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.

Let ${\displaystyle f}${\displaystyle f} and ${\displaystyle g}${\displaystyle g} be functions on either the entire complex plane or the unit disk, where ${\displaystyle g}${\displaystyle g} is meromorphic and ${\displaystyle f}${\displaystyle f} is analytic, such that wherever ${\displaystyle g}${\displaystyle g} has a pole of order ${\displaystyle m}${\displaystyle m}, ${\displaystyle f}${\displaystyle f} has a zero of order ${\displaystyle 2m}${\displaystyle 2m} (or equivalently, such that the product ${\displaystyle fg^{2}}${\displaystyle fg^{2}} is holomorphic), and let ${\displaystyle c_{1},c_{2},c_{3}}${\displaystyle c_{1},c_{2},c_{3}} be constants. Then the surface with coordinates ${\displaystyle (x_{1},x_{2},x_{3})}${\displaystyle (x_{1},x_{2},x_{3})} is minimal, where the ${\displaystyle x_{k}}${\displaystyle x_{k}} are defined using the real part of a complex integral, as follows: $${\displaystyle {\begin{aligned}x_{k}(\zeta )&{}=\mathrm {Re} \left\{\int _{0}^{\zeta }\varphi _{k}(z),円dz\right\}+c_{k},\qquad k=1,2,3\\\varphi _{1}&{}=f(1-g^{2})/2\\\varphi _{2}&{}=if(1+g^{2})/2\\\varphi _{3}&{}=fg\end{aligned}}}$${\displaystyle {\begin{aligned}x_{k}(\zeta )&{}=\mathrm {Re} \left\{\int _{0}^{\zeta }\varphi _{k}(z),円dz\right\}+c_{k},\qquad k=1,2,3\\\varphi _{1}&{}=f(1-g^{2})/2\\\varphi _{2}&{}=if(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 f(z) = 1, g(z) = z^m.

The Weierstrass-Enneper model defines a minimal surface ${\displaystyle X}${\displaystyle X} (${\displaystyle \mathbb {R} ^{3}}${\displaystyle \mathbb {R} ^{3}}) on a complex plane (${\displaystyle \mathbb {C} }${\displaystyle \mathbb {C} }). Let ${\displaystyle \omega =u+vi}${\displaystyle \omega =u+vi} (the complex plane as the ${\displaystyle uv}${\displaystyle uv} space), the Jacobian matrix of the surface can be written as a column of complex entries: $${\displaystyle \mathbf {J} ={\begin{bmatrix}\left(1-g^{2}(\omega )\right)f(\omega )\\i\left(1+g^{2}(\omega )\right)f(\omega )\2円g(\omega )f(\omega )\end{bmatrix}}}$${\displaystyle \mathbf {J} ={\begin{bmatrix}\left(1-g^{2}(\omega )\right)f(\omega )\\i\left(1+g^{2}(\omega )\right)f(\omega )\2円g(\omega )f(\omega )\end{bmatrix}}} where ${\displaystyle f(\omega )}${\displaystyle f(\omega )} and ${\displaystyle g(\omega )}${\displaystyle g(\omega )} are holomorphic functions of ${\displaystyle \omega }${\displaystyle \omega }.

The Jacobian ${\displaystyle \mathbf {J} }${\displaystyle \mathbf {J} } represents the two orthogonal tangent vectors of the surface:^[2] $${\displaystyle \mathbf {X_{u}} ={\begin{bmatrix}\operatorname {Re} \mathbf {J} _{1}\\\operatorname {Re} \mathbf {J} _{2}\\\operatorname {Re} \mathbf {J} _{3}\end{bmatrix}}\;\;\;\;\mathbf {X_{v}} ={\begin{bmatrix}-\operatorname {Im} \mathbf {J} _{1}\\-\operatorname {Im} \mathbf {J} _{2}\\-\operatorname {Im} \mathbf {J} _{3}\end{bmatrix}}}$${\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 $${\displaystyle \mathbf {\hat {n}} ={\frac {\mathbf {X_{u}} \times \mathbf {X_{v}} }{|\mathbf {X_{u}} \times \mathbf {X_{v}} |}}={\frac {1}{|g|^{2}+1}}{\begin{bmatrix}2\operatorname {Re} g\2円\operatorname {Im} g\\|g|^{2}-1\end{bmatrix}}}$${\displaystyle \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円\operatorname {Im} g\\|g|^{2}-1\end{bmatrix}}}

The Jacobian ${\displaystyle \mathbf {J} }${\displaystyle \mathbf {J} } leads to a number of important properties: ${\displaystyle \mathbf {X_{u}} \cdot \mathbf {X_{v}} =0}${\displaystyle \mathbf {X_{u}} \cdot \mathbf {X_{v}} =0}, ${\displaystyle \mathbf {X_{u}} ^{2}=\operatorname {Re} (\mathbf {J} ^{2})}${\displaystyle \mathbf {X_{u}} ^{2}=\operatorname {Re} (\mathbf {J} ^{2})}, ${\displaystyle \mathbf {X_{v}} ^{2}=\operatorname {Im} (\mathbf {J} ^{2})}${\displaystyle \mathbf {X_{v}} ^{2}=\operatorname {Im} (\mathbf {J} ^{2})}, ${\displaystyle \mathbf {X_{uu}} +\mathbf {X_{vv}} =0}${\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: $${\displaystyle {\begin{bmatrix}\mathbf {X_{u}} \cdot \mathbf {X_{u}} &\;\;\mathbf {X_{u}} \cdot \mathbf {X_{v}} \\\mathbf {X_{v}} \cdot \mathbf {X_{u}} &\;\;\mathbf {X_{v}} \cdot \mathbf {X_{v}} \end{bmatrix}}={\begin{bmatrix}1&0\0円&1\end{bmatrix}}}$${\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円&1\end{bmatrix}}}

and the second fundamental form matrix $${\displaystyle {\begin{bmatrix}\mathbf {X_{uu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{uv}} \cdot \mathbf {\hat {n}} \\\mathbf {X_{vu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{vv}} \cdot \mathbf {\hat {n}} \end{bmatrix}}}$${\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 ${\displaystyle \omega _{t}}${\displaystyle \omega _{t}} on the complex plane maps to a point ${\displaystyle \mathbf {X} }${\displaystyle \mathbf {X} } on the minimal surface in ${\displaystyle \mathbb {R} ^{3}}${\displaystyle \mathbb {R} ^{3}} by $${\displaystyle \mathbf {X} ={\begin{bmatrix}\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{1}d\omega \\\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{2}d\omega \\\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{3}d\omega \end{bmatrix}}}$${\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 ${\displaystyle \omega _{0}=0}${\displaystyle \omega _{0}=0} for all minimal surfaces throughout this paper except for Costa's minimal surface where ${\displaystyle \omega _{0}=(1+i)/2}${\displaystyle \omega _{0}=(1+i)/2}.

The classical examples of embedded complete minimal surfaces in ${\displaystyle \mathbb {R} ^{3}}${\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 ${\displaystyle \wp }${\displaystyle \wp }:^[4] $${\displaystyle g(\omega )={\frac {A}{\wp '(\omega )}}}$${\displaystyle g(\omega )={\frac {A}{\wp '(\omega )}}} $${\displaystyle f(\omega )=\wp (\omega )}$${\displaystyle f(\omega )=\wp (\omega )} where ${\displaystyle A}${\displaystyle A} is a constant.^[5]

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

$${\displaystyle \varphi _{1}=e^{-i\alpha }\sinh \left({\frac {\omega }{A}}\right)}$${\displaystyle \varphi _{1}=e^{-i\alpha }\sinh \left({\frac {\omega }{A}}\right)} $${\displaystyle \varphi _{2}=ie^{-i\alpha }\cosh \left({\frac {\omega }{A}}\right)}$${\displaystyle \varphi _{2}=ie^{-i\alpha }\cosh \left({\frac {\omega }{A}}\right)} $${\displaystyle \varphi _{3}=e^{-i\alpha }}$${\displaystyle \varphi _{3}=e^{-i\alpha }} $${\displaystyle \mathbf {X} (\omega )=\operatorname {Re} {\begin{bmatrix}e^{-i\alpha }A\cosh \left({\frac {\omega }{A}}\right)\\ie^{-i\alpha }A\sinh \left({\frac {\omega }{A}}\right)\\e^{-i\alpha }\omega \\\end{bmatrix}}=\cos(\alpha ){\begin{bmatrix}A\cosh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\cos \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\-A\cosh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\sin \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\\operatorname {Re} (\omega )\\\end{bmatrix}}+\sin(\alpha ){\begin{bmatrix}A\sinh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\sin \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\A\sinh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\cos \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\\operatorname {Im} (\omega )\\\end{bmatrix}}}$${\displaystyle \mathbf {X} (\omega )=\operatorname {Re} {\begin{bmatrix}e^{-i\alpha }A\cosh \left({\frac {\omega }{A}}\right)\\ie^{-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 ${\displaystyle \omega =s+i(A\phi )}${\displaystyle \omega =s+i(A\phi )}: $${\displaystyle \mathbf {X} (s,\phi )=\cos(\alpha ){\begin{bmatrix}A\cosh \left({\frac {s}{A}}\right)\cos \left(\phi \right)\\-A\cosh \left({\frac {s}{A}}\right)\sin \left(\phi \right)\\s\\\end{bmatrix}}+\sin(\alpha ){\begin{bmatrix}A\sinh \left({\frac {s}{A}}\right)\sin \left(\phi \right)\\A\sinh \left({\frac {s}{A}}\right)\cos \left(\phi \right)\\A\phi \\\end{bmatrix}}}$${\displaystyle \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}}}

At the extremes, the surface is a catenoid ${\displaystyle (\alpha =0)}${\displaystyle (\alpha =0)} or a helicoid ${\displaystyle (\alpha =\pi /2)}${\displaystyle (\alpha =\pi /2)}. Otherwise, ${\displaystyle \alpha }${\displaystyle \alpha } represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the ${\displaystyle \mathbf {X} _{3}}${\displaystyle \mathbf {X} _{3}} axis in a helical fashion.

One can rewrite each element of second fundamental matrix as a function of ${\displaystyle f}${\displaystyle f} and ${\displaystyle g}${\displaystyle g}, for example ${\textstyle \mathbf {X_{uu}} \cdot \mathbf {\hat {n}} ={\frac {1}{|g|^{2}+1}}{\begin{bmatrix}\operatorname {Re} \left((1-g^{2})f'-2gfg'\right)\\\operatorname {Re} \left((1+g^{2})f'i+2gfg'i\right)\\\operatorname {Re} \left(2gf'+2fg'\right)\\\end{bmatrix}}\cdot {\begin{bmatrix}\operatorname {Re} \left(2g\right)\\\operatorname {Re} \left(-2gi\right)\\\operatorname {Re} \left(|g|^{2}-1\right)\\\end{bmatrix}}=-2\operatorname {Re} (fg')}${\textstyle \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 $${\displaystyle {\begin{bmatrix}-\operatorname {Re} fg'&\;\;\operatorname {Im} fg'\\\operatorname {Im} fg'&\;\;\operatorname {Re} fg'\end{bmatrix}}}$${\displaystyle {\begin{bmatrix}-\operatorname {Re} fg'&\;\;\operatorname {Im} fg'\\\operatorname {Im} fg'&\;\;\operatorname {Re} fg'\end{bmatrix}}}

One of its eigenvectors is $${\displaystyle {\overline {\sqrt {fg'}}}}$${\displaystyle {\overline {\sqrt {fg'}}}} which represents the principal direction in the complex domain.^[6] Therefore, the two principal directions in the ${\displaystyle uv}${\displaystyle uv} space turn out to be $${\displaystyle \phi =-{\frac {1}{2}}\operatorname {Arg} (fg')\pm k\pi /2}$${\displaystyle \phi =-{\frac {1}{2}}\operatorname {Arg} (fg')\pm k\pi /2}

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

• Differential geometry
• Surfaces
• Minimal surfaces

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