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arXiv · 1507.05294

Conformal surface embeddings and extremal length

Jeremy Kahn·Kevin M. Pilgrim·Dylan P. Thurston
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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 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 covers.

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Jeremy Kahn, Kevin M. Pilgrim, Dylan P. Thurston. 2015-07-19. Conformal surface embeddings and extremal length. https://doi.org/10.1017/fms.2019.4

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