Weierstrass–Enneper parameterization

The Weierstrass-Enneper model defines a minimal surface $X$ ($\backslash Reals^3$) on a complex plane ($\backslash Complex$). Let $\backslash omega=u+v\; i$ (the complex plane as the $uv$ space), the Jacobian matrix of the surface can be written as a column of complex entries:

$$\backslash mathbf\{J\}\; =\; \backslash begin\{bmatrix\}$$

\left( 1 - g^2(\omega) \right)f(\omega) \\

i\left( 1+ g^2(\omega) \right)f(\omega) \\

2g(\omega) f(\omega)

\end{bmatrix}

where $f(\backslash omega)$ and $g(\backslash omega)$ are holomorphic functions of $\backslash omega$.

The Jacobian $\backslash mathbf\{J\}$ represents the two orthogonal tangent vectors of the surface:{{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 }}

\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}

The surface normal is given by

\mathbf{\hat{n}} =

\frac{\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}

The Jacobian $\backslash mathbf\{J\}$ leads to a number of important properties: $\backslash mathbf\{X\_u\}\; \backslash cdot\; \backslash mathbf\{X\_v\}=0$, $\backslash mathbf\{X\_u\}^2\; =\; \backslash operatorname\{Re\}(\backslash mathbf\{J\}^2)$, $\backslash mathbf\{X\_v\}^2\; =\; \backslash operatorname\{Im\}(\backslash mathbf\{J\}^2)$, $\backslash mathbf\{X\_\{uu\}\}\; +\; \backslash mathbf\{X\_\{vv\}\}=0$. The proofs can be found in Sharma's essay: The Weierstrass representation always gives a minimal surface.{{cite arXiv|last=Sharma |first=R. |title=The Weierstrass Representation always gives a minimal surface |eprint=1208.5689 |date=2012 |class=math.DG }} The derivatives can be used to construct the first fundamental form matrix:

\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}

and the second fundamental form matrix

$$\backslash begin\{bmatrix\}$$

\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}

Finally, a point $\backslash omega\_t$ on the complex plane maps to a point $\backslash mathbf\{X\}$ on the minimal surface in $\backslash R^3$ by

$$\backslash mathbf\{X\}=\; \backslash begin\{bmatrix\}$$

\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}

where $\backslash omega\_0\; =\; 0$ for all minimal surfaces throughout this paper except for Costa's minimal surface where $\backslash omega\_0=(1+i)/2$.

The classical examples of embedded complete minimal surfaces in $\backslash mathbb\{R\}^3$ with finite topology include the plane, the catenoid, the helicoid, and the Costa's minimal surface. Costa's surface involves Weierstrass's elliptic function $\backslash wp$:{{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 }}

$$g(\backslash omega)=\backslash frac\{A\}\{\backslash wp\text{'}\; (\backslash omega)\}$$

$$f(\backslash omega)=\; \backslash wp(\backslash omega)$$

where $A$ is a constant.{{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 }}

Choosing the functions $f(\backslash omega)\; =\; e^\{-i\; \backslash alpha\}e^\{\backslash omega/A\}$ and $g(\backslash omega)\; =\; e^\{-\backslash omega/A\}$, a one parameter family of minimal surfaces is obtained.

$$\backslash varphi\_1\; =\; e^\{-i\; \backslash alpha\}\; \backslash sinh\backslash left(\backslash frac\{\backslash omega\}\{A\}\backslash right)$$

$$\backslash varphi\_2\; =\; i\; e^\{-i\; \backslash alpha\}\; \backslash cosh\backslash left(\backslash frac\{\backslash omega\}\{A\}\backslash right)$$

$$\backslash varphi\_3\; =\; e^\{-i\; \backslash alpha\}$$

\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}

Choosing the parameters of the surface as $\backslash omega\; =\; s\; +\; i(A\; \backslash phi)$:

$$\backslash mathbf\{X\}(s,\backslash phi)=$$

\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}

At the extremes, the surface is a catenoid $(\backslash alpha\; =\; 0)$ or a helicoid $(\backslash alpha\; =\; \backslash pi/2)$. Otherwise, $\backslash alpha$ represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the $\backslash mathbf\{X\}\_3$ axis in a helical fashion.

File:Helically Rotated Catenary (Helicatenoid).jpg

File:The fundamental domain (C) and the 3D surfaces. The continuous surfaces are made of copies of the fundamental patch (R3).jpg

One can rewrite each element of second fundamental matrix as a function of $f$ and $g$, for example

\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')

And consequently the second fundamental form matrix can be simplified as

$$\backslash begin\{bmatrix\}$$

-\operatorname{Re} f g' & \;\; \operatorname{Im} f g' \\

\operatorname{Im} f g' & \;\; \operatorname{Re} f g'

\end{bmatrix}

File:Lines of curvature make a quadrangulation of the domain.jpg

One of its eigenvectors is $$\backslash overline\{\backslash sqrt\{\; f\; g\text{'}\}\; \}$$ which represents the principal direction in the complex domain.{{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 }} Therefore, the two principal directions in the $uv$ space turn out to be

$$\backslash phi\; =\; -\backslash frac\{1\}\{2\}\; \backslash operatorname\{Arg\}(f\; g\text{'})\; \backslash pm\; k\; \backslash pi\; /2$$

• Associate family
• Bryant surface, found by an analogous parameterization in hyperbolic space

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Category:Differential geometry

Category:Surfaces

Category:Minimal surfaces