Global Existence and Blowup for Geodesics in Universal Teichm\"uller Spaces

TL;DR

This paper investigates the long-term behavior of solutions to geodesic equations on the universal Teichmüller space, proving global smoothness for the Euler-Weil-Petersson case and finite-time blowup for the Wunsch equation, with geometric illustrations.

Contribution

It extends existing results by establishing global existence for the Euler-Weil-Petersson equation at the borderline Sobolev regularity and demonstrating finite-time blowup for the Wunsch equation, with geometric interpretations.

Findings

Euler-Weil-Petersson solutions remain smooth globally

Wunsch equation solutions blow up in finite time

Geometric illustrations via conformal maps and welding

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 $H^{3/2}$ 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.