Kevin Pilgrim (Indiana University)


Title: Conformal surface embeddings and extremal length.

Abstract: Given two Riemann surfaces with boundary and a homotopy class of topological embeddings between them, we show there is a conformal embedding in the homotopy class if and only if the extremal length of every simple multi-curve is decreased under the embedding. For applications to dynamical systems, we need an additional fact: if the ratio is bounded above away from one, then it remains so under passing to any finite cover. I will also briefly mention how under natural conditions the technique of quasiconformal surgery promotes so-called rational-like maps f:f^{-1}(S)→S, where f^{-1}(S)⊂S are planar Riemann surfaces, to rational maps. This is joint work of Jeremy Kahn, Kevin M. Pilgrim, and Dylan P. Thurston;

To attend a zoom session, and for suggestions, questions etc. please contact Galia Dafni (galia.dafni [at], Alexandre Girouard (alexandre.girouard [at], Dmitry Jakobson (dmitry.jakobson [at], Damir Kinzebulatov (damir.kinzebulatov [at] or Maxime Fortier Bourque (maxime.fortier.bourque [at]

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