The holomorphic couch theorem

Abstract

We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class is homotopy equivalent to a point, a circle, a torus, or the unit tangent bundle of the codomain, depending on the induced homomorphism on fundamental groups. Quadratic differentials play a central role in the proof.

The converse of the Schwarz Lemma is false

“…In his paper, Ioffe states that f has to be equal to the inclusion map, but this is false [FB15]. His proof of the above weaker statement is nonetheless correct.…”

“…We can therefore define the map f i on X i by setting f i (x) = g l • e −1 li (x) for any l < i such that x ∈ e li (X l ). The restriction f i • e 0i of f i to X 0 = X is homotopic to f since homotopy classes of maps between surfaces with finite topology are closed [FB15,Corollary 2.7]. If i = ∞, then X i has a finite number of ends all of which are cusps.…”

A positive characterization of rational maps

“…This follows from the fact that any two orientation-preserving homotopic embeddings from one connected surface to another are isotopic, which in turn follows from work of Epstein [Eps66] by looking at the boundary curves [Put16]. It is also proved as a side effect of work of Fortier Bourque on conformal embeddings [FB15]. Definition 2.9.…”