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 {\displaystyle f} and g {\displaystyle g} be functions on either the entire complex plane or the unit disk, where g {\displaystyle g} is meromorphic and f {\displaystyle f} is analytic, such that wherever g {\displaystyle g} has a pole of order m {\displaystyle m} , f {\displaystyle f} has a zero of order 2 m {\displaystyle 2m} (or equivalently, such that the product f g 2 {\displaystyle fg^{2}} is holomorphic), and let c 1 , c 2 , c 3 {\displaystyle c_{1},c_{2},c_{3}} be constants. Then the surface with coordinates ( x 1 , x 2 , x 3 ) {\displaystyle (x_{1},x_{2},x_{3})} is minimal, where the x k {\displaystyle x_{k}} are defined using the real part of a complex integral, as follows:
x k ( ζ )
= R e { ∫ 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 {\displaystyle {\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. For example, Enneper's surface has f(z) = 1, g(z) = zm.
Parametric surface of complex variables The Weierstrass-Enneper model defines a minimal surface X {\displaystyle X} ( R 3 {\displaystyle \mathbb {R} ^{3}} ) on a complex plane ( C {\displaystyle \mathbb {C} } ). Let ω = u + v i {\displaystyle \omega =u+vi} (the complex plane as the u v {\displaystyle uv} space), the Jacobian matrix of the surface can be written as a column of complex entries:
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