Extremal Length and Uniformization

@inproceedings{Gardiner2017ExtremalLA,
  title={Extremal Length and Uniformization},
  author={Frederick P. Gardiner},
  year={2017}
}
The classical theorem that identifies Riemann surfaces of negative Euler characteristic with surfaces carrying a hyperbolic structure is called uniformization. For surfaces that have complex structure it gives simultaneously a common parametrization by a parameter varying in a simply connected domain. For topological reasons the covering domain is connected and simply connected and, this being so, the uniformization theorem says that it must be conformal to either the Riemann sphere, the… 
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