(\omega)</math>
where <math>A</math> is a constant.<ref>{{cite book |last1=Abbena |first1=E. |last2=Salamon |first2=S. |last3=Gray |first3=A. |chapter=Minimal Surfaces via Complex Variables |title=Modern Differential Geometry of Curves and Surfaces with Mathematica |publisher=CRC Press |location=Boca Raton |year=2006 |isbn=1-58488-448-7 |pages=719–766 }}</ref>
=== Helicatenoid ===
Choosing the functions <math>f(\omega) = e^{-i \alpha}e^{\omega/A}</math> and <math>g(\omega) = e^{-\omega/A}</math>, a one parameter family of minimal surfaces is obtained.
<math display="block">\varphi_1 = e^{-i \alpha} \sinh\left(\frac{\omega}{A}\right)</math>
<math display="block">\varphi_2 = i e^{-i \alpha} \cosh\left(\frac{\omega}{A}\right)</math>
<math display="block">\varphi_3 = e^{-i \alpha}</math>
<math display="block">
\mathbf{X}(\omega)  =
\operatorname{Re}
\begin{bmatrix}
e^{-i\alpha} A \cosh \left( \frac{\omega}{A} \right)  \\
i e^{-i\alpha} A \sinh \left( \frac{\omega}{A} \right) \\
e^{-i\alpha} \omega \\
\end{bmatrix}
=
\cos(\alpha)
\begin{bmatrix}
A \cosh \left( \frac{\operatorname{Re}(\omega)}{A} \right) \cos \left( \frac{\operatorname{Im}(\omega)}{A} \right)\\
- A \cosh \left( \frac{\operatorname{Re}(\omega)}{A} \right) \sin \left( \frac{\operatorname{Im}(\omega)}{A} \right) \\
\operatorname{Re}(\omega) \\
\end{bmatrix} +
\sin(\alpha)
\begin{bmatrix}
A \sinh \left( \frac{\operatorname{Re}(\omega)}{A} \right) \sin \left( \frac{\operatorname{Im}(\omega)}{A} \right)\\
A \sinh \left( \frac{\operatorname{Re}(\omega)}{A} \right) \cos \left( \frac{\operatorname{Im}(\omega)}{A} \right) \\
\operatorname{Im}(\omega) \\
\end{bmatrix}
</math>
Choosing the parameters of the surface as <math>\omega = s + i(A \phi)</math>:
<math display="block">\mathbf{X}(s,\phi)=
\cos(\alpha)
\begin{bmatrix}
A \cosh \left( \frac{s}{A} \right) \cos \left( \phi \right)\\
- A \cosh \left( \frac{s}{A} \right) \sin \left( \phi \right) \\
s \\
\end{bmatrix} +
\sin(\alpha)
\begin{bmatrix}
A \sinh \left( \frac{s}{A} \right) \sin \left( \phi \right)\\
A \sinh \left( \frac{s}{A} \right) \cos \left( \phi \right) \\
A \phi \\
\end{bmatrix}</math>
At the extremes, the surface is a catenoid <math>(\alpha = 0)</math> or a helicoid <math>(\alpha = \pi/2)</math>. Otherwise, <math>\alpha</math> represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the <math>\mathbf{X}_3</math> axis in a helical fashion.
[[File:Helically Rotated Catenary (Helicatenoid).jpg|thumb|A catenary that spans periodic points on a helix, subsequently rotated along the helix to produce a minimal surface.]]
[[File:The fundamental domain (C) and the 3D surfaces. The continuous surfaces are made of copies of the fundamental patch (R3).jpg|thumb|The fundamental domain (C) and the 3D surfaces. The continuous surfaces are made of copies of the fundamental patch (R3)]]
==Lines of curvature==
One can rewrite each element of second [[Fundamental matrix (linear differential equation)|fundamental matrix]] as a function of <math>f</math> and <math>g</math>, for example
<math display="block">
\mathbf{X_{uu}}  \cdot \mathbf{\hat{n}} =
\frac{1}{|g|^2+1}
\begin{bmatrix}
\operatorname{Re}  \left(  ( 1- g^2 ) f' - 2gfg'\right)  \\
\operatorname{Re}  \left( ( 1+ g^2 ) f'i+ 2gfg'i \right) \\
\operatorname{Re}  \left(  2gf' +2fg' \right) \\
\end{bmatrix}
\cdot
\begin{bmatrix}
\operatorname{Re} \left( 2g   \right) \\
\operatorname{Re} \left( -2gi   \right) \\
\operatorname{Re} \left( |g|^2-1   \right) \\
\end{bmatrix}
= -2\operatorname{Re} (fg')
</math>
And consequently the second fundamental form matrix can be simplified as
<math display="block">\begin{bmatrix}
-\operatorname{Re} f g'  & \;\;  \operatorname{Im} f g' \\
\operatorname{Im} f g'  & \;\;  \operatorname{Re} f g'
\end{bmatrix} </math>
[[File:Lines of curvature make a quadrangulation of the domain.jpg|thumb|Lines of curvature make a quadrangulation of the domain]]
One of its eigenvectors is <math display="block"> \overline{\sqrt{ f g'} }</math> which represents the principal direction in the complex domain.<ref>{{cite journal |last1=Hua |first1=H. |last2=Jia |first2=T. |year=2018 |title=Wire cut of double-sided minimal surfaces |journal=The Visual Computer |volume=34 |issue=6–8 |pages=985–995 |doi=10.1007/s00371-018-1548-0 |s2cid=13681681 }}</ref> Therefore, the two principal directions in the <math>uv</math> space turn out to be
<math display="block"> \phi = -\frac{1}{2} \operatorname{Arg}(f g') \pm k \pi /2</math>
==See also==
* [[Associate family]]
* [[Bryant surface]], found by an analogous parameterization in [[hyperbolic space]]
==References==
{{reflist}}
{{DEFAULTSORT:Weierstrass-Enneper parameterization}}
[[Category:Differential geometry]]
[[Category:Surfaces]]
[[Category:Minimal surfaces]]