The space of Poincar\'e type K\"ahler metrics on the complement of a   divisor
  Consider a divisor D with simple normal crossings in a compact K\"ahler manifold X. We show in this article that a K\"ahler metric in an arbitrary class, with constant scalar curvature and cusp singularities along the divisor is unique in this class when K[D] is ample. This we do by generalizing Chen's construction of approximate geodesics in the space of K\"ahler metrics, and proving an approximate version of the Calabi-Yau theorem, both independently of the ampleness of K[D]. 
Hugues Auvray
math.DG