Editing Liouville's theorem (complex analysis) (section)

== On compact Riemann surfaces ==
Any holomorphic function on a [[compact space|compact]] [[Riemann surface]] is necessarily constant.<ref>a concise course in complex analysis and Riemann surfaces, Wilhelm Schlag, corollary 4.8, p.77 http://www.math.uchicago.edu/~schlag/bookweb.pdf {{Webarchive|url=https://web.archive.org/web/20170830063422/http://www.math.uchicago.edu/~schlag/bookweb.pdf |date=2017-08-30 }}</ref>

Let <math>f(z)</math> be holomorphic on a compact Riemann surface <math>M</math>. By compactness, there is a point <math>p_0 \in M</math> where <math>|f(p)|</math> attains its maximum. Then we can find a chart from a neighborhood of <math>p_0</math> to the unit disk <math>\mathbb{D}</math> such that <math>f(\varphi^{-1}(z))</math> is holomorphic on the unit disk and has a maximum at <math>\varphi(p_0) \in \mathbb{D}</math>, so it is constant, by the [[maximum modulus principle]].