Criteria for univalence and quasiconformal extension of harmonic   mappings in terms of the Schwarzian DERIVATIVE
  We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane. 
Rodrigo Hern\'andez and Mar\'ia J. Mart\'in
math.CV