The holomorphic couch theorem
Complex Variables
2019-10-16 v4 Geometric Topology
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.
Cite
@article{arxiv.1503.05473,
title = {The holomorphic couch theorem},
author = {Maxime Fortier Bourque},
journal= {arXiv preprint arXiv:1503.05473},
year = {2019}
}
Comments
70 pages, 13 figures. Sections 4 and 8 modified following referee's report