From rubber bands to rational maps: A research report

TL;DR

This paper develops a parallel theory of elastic graphs and conformal surfaces to characterize hyperbolic critically finite rational maps through a positive criterion involving elastic graph self-embeddings, complementing Thurston's negative criterion.

Contribution

It introduces a new positive criterion for identifying hyperbolic rational maps using elastic graphs and conformal surface theory, extending Thurston's work.

Findings

Characterization of hyperbolic rational maps via elastic graph self-embedding

Development of a theory linking elastic graphs with conformal surfaces

Application of the theory to classify branched self-coverings of the sphere

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. One 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.