The holomorphic couch theorem

TL;DR

This paper proves that homotopic conformal embeddings between finite topological Riemann surfaces are isotopic through conformal embeddings and characterizes the deformation space based on fundamental group homomorphisms.

Contribution

It establishes a homotopy-to-isotopy equivalence for conformal embeddings and describes the deformation space structure using quadratic differentials.

Findings

Homotopic conformal embeddings are isotopic.

Deformation space is a point, circle, torus, or unit tangent bundle.

Quadratic differentials are key to the proof.

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 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.