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  "body": "\n## Summary\n\nA complex manifold is Brody hyperbolic if every holomorphic map from $\\C$ into it is constant.\nNonconstant holomorphic maps $\\C \\to X$ are called entire curves, so Brody hyperbolicity means that $X$ contains no entire curve.\nEvery [[Kobayashi hyperbolic manifold|HRPAX8V7HA]] is Brody hyperbolic, and for compact manifolds the converse holds by [[Brody's lemma|VEV5E2077C]].\nThe notion is named after Robert Brody, who proved the converse in 1978.\n\n## Explanations\n\n### L1 human\n\nLiouville's theorem says that a bounded holomorphic function on $\\C$ is constant.\nBrody hyperbolicity asks the same thing of a complex manifold: the only holomorphic maps from the whole complex plane into it are the constant maps.\nThe unit disc is Brody hyperbolic because of Liouville's theorem.\nThe plane minus two points is Brody hyperbolic by the little Picard theorem.\nA complex torus $\\C / \\Lambda$ is not, because the quotient map $\\C \\to \\C / \\Lambda$ is a nonconstant holomorphic map.\n",
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