Conformal surface embeddings and extremal length

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 finite covers.

Closed continuations of Riemann surfaces

“…A marked closed Riemann surface S ∈ T g belongs to M(R 0 ) if and only if Ext(S, H R 0 (ϕ)) ≦ 1 for all ϕ ∈ A L (R 0 ). Theorem 3, which will be established in §12, is in line with a theorem in the recent paper Kahn-Pilgrim-Thurston [15], where the existence of conformal embeddings of a topologically finite Riemann surface into another homotopic to a given homeomorphic injection is characterized in terms of extremal lengths of multi-curves. Though we do not appeal to their paper, it did suggest that we examine the complement T g \ M(R 0 ) to study M(R 0 ).…”

“…Let S ∈ M(R 0 ). Then we infer from Theorem 7 that inequality (15) holds for all ϕ ∈ A L (R 0 ). To show the converse assume that S ∈ T g \ M(R 0 ).…”

“…Theorem 20. Let R 0 be a marked finite open Riemann surface of genus g. Then a point S ∈ T g belongs to M(R 0 ) if and only if (15) Ext(S, H R 0 (ϕ)) ≦ ϕ R 0 for all ϕ ∈ A L (R 0 ).…”

“…Owing to Theorem 20 and Proposition 15 we can replace U with a smaller one so that U ∩ M(R 0 ) is the set of S ∈ U such that (15) holds for all ϕ ∈ A L (R 0 ) with H R 0 (ϕ) ∈ M F (Σ g ; M(R 0 ), U ∩ ∂M(R 0 )). We then conclude from (29) that the cone…”

On sets of marked once-holed tori allowing holomorphic mappings into Riemann surfaces with marked handle

“…In [8] we examined the set of once-holed tori that can be conformally embedded into a given Riemann surface of positive genus. For topologically finite surfaces Kahn-Pilgrim-Thurston [4] has recently given a characterization for the existence of conformal embeddings in terms of extremal lengths.…”