The Cauchy problem for wave maps on hyperbolic space in dimensions $d \geq 4$

TL;DR

This paper proves global well-posedness and scattering for wave maps from hyperbolic space into Riemannian manifolds in dimensions four and higher, using a novel caloric gauge approach based on harmonic map heat flow.

Contribution

It introduces a new caloric gauge formulation for wave maps on hyperbolic space, simplifying the analysis by reducing to scalar wave equations and enabling Strichartz estimates.

Findings

Established global well-posedness for small data in critical Sobolev space.

Proved scattering for wave maps in hyperbolic space in dimensions $d \\geq 4$.

Demonstrated the effectiveness of the caloric gauge over Coulomb gauge in curved domains.

Abstract

We establish global well-posedness and scattering for wave maps from $d$-dimensional hyperbolic space into Riemannian manifolds of bounded geometry for initial data that is small in the critical Sobolev space for $d \geq 4$. The main theorem is proved using the moving frame approach introduced by Shatah and Struwe. However, rather than imposing the Coulomb gauge we formulate the wave maps problem in Tao's caloric gauge, which is constructed using the harmonic map heat flow. In this setting the caloric gauge has the remarkable property that the main `gauged' dynamic equations reduce to a system of nonlinear scalar wave equations on $\mathbb{H}^{d}$ that are amenable to Strichartz estimates rather than tensorial wave equations (which arise in other gauges such as the Coulomb gauge) for which useful dispersive estimates are not known. This last point makes the heat flow approach crucial in the context of wave maps on curved domains.