Conformal surface embeddings and extremal length
arXiv:1507.05294 · doi:10.1017/fms.2019.4
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.
32 pages, 6 figures; v3: New Section 3.4, improved Example 4.4, other improvements throughout