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.
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Acknowledgements
I would like to thank Matt Bainbridge, Steven Gortler, Richard Kenyon, Sarah Koch, Tan Lei, Dan Margalit, and Giulio Tiozzo for many helpful conversations. I would also like to thank especially Maxime Fortier Bourque, who pointed me towards Ioffe’s theorem [23] and had numerous other insights. This project grew out of extensive conversations with Kevin Pilgrim, who helped shape my understanding of the subject in many ways. Many of the arguments were developed jointly with him. Notably, he communicated Theorem 8.4 to me, and Theorem 4 is joint work with him. Theorem 5 is joint work with Jeremy Kahn, who also contributed substantially throughout. Above all, I would like to thank William Thurston for introducing me to the subject and insisting on understanding deeply. This work was partially supported by NSF grants DMS-1358638 and DMS-1507244.
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Dedicated to the memory of William P. Thurston, 1946–2012
This work was partially supported by NSF grants DMS-1358638 and DMS-1507244.
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Thurston, D.P. From rubber bands to rational maps: a research report. Res Math Sci 3, 15 (2016). https://doi.org/10.1186/s40687-015-0039-4
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DOI: https://doi.org/10.1186/s40687-015-0039-4