On the global well-posedness of the one-dimensional Schrodinger map flow
  We establish the global well-posedness of the initial value problem for the Schrodinger map flow for maps from the real line into Kahler manifolds and for maps from the circle into Riemann surfaces. This partially resolves a conjecture of W.-Y. Ding. 
Igor Rodnianski, Yanir A. Rubinstein, Gigliola Staffilani