Date of Degree
conformal embeddings; extremal length; h-principle; quadratic differentials; Teichmuller's theorem
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 deformation retracts into 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.
Fortier Bourque, Maxime, "The holomorphic couch theorem" (2015). CUNY Academic Works.