Geodesics in the space of Kähler metrics

Abstract

Let $(X,\omega )$ be a compact Kähler manifold. As discovered in the late 1980s by Mabuchi, the set ${\mathcal{H}}_0$ of Kähler forms cohomologous to $\omega$ has the natural structure of an infinite-dimensional Riemannian manifold. We address the question of whether any two points in ${\mathcal{H}}_0$ can be connected by a smooth geodesic and show that the answer, in general, is “no.”