Gradient NLW on curved background in 4+1 dimensions
  We obtain a sharp local well-posedness result for the Gradient   Nonlinear Wave Equation on a nonsmooth curved background. In the process we introduce variable coefficient versions of Bourgain's $X^{s,b}$ spaces, and use a trilinear multiscale wave packet decomposition in order to prove a key trilinear estimate. 
Dan-Andrei Geba and Daniel Tataru