A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kahler geometry

For a metric on the anticanonical bundle, , of a Fano manifold we consider the volume of In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on . Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on , even with very low regularity assumptions on the geodesic. As a consequence we get a simplified proof of the Bando-Mabuchi uniqueness theorem for Kahler-Einstein metrics. A generalization of this theorem to 'twisted' Kahler-Einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being Fano is also given. We moreover discuss a generalization of the main result to other bundles than , and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kahler-Ricci solitons.