Conformal surface embeddings and extremal length
Jeremy Kahn
Brown University, Providence, USAKevin M. Pilgrim
Indiana University, Bloomington, USADylan P. Thurston
Indiana University, Bloomington, USA
Abstract
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