Global regularity of critical Schr\"odinger maps: subthreshold dispersed energy

TL;DR

This paper proves that for energy-dispersed initial data below a certain threshold, the energy-critical Schrödinger map from R^2 to S^2 admits a unique global smooth solution, advancing previous results.

Contribution

It introduces new estimates on the commutator of Schrödinger maps and harmonic map heat flows to establish global regularity for subthreshold dispersed energy.

Findings

Global smooth solutions exist for subthreshold dispersed energy.

Improved upon earlier conditional regularity results.

Utilized novel commutator estimates between Schrödinger and harmonic map flows.

Abstract

We consider the energy-critical Schroedinger map initial value problem with smooth initial data from R^2 into the sphere S^2. Given sufficiently energy-dispersed data with subthreshold energy, we prove that the system admits a unique global smooth solution. This improves earlier analogous conditional results. The key behind this improvement lies in exploiting estimates on the commutator of the Schroedinger map and harmonic map heat flows.