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; https://arxiv.org/abs/1507.05294

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