Rodnianski, Igor; Rubinstein, Yanir A.; Staffilani, Gigliola On the global well-posedness of the one-dimensional Schrödinger map flow. (English) Zbl 1191.35258 Anal. PDE 2, No. 2, 187-209 (2009). In 2002, W.-Y. Ding conjectured that the Schrödinger map flow is globally well-posed for maps from one dimensional domains into compact Kähler manifolds. The present paper validates Ding’s conjecture for maps from the real line and for maps from the circle into Riemann surfaces. The general idea in both cases is to get a priori estimates on the short time solution to an equivalent system of nonlinear equations which, with non-trivial work, can be extended to stronger norms and these latter estimates imply global well-posedness to the map flow. Despite the similar approach, passing from the case when the domain is the real line to the domain being a circle one loses the simply connectedness, and the compactness, rendering the well-posedness for maps from the circle into Riemann surfaces more difficult in an essential way. Reviewer: Alina Stancu (Lowell) Cited in 23 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 35B10 Periodic solutions to PDEs 32Q15 Kähler manifolds 42B35 Function spaces arising in harmonic analysis 15A23 Factorization of matrices 35B45 A priori estimates in context of PDEs Keywords:Schrödinger flow; periodic NLS; cubic NLS; Strichartz estimates; Kähler manifolds PDFBibTeX XMLCite \textit{I. Rodnianski} et al., Anal. PDE 2, No. 2, 187--209 (2009; Zbl 1191.35258) Full Text: DOI arXiv Link