Global existence of small equivariant wave maps on rotationally symmetric manifolds

TL;DR

This paper proves the global existence of small equivariant wave maps on a new class of rotationally symmetric manifolds called admissible manifolds, which satisfy certain smoothing and Strichartz estimates, extending results to asymptotically flat and hyperbolic spaces.

Contribution

Introduces the class of admissible manifolds and establishes global existence of wave maps for small data in critical regularity on these manifolds.

Findings

Global existence of wave maps on admissible manifolds

Includes asymptotically flat and hyperbolic spaces as special cases

Establishes smoothing and Strichartz estimates for the wave flow

Abstract

We introduce a class of rotationally invariant manifolds, which we call \emph{admissible}, on which the wave flow satisfies smoothing and Strichartz estimates. We deduce the global existence of equivariant wave maps from admissible manifolds to general targets, for small initial data of critical regularity $H^{\frac n2}$. The class of admissible manifolds includes in particular asymptotically flat manifolds and perturbations of real hyperbolic spaces $\mathbb{H}^{n}$ for $n\ge3$.