A splitting theorem for extremal Kaehler metrics
  Based on recent work of S. K. Donaldson and T. Mabuchi, we prove that any extremal Kaehler metric in the sense of E. Calabi, defined on the product of polarized compact complex projective manifolds is the product of extremal Kaehler metrics on each factor, provided that the integral Futaki invariants of the polarized manifold vanish or its automorphism group satisfies a constraint. This extends a result of S.-T. Yau about the splitting of a Kaehler-Einstein metric on the product of compact complex manifolds to the more general setting of extremal Kaehler metrics. 
Vestislav Apostolov and Hongnian Huang
math.DG math.AG