The isometries of the space of Kahler metrics

Given a compact Kahler manifold, we prove that all global isometries of the space of Kahler metrics are induced by biholomorphisms and anti-biholomorphisms of the manifold. In particular, there exist no global symmetries for Mabuchi's metric. Moreover, we show that the Mabuchi completion does not even admit local symmetries. Closely related to these findings, we provide a large class of metric geodesic segments that cannot be extended at one end, exhibiting the first such examples in the literature.