Global Existence and Blowup for Geodesics in Universal Teichmüller Spaces

Abstract

In this paper we prove that all initially-smooth solutions of the Euler-Weil-Petersson equation, which describes geodesics on the universal Teichmüller 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 Teichmüller curve under the Velling-Kirillov metric, must blow up in finite time due to wave breaking, extending work of Castro-Córdoba 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 Teichmüller theory.

References in corpus (3)

• The geometry of a vorticity model equation
• Geometric investigations of a vorticity model equation
• Velling-Kirillov metric on the universal Teichmuller curve

Cited by in corpus (1)

• Global Lagrangian solutions of the Camassa-Holm equation