Weierstrass–Enneper parameterization

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.

Weierstrass parameterization facilities fabrication of periodic minimal surfaces

Let ƒ and g be functions on either the entire complex plane or the unit disk, where g is meromorphic and ƒ 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 ƒg² is holomorphic), and let c₁, c₂, c₃ be constants. Then the surface with coordinates (x₁,x₂,x₃) is minimal, where the x_k are defined using the real part of a complex integral, as follows:

${\displaystyle {\begin{aligned}x_{k}(\zeta )&{}=\Re \left\{\int _{0}^{\zeta }\varphi _{k}(z)\,dz\right\}+c_{k},\qquad k=1,2,3\\\varphi _{1}&{}=f(1-g^{2})/2\\\varphi _{2}&{}=\mathbf {i} f(1+g^{2})/2\\\varphi _{3}&{}=fg\end{aligned}}}$

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 ƒ(z) = 1, g(z) = z^m.

Parametric surface of complex variables[edit]

The Weierstrass-Enneper model defines a minimal surface ${\displaystyle X}$ (${\displaystyle \mathbb {R} ^{3}}$) on a complex plane (${\displaystyle \mathbb {C} }$). Let ${\displaystyle \omega =u+vi}$ (the complex plane as the ${\displaystyle uv}$ space), we write the Jacobian matrix of the surface as a column of complex entries:

${\displaystyle \mathbf {J} ={\begin{bmatrix}\left(1-g^{2}(\omega )\right)f(\omega )\\i\left(1+g^{2}(\omega )\right)f(\omega )\\2g(\omega )f(\omega )\end{bmatrix}}}$

Where ${\displaystyle f(\omega )}$ and ${\displaystyle g(\omega )}$ are holomorphic functions of ${\displaystyle \omega }$.

The Jacobian ${\displaystyle \mathbf {J} }$ represents the two orthogonal tangent vectors of the surface:^[2]

${\displaystyle \mathbf {X_{u}} ={\begin{bmatrix}\operatorname {Re} \mathbf {J} _{1}\\\operatorname {Re} \mathbf {J} _{2}\\\operatorname {Re} \mathbf {J} _{3}\end{bmatrix}}\;\;\;\;\mathbf {X_{v}} ={\begin{bmatrix}-\operatorname {Im} \mathbf {J} _{1}\\-\operatorname {Im} \mathbf {J} _{2}\\-\operatorname {Im} \mathbf {J} _{3}\end{bmatrix}}}$

The surface normal is given by

${\displaystyle \mathbf {\hat {n}} =\mathbf {X_{u}} \times \mathbf {X_{v}} ={\frac {1}{|g|^{2}+1}}{\begin{bmatrix}2\operatorname {Re} g\\2\operatorname {Im} g\\|g|^{2}-1\end{bmatrix}}}$

The Jacobian ${\displaystyle \mathbf {J} }$ leads to a number of important properties: ${\displaystyle \mathbf {X_{u}} \cdot \mathbf {X_{v}} =0}$, ${\displaystyle \mathbf {X_{u}} ^{2}=\mathbf {X_{v}} ^{2}}$, ${\displaystyle \mathbf {X_{uu}} +\mathbf {X_{vv}} =0}$. 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:

${\displaystyle {\begin{bmatrix}\mathbf {X_{u}} \cdot \mathbf {X_{u}} &\;\;\mathbf {X_{u}} \cdot \mathbf {X_{v}} \\\mathbf {X_{v}} \cdot \mathbf {X_{u}} &\;\;\mathbf {X_{v}} \cdot \mathbf {X_{v}} \end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}}$

and the second fundamental form matrix

${\displaystyle {\begin{bmatrix}\mathbf {X_{uu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{uv}} \cdot \mathbf {\hat {n}} \\\mathbf {X_{vu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{vv}} \cdot \mathbf {\hat {n}} \end{bmatrix}}}$

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

${\displaystyle \mathbf {X} ={\begin{bmatrix}\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{1}d\omega \\\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{2}d\omega \\\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{3}d\omega \end{bmatrix}}}$

where ${\displaystyle \omega _{0}=0}$ for all minimal surfaces throughout this paper except for Costa's minimal surface where ${\displaystyle \omega _{0}=(1+i)/2}$.

Embedded minimal surfaces and examples[edit]

The classical examples of embedded complete minimal surfaces in ${\displaystyle \mathbb {R} ^{3}}$ with finite topology include the plane, the catenoid, the helicoid, and the Costa's minimal surface. Costa's surface involves Weierstrass's elliptic function ${\displaystyle \wp }$:^[4]

${\displaystyle g(\omega )={\frac {A}{\wp '(\omega )}}}$

${\displaystyle f(\omega )=\wp (\omega )}$

where ${\displaystyle A}$ is a constant ^[5] (Chapter 22).

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

Lines of curvature[edit]

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

${\displaystyle \mathbf {X_{uu}} \cdot \mathbf {\hat {n}} ={\frac {1}{|g|^{2}+1}}{\begin{bmatrix}\operatorname {Re} \left((1-g^{2})f'-2gfg'\right)\\\operatorname {Re} \left((1+g^{2})f'i+2gfg'i\right)\\\operatorname {Re} \left(2gf'+2fg'\right)\\\end{bmatrix}}\cdot {\begin{bmatrix}\operatorname {Re} \left(2g\right)\\\operatorname {Re} \left(-2gi\right)\\\operatorname {Re} \left(|g|^{2}-1\right)\\\end{bmatrix}}=-2\operatorname {Re} (fg')}$

And consequently we can simplify the second fundamental form matrix as

${\displaystyle {\begin{bmatrix}-\operatorname {Re} fg'&\;\;\operatorname {Im} fg'\\\operatorname {Im} fg'&\;\;\operatorname {Re} fg'\end{bmatrix}}}$

Lines of curvature make a quadrangulation of the domain

One of its eigenvectors is

${\displaystyle {\overline {\sqrt {fg'}}}}$

which represents the principal direction in the complex domain.^[6] Therefore, the two principal directions in the ${\displaystyle uv}$ space turn out to be

${\displaystyle \phi =-{\frac {1}{2}}Arg(fg')\pm k\pi /2}$

See also[edit]

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

References[edit]