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  "body": "\n## Statement\n\nLet $\\mu \\colon \\C \\to \\C$ be measurable with $\\norm{\\mu}_\\infty \u003c 1$.\nLet $f^\\mu \\colon \\C \\to \\C$ be the unique [[quasiconformal|SB8KG46NHP]] homeomorphism that fixes $0$ and $1$ and solves the [[Beltrami equation|9J7GFJ1GPF]] $f_{\\bar z} = \\mu f_z$ almost everywhere.\nIts existence and uniqueness is the [[measurable Riemann mapping theorem|TZHD2WKNSD]].\n\nLet $U \\subset \\C$ be open.\nLet $t \\mapsto \\mu_t$ be a map from $U$ to the open unit ball of $L^\\infty(\\C)$ that is holomorphic as a map into the Banach space $L^\\infty(\\C)$.\nThen for each $z \\in \\C$, the function $t \\mapsto f^{\\mu_t}(z)$ is [[holomorphic|0NAHXR1FRH]] on $U$.\n\n## Remarks\n\nFor $\\mu$ with compact support, the proof writes the derivative $\\partial_{\\bar z}$ of the principal solution as a Neumann series in the [[Beurling transform|TX32GDQ330]] $T$: $\\mu + \\mu T \\mu + \\mu T(\\mu T \\mu) + \\cdots$.\nThe series converges in $L^p$ for some $p \u003e 2$, because $T$ is an isometry of $L^2$ and its norm on $L^p$ tends to $1$ as $p \\to 2$.\nEach term depends holomorphically on $\\mu$, and the convergence is locally uniform.\nThe same argument gives real-analytic dependence when $\\mu_t$ depends real-analytically on a real parameter.\n\nExample: for $\\mu_t = t \\chi$ with $\\chi$ the indicator of a disc and $\\abs{t} \u003c 1$, the normalized solutions $f^{\\mu_t}$ form a holomorphic family of quasiconformal maps through $f^{0} = \\id$.\n\nThe condition $\\norm{\\mu_t}_\\infty \u003c 1$ is needed: when $\\norm{\\mu}_\\infty = 1$ the Beltrami equation is degenerate, and a homeomorphic solution need not exist.\n\n## Explanations\n\n### L1 human\n\nA Beltrami coefficient $\\mu$ prescribes how a map should distort infinitesimal circles into ellipses.\nThe measurable Riemann mapping theorem gives exactly one normalized map with this distortion.\nThe Ahlfors–Bers theorem adds that if the prescribed distortion moves holomorphically with a complex parameter, then so does the map.\n\n### L2 human\n\nThis holomorphic dependence is what makes [[Teichmüller space|ZZGC6BZNJ5]] a complex manifold: Beltrami coefficients give coordinates, and the solution maps vary holomorphically with them.\nIt is also the tool behind holomorphic motions and quasiconformal surgery in complex dynamics, where one deforms a rational map by moving an invariant Beltrami coefficient in a holomorphic family.\n",
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