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 $f$ 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 $fg^{2}$ is holomorphic), and let $c_{1},c_{2},c_{3}$ be constants. Then the surface with coordinates $(x_{1},x_{2},x_{3})$ is minimal, where the $x_{k}$ are defined using the real part of a complex integral, as follows:

{\begin{aligned}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}&{}=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 $f (z) = 1$ , $g (z) = z m$ .

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Parametric surface of complex variables

The Weierstrass-Enneper model defines a minimal surface $X$ ( $\mathbb {R} ^{3}$ ) on a complex plane ( $\mathbb {C}$ ). 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 {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

f(\omega )

and

g(\omega )

are holomorphic functions of

\omega

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

\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}} }{|\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 $\mathbf {J}$ leads to a number of important properties: $\mathbf {X_{u}} \cdot \mathbf {X_{v}} =0$ , $\mathbf {X_{u}} ^{2}=\operatorname {Re} (\mathbf {J} ^{2})$ , $\mathbf {X_{v}} ^{2}=\operatorname {Im} (\mathbf {J} ^{2})$ , $\mathbf {X_{uu}} +\mathbf {X_{vv}} =0$ . 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:

{\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

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

\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

\omega _{0}=0

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

\omega _{0}=(1+i)/2

Embedded minimal surfaces and examples

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

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

f(\omega )=\wp (\omega )

where

A

is a constant.^[5]

Helicatenoid

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

\varphi _{1}=e^{-i\alpha }\sinh \left({\frac {\omega }{A}}\right)

\varphi _{2}=ie^{-i\alpha }\cosh \left({\frac {\omega }{A}}\right)

\varphi _{3}=e^{-i\alpha }

\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 $\omega =s+i(A\phi )$ :

\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 $(\alpha =0)$ 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 $\mathbf {X} _{3}$ 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 {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

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

One of its eigenvectors is

{\overline {\sqrt {fg'}}}

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 {1}{2}}\operatorname {Arg} (fg')\pm k\pi /2

References

^ Dierkes, U.; Hildebrandt, S.; Küster, A.; Wohlrab, O. (1992). Minimal surfaces. Vol. I. Springer. p. 108. ISBN 3-540-53169-6.
^ Andersson, S.; Hyde, S. T.; Larsson, K.; Lidin, S. (1988). "Minimal Surfaces and Structures: From Inorganic and Metal Crystals to Cell Membranes and Biopolymers". Chem. Rev. 88 (1): 221–242. doi:10.1021/cr00083a011.
^ Sharma, R. (2012). "The Weierstrass Representation always gives a minimal surface". arXiv:1208.5689 [math.DG].
^ Lawden, D. F. (2011). Elliptic Functions and Applications. Applied Mathematical Sciences. Vol. 80. Berlin: Springer. ISBN 978-1-4419-3090-3.
^ Abbena, E.; Salamon, S.; Gray, A. (2006). "Minimal Surfaces via Complex Variables". Modern Differential Geometry of Curves and Surfaces with Mathematica. Boca Raton: CRC Press. pp. 719–766. ISBN 1-58488-448-7.
^ Hua, H.; Jia, T. (2018). "Wire cut of double-sided minimal surfaces". The Visual Computer. 34 (6–8): 985–995. doi:10.1007/s00371-018-1548-0. S2CID 13681681.

This page was last edited on 3 June 2024, at 20:44

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Transcription

Parametric surface of complex variables

Embedded minimal surfaces and examples

Helicatenoid

Lines of curvature

See also

References