Equivariant Schr\"odinger Maps in two spatial dimensions

TL;DR

This paper proves that equivariant solutions to the Schrödinger map problem from two-dimensional space to the sphere with energy below a critical threshold are globally well-behaved and scatter over time.

Contribution

It establishes global existence and scattering for equivariant Schrödinger maps in 2+1 dimensions with subcritical energy levels.

Findings

Solutions are global in time.

Solutions scatter as time approaches infinity.

Energy threshold of 4π is critical for behavior.

Abstract

We consider equivariant solutions for the Schr\"odinger map problem from $\mathbb{R}^{2+1}$ to $\mathbb{S}^2$ with energy less than $4\pi$ and show that they are global in time and scatter.