Weierstrass–Enneper parameterization explained

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:

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, .

Parametric surface of complex variables

The Weierstrass-Enneper model defines a minimal surface

X (\Reals ^{3} ) on a complex plane (\Complex ). Let \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:$$\backslash mathbf\; =\; \backslash begin\backslash left(1\; -\; g^2(\backslash omega)\; \backslash right)f(\backslash omega)\; \backslash \backslash \; i\backslash left(1+\; g^2(\backslash omega)\; \backslash right)f(\backslash omega)\; \backslash \backslash \; 2g(\backslash omega)\; f(\backslash omega)\; \backslash end$$where f(\omega) and g(\omega) are holomorphic functions of \omega .

The Jacobian

J represents the two orthogonal tangent vectors of the surface:^[2] $$\backslash mathbf\; =\; \backslash begin\backslash operatorname\backslash mathbf\_1\; \backslash \backslash \backslash operatorname\backslash mathbf\_2\; \backslash \backslash \backslash operatorname\; \backslash mathbf\_3\; \backslash end\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash mathbf\; =\; \backslash begin-\backslash operatorname\backslash mathbf\_1\; \backslash \backslash -\backslash operatorname\backslash mathbf\_2\; \backslash \backslash -\backslash operatorname\; \backslash mathbf\_3\; \backslash end$$

The surface normal is given by$$\backslash mathbf\; =\backslash frac$$

=\frac

^2+1

\begin2\operatorname g \\2\operatorname g \\| g|^2-1\end

The Jacobian

J leads to a number of important properties: X _{u} ⋅ X _{v} =0 , X _{u} ^{2} = \operatorname{Re}( J ^{2)} , X _{v} ^{2} = \operatorname{Im}( J ^{2)} , X _{uu} + 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:

and the second fundamental form matrix

Finally, a point

\omega _{t} on the complex plane maps to a point X on the minimal surface in \R ^{3} by$$\backslash mathbf=\; \backslash begin\backslash operatorname\; \backslash int\_^\backslash mathbf\_1\; d\backslash omega\backslash \backslash \backslash operatorname\; \backslash int\_^\; \backslash mathbf\_2\; d\backslash omega\backslash \backslash \backslash operatorname\; \backslash int\_^\; \backslash mathbf\_3\; d\backslash omega\; \backslash end$$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

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] $$g(\backslash omega)=\backslash frac$$$$f(\backslash omega)=\; \backslash wp(\backslash omega)$$where A is a constant.^[5]

Helicatenoid

Choosing the functions

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

$$\backslash varphi\_1\; =\; e^\; \backslash sinh\backslash left(\backslash frac\backslash right)$$$$\backslash varphi\_2\; =\; i\; e^\; \backslash cosh\backslash left(\backslash frac\backslash right)$$$$\backslash varphi\_3\; =\; e^$$$$\backslash mathbf(\backslash omega)\; =\; \backslash operatorname\; \backslash begine^\; A\; \backslash cosh\; \backslash left(\backslash frac\; \backslash right)\; \backslash \backslash i\; e^\; A\; \backslash sinh\; \backslash left(\backslash frac\; \backslash right)\; \backslash \backslash e^\; \backslash omega\; \backslash \backslash \backslash end=\backslash cos(\backslash alpha)\; \backslash beginA\; \backslash cosh\; \backslash left(\backslash frac\; \backslash right)\; \backslash cos\; \backslash left(\backslash frac\; \backslash right)\backslash \backslash -\; A\; \backslash cosh\; \backslash left(\backslash frac\; \backslash right)\; \backslash sin\; \backslash left(\backslash frac\; \backslash right)\; \backslash \backslash \backslash operatorname(\backslash omega)\; \backslash \backslash \backslash end\; +\backslash sin(\backslash alpha)\; \backslash beginA\; \backslash sinh\; \backslash left(\backslash frac\; \backslash right)\; \backslash sin\; \backslash left(\backslash frac\; \backslash right)\backslash \backslash A\; \backslash sinh\; \backslash left(\backslash frac\; \backslash right)\; \backslash cos\; \backslash left(\backslash frac\; \backslash right)\; \backslash \backslash \backslash operatorname(\backslash omega)\; \backslash \backslash \backslash end$$

Choosing the parameters of the surface as

\omega = s + i(A \phi) :$$\backslash mathbf(s,\backslash phi)=\backslash cos(\backslash alpha)\; \backslash beginA\; \backslash cosh\; \backslash left(\backslash frac\; \backslash right)\; \backslash cos\; \backslash left(\backslash phi\; \backslash right)\backslash \backslash -\; A\; \backslash cosh\; \backslash left(\backslash frac\; \backslash right)\; \backslash sin\; \backslash left(\backslash phi\; \backslash right)\; \backslash \backslash s\; \backslash \backslash \backslash end\; +\backslash sin(\backslash alpha)\; \backslash beginA\; \backslash sinh\; \backslash left(\backslash frac\; \backslash right)\; \backslash sin\; \backslash left(\backslash phi\; \backslash right)\backslash \backslash A\; \backslash sinh\; \backslash left(\backslash frac\; \backslash right)\; \backslash cos\; \backslash left(\backslash phi\; \backslash right)\; \backslash \backslash A\; \backslash phi\; \backslash \backslash \backslash end$$

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 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$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; \backslash frac$$

^2+1

\begin\operatorname \left((1- g^2) f' - 2gfg'\right) \\\operatorname \left((1+ g^2) f'i+ 2gfg'i \right) \\\operatorname \left(2gf' +2fg' \right) \\\end\cdot \begin \operatorname \left(2g \right) \\ \operatorname \left(-2gi \right) \\ \operatorname \left(|g|^2-1 \right) \\ \end = -2\operatorname (fg')

And consequently the second fundamental form matrix can be simplified asOne of its eigenvectors is $$\backslash overline$$ which represents the principal direction in the complex domain.^[6] Therefore, the two principal directions in the

uv space turn out to be $$\backslash phi\; =\; -\backslash frac\; \backslash operatorname(f\; g\text{'})\; \backslash pm\; k\; \backslash pi\; /2$$

See also

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

Notes and References

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