From rubber bands to rational maps: a research report

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.

From curves to currents

“…There is a parallel notion of extremal length with respect to elastic graphs, as introduced by the second author [56], just as there is for ordinary lengths (Subsection 4.4).…”

“…Extremal length fits into a family of energies for graphs, as explored by the second author in [56, Appendix A]. For a metric graph with metric g , a constant p with and C a curve on , define where the norm is taken with respect to the metric g .…”

Elastic Graphs

“…Many of the results of this paper were announced as part of an earlier research report [Thu16a], which also contains many related open problems.…”

The holomorphic couch theorem

“…The holomorphic couch problem arose in the context of renormalization in complex dynamics. Although Theorem 1.1 does not have any direct application to dynamics, some of the tools used here do (see [Thu16]).…”