Conformal surface embeddings and extremal length
Complex Variables
2023-08-21 v3
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.
Cite
@article{arxiv.1507.05294,
title = {Conformal surface embeddings and extremal length},
author = {Jeremy Kahn and Kevin M. Pilgrim and Dylan P. Thurston},
journal= {arXiv preprint arXiv:1507.05294},
year = {2023}
}
Comments
32 pages, 6 figures; v3: New Section 3.4, improved Example 4.4, other improvements throughout