Weak geodesics in the space of Kähler metrics

Abstract

Given a compact Kähler manifold (X,ω_0), according to Mabuchi, the set of Kähler forms cohomologous to ω_0 has the natural structure of an infinite dimensional Riemannian manifold. We address the question whether points in this space can be joined by a geodesic, and strengthening previous findings of the second author with Vivas, we show that this cannot always be done even with a certain type of generalized geodesics.