Conformal surface embeddings and extremal length

  • Jeremy Kahn

    Brown University, Providence, USA
  • Kevin M. Pilgrim

    Indiana University, Bloomington, USA
  • Dylan P. Thurston

    Indiana University, Bloomington, USA
Conformal surface embeddings and extremal length cover
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Given two Riemann surfaces with boundary and a homotopy class of topological embeddings between them, there is a conformal embedding in the homotopy class if and only if the extremal length of every simple closed multi-curve is decreased under the embedding. Furthermore, the homotopy class has a conformal embedding that misses an open disk if and only if extremal lengths are decreased by a definite ratio. This ratio remains bounded away from one under finite covers.

Cite this article

Jeremy Kahn, Kevin M. Pilgrim, Dylan P. Thurston, Conformal surface embeddings and extremal length. Groups Geom. Dyn. 16 (2022), no. 2, pp. 403–435

DOI 10.4171/GGD/673