Scattering for wave maps exterior to a ball

TL;DR

This paper studies the behavior of wave maps outside a ball, showing that degree-zero maps scatter to zero and positive degree maps are asymptotically stable, advancing understanding of wave map dynamics in exterior domains.

Contribution

It establishes scattering for degree-zero wave maps and proves asymptotic stability of harmonic maps for positive degrees in an exterior domain setting.

Findings

Degree-zero maps scatter to zero regardless of energy.

Positive degree harmonic maps are asymptotically stable.

Results apply to 1-equivariant wave maps outside a ball.

Abstract

We consider 1-equivariant wave maps from \R \times (\R^3 \setminus B) to S^3 where B is a ball centered at 0, and the boundary of B gets mapped to a fixed point on S^3. We show that 1-equivariant maps of degree zero scatter to zero irrespective of their energy. For positive degrees, we prove asymptotic stability of the unique harmonic maps in the energy class determined by the degree.