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For $$\phi $$ϕ a metric on the anticanonical bundle, $$-K_X$$-KX, of a Fano manifold $$X$$X we consider the volume of $$X$$X$$\begin{aligned} \int _X e^{-\phi }. \end{aligned}$$∫Xe-ϕ.In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on $$-K_X$$-KX. Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $$X$$X, even with very low regularity assumptions on the geodesic. As a consequence we get a simplified proof of the Bando–Mabuchi uniqueness theorem for Kähler–Einstein metrics. A generalization of this theorem to ‘twisted’ Kähler–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 $$-K_X$$-KX, and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kähler–Ricci solitons.

A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry