Equivariant Schr\"odinger Maps in two spatial dimensions: the $\H^2$ target

TL;DR

This paper proves that equivariant solutions to the Schr"odinger map problem from 2+1 dimensions to the hyperbolic plane are globally well-behaved and scatter over time.

Contribution

It establishes the global existence and scattering for equivariant Schr"odinger maps into ^2, a result not previously known for this setting.

Findings

Equivariant solutions are global in time.

Solutions scatter as time progresses.

Finite energy solutions exhibit well-behaved long-term dynamics.

Abstract

We consider equivariant solutions for the Schr\"odinger map problem from $\R^{2+1}$ to $\H^2$ with finite energy and show that they are global in time and scatter.