The Cauchy Problem for Wave Maps on a Curved Background

TL;DR

This paper proves global existence and uniqueness of small data wave maps on a curved background, extending previous results to non-flat geometries using advanced Strichartz estimates for variable coefficient wave equations.

Contribution

It extends the theory of wave maps to curved backgrounds by establishing global well-posedness for small initial data on perturbed Euclidean spaces.

Findings

Proves global existence and uniqueness for small data wave maps on curved backgrounds.

Utilizes Strichartz estimates for variable coefficient wave equations.

Extends prior Euclidean results to Riemannian manifolds with small metric perturbations.

Abstract

We consider the Cauchy problem for wave maps u: \R times M \to N for Riemannian manifolds, (M, g) and (N, h). We prove global existence and uniqueness for initial data that is small in the critical Sobolev norm in the case (M, g) = (\R^4, g), where g is a small perturbation of the Euclidean metric. The proof follows the method introduced by Statah and Struwe for proving global existence and uniqueness of small data wave maps u : \R \times \R^d \to N in the critical norm, for d at least 4. In our argument we employ the Strichartz estimates for variable coefficient wave equations established by Metcalfe and Tataru.