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Periodic Schrödinger map flow on Kähler manifolds

Sheng Wang, Yi Zhou

Abstract

Wei-Yue Ding \cite{Ding 2002} proposeed a proposition about Schrödinger 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 Schrödinger map flow for maps from the real line into Kähler 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 Schrödinger map flow into nonlinear Schrödinger-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 Schrödinger map flow for maps from the circle into Kähler manifolds is globally regular. So far, the Wei-Yue Ding's conjecture has been completely solved.

Periodic Schrödinger map flow on Kähler manifolds

Abstract

Wei-Yue Ding \cite{Ding 2002} proposeed a proposition about Schrödinger 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 Schrödinger map flow for maps from the real line into Kähler 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 Schrödinger map flow into nonlinear Schrödinger-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 Schrödinger map flow for maps from the circle into Kähler manifolds is globally regular. So far, the Wei-Yue Ding's conjecture has been completely solved.
Paper Structure (8 sections, 6 theorems, 69 equations)

This paper contains 8 sections, 6 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Conjecture Ding 2002
  3. Outline of the article.
  4. Notations and Preliminaries
  5. law of equilibrium
  6. New div-curl type lemma
  7. Some calculus from law of equilibrium
  8. Global well-posedness for the Schrödinger map flow

Key Result

Theorem 1.1

Suppose $\left(M, g\right)$ is an $m$-dimensional complete Riemannian manifold and $\left(N, J, h\right)$ is a complete Kähler manifold with bounded geometry. If the initial data $u_0\in H^k\left(M, N\right)$ with $k>\frac{m}{2}+1$, then the system schf admits a unique flow $u\in C^0 \left(\left[0, where the constants $C_1$ and $C_2$ only depend on the geometry of the manifold $N$ and $\left \Ver

Theorems & Definitions (10)

  • Theorem 1.1: Local well-posedness
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Remark 4.3