  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.
