Global Existence and Blowup for Geodesics in Universal Teichm\"uller Spaces
Abstract
In this paper we prove that all initially-smooth solutions of the Euler-Weil-Petersson equation, which describes geodesics on the universal Teichm\"uller space under the Weil-Petersson metric, will remain smooth for all time. This extends the work of Escher-Kolev for strong Riemannian metrics to the borderline case of metrics. In addition we show that all initially-smooth solutions of the Wunsch equation, which describes geodesics on the universal Teichm\"uller curve under the Velling-Kirillov metric, must blow up in finite time due to wave breaking, extending work of Castro-C\'ordoba and Bauer-Kolev-Preston. Finally we illustrate these phenomena in terms of conformal maps of the unit disc, using the conformal welding representation of circle diffeomorphisms which is natural in Teichm\"uller theory.
Keywords
Cite
@article{arxiv.1603.07061,
title = {Global Existence and Blowup for Geodesics in Universal Teichm\"uller Spaces},
author = {Stephen C. Preston and Pearce Washabaugh},
journal= {arXiv preprint arXiv:1603.07061},
year = {2016}
}