A controlling norm for energy-critical Schr\"odinger maps

TL;DR

This paper proves that energy-critical Schrödinger maps into spheres or hyperbolic planes can be globally continued if a specific space-time norm remains bounded, simplifying the large data well-posedness problem.

Contribution

It establishes a norm-based continuation criterion for energy-critical Schrödinger maps, linking global well-posedness to controlling a key space-time norm.

Findings

Unique solutions can be extended as long as the L^4 norm stays bounded.

Reduces large data global well-posedness to norm control problem.

Provides a criterion for continuation of solutions in energy-critical Schrödinger maps.

Abstract

We consider energy-critical Schroedinger maps with target either the sphere S^2 or hyperbolic plane H^2 and establish that a unique solution may be continued so long as a certain space-time L^4 norm remains bounded. This reduces the large data global wellposedness problem to that of controlling this norm.