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  "body": "\n## Statement\n\nLet $U \\subset \\C$ be a [[simply connected domain|0QW64ARSHB]] and let $z_0 \\in U$.\nLet $f$ be a [[holomorphic function|0NAHXR1FRH]] on $U$ and $g$ a [[meromorphic function|0ZZ7ZYDR8W]] on $U$.\nAssume that the zeros of $f$ are exactly the poles of $g$.\nAssume also that at a pole of $g$ of order $m$, the function $f$ has a zero of order exactly $2m$.\nFor $z \\in U$, let\n$$x_1(z) = \\operatorname{Re} \\int_{z_0}^{z} \\frac{f(w)\\,(1 - g(w)^2)}{2} \\, dw, \\quad x_2(z) = \\operatorname{Re} \\int_{z_0}^{z} \\frac{i f(w)\\,(1 + g(w)^2)}{2} \\, dw, \\quad x_3(z) = \\operatorname{Re} \\int_{z_0}^{z} f(w) g(w) \\, dw.$$\nThen $x = (x_1, x_2, x_3) \\colon U \\to \\R^3$ is a [[conformal|0YYT36R4YD]] immersion whose image is a [[minimal surface|41X0YW2R9N]].\nConversely, after a rotation of $\\R^3$ if needed, every conformal minimal immersion of $U$ into $\\R^3$ arises in this way.\n\n## Remarks\n\nThe idea of the proof: for a conformal immersion $x$, write $\\varphi_k = 2\\,\\partial x_k / \\partial z$.\nThe surface is minimal exactly when each $x_k$ is harmonic, that is, when each $\\varphi_k$ is holomorphic.\nConformality is the equation $\\varphi_1^2 + \\varphi_2^2 + \\varphi_3^2 = 0$.\nThe 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.\nThe condition on zeros and poles makes the integrands holomorphic and keeps $(\\varphi_1, \\varphi_2, \\varphi_3)$ nonzero, so $x$ is an immersion.\n\nThe data $f = 2$ and $g(z) = z$ on $\\C$ give the [[Enneper surface|73XN6AS0WX]].\nThe function $g$ is the Gauss map of the surface followed by stereographic projection.\nSimple connectivity of $U$ makes the integrals independent of the path; on other domains one must check that the real parts of the periods vanish.\n\n## Explanations\n\n### L1 human\n\nThis formula produces minimal surfaces from two functions of a complex variable.\nChoose the functions, compute three integrals, and take real parts: the result is always a minimal surface.\nSo the study of minimal surfaces in space turns into a study of complex functions.\n\n### L2 human\n\nThe representation is the main tool for constructing complete minimal surfaces, such as [[Costa's minimal surface|EF0QMQB2J1]].\nOn 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.\nBecause $g$ is the Gauss map, curvature questions become questions about meromorphic functions.\nFor example, the total curvature of a complete minimal surface of finite total curvature is $-4\\pi$ times the degree of $g$.\n",
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