Zolotarev's lemma: Difference between revisions
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In [[mathematics]], the '''Weierstrass–Enneper parameterization''' of [[minimal surface]]s is a classical piece of [[differential geometry]]. | |||
[[Alfred Enneper]] and [[Karl Weierstrass]] studied minimal surfaces as far back as 1863. | |||
Let ƒ and ''g'' be functions on either the entire complex plane or the unit disk, where ''g'' is [[meromorphic function|meromorphic]] and ƒ is [[analytic function|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 ƒ''g''<sup>2</sup> is holomorphic), and let ''c''<sub>1</sub>, ''c''<sub>2</sub>, ''c''<sub>3</sub> be constants. Then the surface with coordinates (''x''<sub>1</sub>,''x''<sub>2</sub>,''x''<sub>3</sub>) is minimal, where the ''x''<sub>''k''</sub> are defined using the real part of a complex integral, as follows: | |||
:<math>\begin{align} | |||
x_k(\zeta) &{}= \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 &{}= \bold{i} f(1+g^2)/2 \\ | |||
\varphi_3 &{}= fg | |||
\end{align}</math> | |||
The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.<ref name="DHWK">Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O. ''Minimal surfaces'', vol. I, p. 108. Springer 1992. ISBN 3-540-53169-6</ref> | |||
For example, [[Enneper's surface]] has ƒ(''z'') = 1, ''g''(''z'') = ''z''. | |||
==See also== | |||
* [[Associate family]] | |||
* [[Bryant surface]], found by an analogous parameterization in [[hyperbolic space]] | |||
==References== | |||
{{reflist}} | |||
{{DEFAULTSORT:Weierstrass-Enneper parameterization}} | |||
[[Category:Differential geometry]] | |||
[[Category:Surfaces]] | |||
[[Category:Minimal surfaces]] | |||
{{differential-geometry-stub}} | |||
Revision as of 20:32, 14 March 2013
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 ƒ and g be functions on either the entire complex plane or the unit disk, where g is meromorphic and ƒ 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 ƒg2 is holomorphic), and let c1, c2, c3 be constants. Then the surface with coordinates (x1,x2,x3) is minimal, where the xk 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 ƒ(z) = 1, g(z) = z.
See also
- Associate family
- Bryant surface, found by an analogous parameterization in hyperbolic space
References
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Template:Differential-geometry-stub
- ↑ Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O. Minimal surfaces, vol. I, p. 108. Springer 1992. ISBN 3-540-53169-6