Schr\"odinger Maps and their associated Frame Systems

TL;DR

This paper proves the equivalence between Schr"odinger maps into spheres or hyperbolic spaces and their gauge-invariant equations, establishing global weak solutions in two dimensions and extending results to certain symmetric manifolds.

Contribution

It establishes the equivalence of Schr"odinger maps and gauge-invariant equations and proves the existence of global weak solutions in two dimensions, extending to symmetric manifolds.

Findings

Equivalence between Schr"odinger maps and gauge-invariant equations.

Existence of global weak solutions in two dimensions.

Extension to maps into compact hermitian symmetric manifolds.

Abstract

In this paper we establish the equivalence of solutions between Schr\"odinger map into $\mathbb{S}^2$ or $ \mathbb{H}^2$ and their associated gauge invariant Schr\"odinger equations. We also establish the existence of global weak solutions into $\mathbb{H}^2$ in two space dimensions. We extend these ideas for maps into compact hermitian symmetric manifolds with trivial first cohomology.