Geodesic
Conformal surface embeddings and extremal length
Jeremy Kahn1 ; Kevin M. Pilgrim2 orcid ; Dylan P. Thurston2 orcid
1Brown University, Providence, USA
2Indiana University, Bloomington, USA
Groups, geometry, and dynamics, Tome 16 (2022) no. 2, pp. 403-435

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DOI

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.
DOI : 10.4171/ggd/673
Classification : 30-XX, 31-XX, 32-XX
Mots-clés : Riemann surfaces with boundary, conformal embeddings, extremal length

Jeremy Kahn  1   ; Kevin M. Pilgrim  2   ; Dylan P. Thurston  2

1 Brown University, Providence, USA
2 Indiana University, Bloomington, USA 
Jeremy Kahn; Kevin M. Pilgrim; Dylan P. Thurston. Conformal surface embeddings and extremal length. Groups, geometry, and dynamics, Tome 16 (2022) no. 2, pp. 403-435. doi: 10.4171/ggd/673
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     title = {Conformal surface embeddings and extremal length},
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     pages = {403--435},
     year = {2022},
     volume = {16},
     number = {2},
     doi = {10.4171/ggd/673},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/673/}
}
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