Differential geometry Minimal and constant mean curvature surfaces Theorem

Weierstrass–Enneper parameterization

Builds on Minimal surface Holomorphic function Meromorphic function Simply connected domain Conformal map

Statement

Let $U \subset \C$ be a simply connected domain and let $z_0 \in U$ . Let $f$ be a holomorphic function on $U$ and $g$ a meromorphic function on $U$ . Assume that the zeros of $f$ are exactly the poles of $g$ . Assume also that at a pole of $g$ of order $m$ , the function $f$ has a zero of order exactly $2m$ . For $z \in U$ , let $x_1(z) = \operatorname{Re} \int_{z_0}^{z} \frac{f(w)\,(1 - g(w)^2)}{2} \, dw, \quad x_2(z) = \operatorname{Re} \int_{z_0}^{z} \frac{i f(w)\,(1 + g(w)^2)}{2} \, dw, \quad x_3(z) = \operatorname{Re} \int_{z_0}^{z} f(w) g(w) \, dw.$ Then $x = (x_1, x_2, x_3) \colon U \to \R^3$ is a conformal immersion whose image is a minimal surface. Conversely, after a rotation of $\R^3$ if needed, every conformal minimal immersion of $U$ into $\R^3$ arises in this way.

Remarks

The idea of the proof: for a conformal immersion $x$ , write $\varphi_k = 2\,\partial x_k / \partial z$ . The surface is minimal exactly when each $x_k$ is harmonic, that is, when each $\varphi_k$ is holomorphic. Conformality is the equation $\varphi_1^2 + \varphi_2^2 + \varphi_3^2 = 0$ . The substitution $f = \varphi_1 - i\varphi_2$ and $g = \varphi_3 / (\varphi_1 - i\varphi_2)$ solves this equation, and it gives back the integrands above. The condition on zeros and poles makes the integrands holomorphic and keeps $(\varphi_1, \varphi_2, \varphi_3)$ nonzero, so $x$ is an immersion.

The data $f = 2$ and $g(z) = z$ on $\C$ give the Enneper surface. The function $g$ is the Gauss map of the surface followed by stereographic projection. Simple connectivity of $U$ makes the integrals independent of the path; on other domains one must check that the real parts of the periods vanish.

Going deeper

For a student

The representation is the main tool for constructing complete minimal surfaces, such as Costa's minimal surface. On a Riemann surface that is not simply connected, the same formulas work once the real periods of the three forms vanish; this period problem is usually the hardest step. Because $g$ is the Gauss map, curvature questions become questions about meromorphic functions. For example, the total curvature of a complete minimal surface of finite total curvature is $-4\pi$ times the degree of $g$ .

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Conformal map concept
Holomorphic function concept
Meromorphic function concept
Minimal surface concept
Simply connected domain concept
Weierstrass–Enneper parameterization statement

Tag	`YWV6J23AP6`
URI	`m2://YWV6J23AP6`
Kind	statement, theorem
Writing	W0 stub
MSC	53A10, 30F45
Status	This copy of the site has no build data. Run `m2 status YWV6J23AP6`.
About	Minimal surface
Uses	Minimal surface, Holomorphic function, Meromorphic function, Simply connected domain, Conformal map
Same as	Wikidata Q7979830 Wikipedia Weierstrass–Enneper parameterization
Changed	2 November 2026

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