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Schr{ö}dinger maps to a K{ä}hler manifold in two dimensions

Benjamin Dodson, Jeremy L. Marzuola

TL;DR

This paper proves global well-posedness and scattering for Schrödinger maps from 2D Euclidean space into a general compact Kähler target under small initial data in a Besov space. The authors develop an extrinsic Besov framework and employ harmonic map heat flow to construct a caloric gauge, translating the geometric flow into a gauged Schrödinger system with derivative fields and connections. Through bootstrap arguments, L^4 bounds, and bilinear Morawetz-type estimates, they control quasilinear interactions and demonstrate scattering to a linearized profile around a constant map. The work extends prior small-data results to general Kahler targets in 2D with Besov data, highlighting gauge- and heat-flow-based methods that avoid reliance on energy-critical thresholds and enabling broader applicability.

Abstract

We prove a global well--posedness and scattering result for Schr{ö}dinger maps to a general K{ä}hler manifold with small initial data in a Besov space.

Schr{ö}dinger maps to a K{ä}hler manifold in two dimensions

TL;DR

This paper proves global well-posedness and scattering for Schrödinger maps from 2D Euclidean space into a general compact Kähler target under small initial data in a Besov space. The authors develop an extrinsic Besov framework and employ harmonic map heat flow to construct a caloric gauge, translating the geometric flow into a gauged Schrödinger system with derivative fields and connections. Through bootstrap arguments, L^4 bounds, and bilinear Morawetz-type estimates, they control quasilinear interactions and demonstrate scattering to a linearized profile around a constant map. The work extends prior small-data results to general Kahler targets in 2D with Besov data, highlighting gauge- and heat-flow-based methods that avoid reliance on energy-critical thresholds and enabling broader applicability.

Abstract

We prove a global well--posedness and scattering result for Schr{ö}dinger maps to a general K{ä}hler manifold with small initial data in a Besov space.
Paper Structure (16 sections, 15 theorems, 337 equations)

This paper contains 16 sections, 15 theorems, 337 equations.

Table of Contents

  1. Introduction
  2. Outline of argument
  3. A warmup result
  4. Scattering
  5. Harmonic Map Heat Flow
  6. The Gauged Schrödinger Map Equation
  7. The Coulomb Gauge
  8. The Caloric Gauge
  9. Smallness in Besov norm leads to smallness in the gauge
  10. Bounds on terms in the Caloric Gauge
  11. $L^4$ bounds
  12. Two dimensional Schrödinger map flow
  13. Proof of Lemma $\ref{['lmass']}$
  14. Bilinear estimate : Proof of Lemma $\ref{['lbilinear']}$
  15. Proof of Lemma $\ref{['lL4']}$
  16. ...and 1 more sections

Key Result

Theorem 1

For data that is small in $\dot{H}^{d/2}$ norm, global well--posedness for $(1.5)$ has been proved in all dimensions $d \geq 2$.

Theorems & Definitions (38)

  • Remark 1
  • Theorem 1: Small data global well--posedness
  • Definition 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Theorem 3: Local in Time $H^3$ result from mcgahagan2007approximation
  • Remark 4
  • Remark 5
  • Lemma 1: Change in mass
  • ...and 28 more