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 $2 m$ (or equivalently, such that the product $f g^{2}$ 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: $\begin{matrix} x_{k} (ζ) & = R e {\int_{0}^{ζ} φ_{k} (z) d z} + c_{k}, k = 1, 2, 3 \\ φ_{1} & = f (1 - g^{2}) / 2 \\ φ_{2} & = i f (1 + g^{2}) / 2 \\ φ_{3} & = f g \end{matrix}$

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 $f (z) = 1$ , $g (z) = z m$ .

Parametric surface of complex variables

The Weierstrass-Enneper model defines a minimal surface $X$ ( $ℝ^{3}$ ) on a complex plane ( $ℂ$ ). Let $ω = u + v i$ (the complex plane as the $u v$ space), the Jacobian matrix of the surface can be written as a column of complex entries: $𝐉 = [\begin{matrix} (1 - g^{2} (ω)) f (ω) \\ i (1 + g^{2} (ω)) f (ω) \\ 2 g (ω) f (ω) \end{matrix}]$ where $f (ω)$ and $g (ω)$ are holomorphic functions of $ω$ .

The Jacobian $𝐉$ represents the two orthogonal tangent vectors of the surface:^[2] $𝐗_{𝐮} = [\begin{matrix} Re 𝐉_{1} \\ Re 𝐉_{2} \\ Re 𝐉_{3} \end{matrix}] 𝐗_{𝐯} = [\begin{matrix} - Im 𝐉_{1} \\ - Im 𝐉_{2} \\ - Im 𝐉_{3} \end{matrix}]$

The surface normal is given by $\hat{𝐧} = \frac{𝐗_{𝐮} \times 𝐗_{𝐯}}{| 𝐗_{𝐮} \times 𝐗_{𝐯} |} = \frac{1}{| g |^{2} + 1} [\begin{matrix} 2 Re g \\ 2 Im g \\ | g |^{2} - 1 \end{matrix}]$

The Jacobian $𝐉$ leads to a number of important properties: $𝐗_{𝐮} \cdot 𝐗_{𝐯} = 0$ , ${𝐗_{𝐮}}^{2} = Re (𝐉^{2})$ , ${𝐗_{𝐯}}^{2} = Im (𝐉^{2})$ , $𝐗_{𝐮 𝐮} + 𝐗_{𝐯 𝐯} = 0$ . 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: $[\begin{matrix} 𝐗_{𝐮} \cdot 𝐗_{𝐮} & 𝐗_{𝐮} \cdot 𝐗_{𝐯} \\ 𝐗_{𝐯} \cdot 𝐗_{𝐮} & 𝐗_{𝐯} \cdot 𝐗_{𝐯} \end{matrix}] = | f |^{2} (1 + | g |^{2})^{2} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}],$

which is isotropic and reflects the conformal property of the parametrization, and the second fundamental form matrix $[\begin{matrix} 𝐗_{𝐮 𝐮} \cdot \hat{𝐧} & 𝐗_{𝐮 𝐯} \cdot \hat{𝐧} \\ 𝐗_{𝐯 𝐮} \cdot \hat{𝐧} & 𝐗_{𝐯 𝐯} \cdot \hat{𝐧} \end{matrix}]$

Finally, a point $ω_{t}$ on the complex plane maps to a point $𝐗$ on the minimal surface in $ℝ^{3}$ by $𝐗 = [\begin{matrix} Re \int_{ω_{0}}^{ω_{t}} 𝐉_{1} d ω \\ Re \int_{ω_{0}}^{ω_{t}} 𝐉_{2} d ω \\ Re \int_{ω_{0}}^{ω_{t}} 𝐉_{3} d ω \end{matrix}]$ where $ω_{0} = 0$ for all minimal surfaces throughout this paper except for Costa's minimal surface where $ω_{0} = (1 + i) / 2$ .

Embedded minimal surfaces and examples

The classical examples of embedded complete minimal surfaces in $ℝ^{3}$ with finite topology include the plane, the catenoid, the helicoid, and the Costa's minimal surface. Costa's surface involves Weierstrass's elliptic function $℘$ :^[4] $g (ω) = \frac{A}{℘^{'} (ω)}$ $f (ω) = ℘ (ω)$ where $A$ is a constant.^[5]

Helicatenoid

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

$φ_{1} = e^{- i α} \sinh (\frac{ω}{A})$ $φ_{2} = i e^{- i α} \cosh (\frac{ω}{A})$ $φ_{3} = e^{- i α}$ $𝐗 (ω) = Re [\begin{matrix} e^{- i α} A \cosh (\frac{ω}{A}) \\ i e^{- i α} A \sinh (\frac{ω}{A}) \\ e^{- i α} ω \end{matrix}] = \cos (α) [\begin{matrix} A \cosh (\frac{Re (ω)}{A}) \cos (\frac{Im (ω)}{A}) \\ - A \cosh (\frac{Re (ω)}{A}) \sin (\frac{Im (ω)}{A}) \\ Re (ω) \end{matrix}] + \sin (α) [\begin{matrix} A \sinh (\frac{Re (ω)}{A}) \sin (\frac{Im (ω)}{A}) \\ A \sinh (\frac{Re (ω)}{A}) \cos (\frac{Im (ω)}{A}) \\ Im (ω) \end{matrix}]$

Choosing the parameters of the surface as $ω = s + i (A ϕ)$ : $𝐗 (s, ϕ) = \cos (α) [\begin{matrix} A \cosh (\frac{s}{A}) \cos (ϕ) \\ - A \cosh (\frac{s}{A}) \sin (ϕ) \\ s \end{matrix}] + \sin (α) [\begin{matrix} A \sinh (\frac{s}{A}) \sin (ϕ) \\ A \sinh (\frac{s}{A}) \cos (ϕ) \\ A ϕ \end{matrix}]$

At the extremes, the surface is a catenoid $(α = 0)$ or a helicoid $(α = π / 2)$ . Otherwise, $α$ represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the $𝐗_{3}$ 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 $𝐗_{𝐮 𝐮} \cdot \hat{𝐧} = \frac{1}{| g |^{2} + 1} [\begin{matrix} Re ((1 - g^{2}) f^{'} - 2 g f g^{'}) \\ Re ((1 + g^{2}) f^{'} i + 2 g f g^{'} i) \\ Re (2 g f^{'} + 2 f g^{'}) \end{matrix}] \cdot [\begin{matrix} Re (2 g) \\ Re (- 2 g i) \\ Re (| g |^{2} - 1) \end{matrix}] = - 2 Re (f g^{'})$

And consequently the second fundamental form matrix can be simplified as $[\begin{matrix} - Re f g^{'} & Im f g^{'} \\ Im f g^{'} & Re f g^{'} \end{matrix}]$

One of its eigenvectors is $\overline{\sqrt{f g^{'}}}$ which represents the principal direction in the complex domain.^[6] Therefore, the two principal directions in the $u v$ space turn out to be $ϕ = - \frac{1}{2} Arg (f g^{'}) \pm k π / 2$

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.

[DHWK-1] Dierkes, U.; Hildebrandt, S.; Küster, A.; Wohlrab, O. (1992). Minimal surfaces. Vol. I. Springer. p. 108. ISBN 3-540-53169-6.

[2] 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.

[3] Sharma, R. (2012). "The Weierstrass Representation always gives a minimal surface". arXiv:1208.5689 [math.DG].

[4] Lawden, D. F. (2011). Elliptic Functions and Applications. Applied Mathematical Sciences. Vol. 80. Berlin: Springer. ISBN 978-1-4419-3090-3.

[5] 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.

[6] 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.

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Parametric surface of complex variables

Embedded minimal surfaces and examples

Helicatenoid

Lines of curvature

See also

References