Periodic Schr\"odinger map flow on K\"ahler manifolds

TL;DR

This paper proves the global regularity of Schr"odinger map flow from the circle into K"ahler manifolds, confirming Wei-Yue Ding's conjecture through new analytical techniques and energy estimates.

Contribution

It introduces a new div-curl type lemma and combines it with energy and momentum laws to establish global regularity for the flow.

Findings

Proved global regularity of Schr"odinger map flow into K"ahler manifolds from the circle.

Derived a new div-curl type lemma for analysis.

Confirmed Wei-Yue Ding's conjecture.

Abstract

Wei-Yue Ding \cite{Ding 2002} proposeed a proposition about Schr\"odinger map flow in 2002 International Congress of Mathematicians in Beijing, which is called Wei-Yue Ding conjecture by Rodnianski-Rubinstein-Staffilani \cite{Rodnianski 2009}. They proved \cite{Rodnianski 2009} that Schr\"odinger map flow for maps from the real line into K\"ahler manifolds and for maps from the circle into Riemann surfaces is globally well-posed which is the first significant advance in this conjecture by translating the Schr\"odinger map flow into nonlinear Schr\"odinger-type equations or (systems) and partially solved this conjecture. In this article, we will derive a new div-curl type lemma and combined it with energy and ``momentum" balance law to get some space-time estimates. Based on this, we prove the Schr\"odinger map flow for maps from the circle into K\"ahler manifolds is globally regular. So far, the Wei-Yue Ding's conjecture has been completely solved.