Open Access

From rubber bands to rational maps: a research report

Research in the Mathematical Sciences20163:15

https://doi.org/10.1186/s40687-015-0039-4

Received: 18 June 2015

Accepted: 13 August 2015

Published: 15 May 2016

Abstract

This research report outlines work, partially joint with Jeremy Kahn and Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal surfaces with boundary. On one hand, this lets us tell when one rubber band network is looser than another and, on the other hand, tell when one conformal surface embeds in another. We apply this to give a new characterization of hyperbolic critically finite rational maps among branched self-coverings of the sphere, by a positive criterion: a branched covering is equivalent to a hyperbolic rational map if and only if there is an elastic graph with a particular “self-embedding” property. This complements the earlier negative criterion of W. Thurston.

Keywords

Complex dynamics Dirichlet energy Elastic graphs Extremal length Measured foliations Riemann surfaces Rational maps Surface embeddings

Notes