Morse theory and geodesics in the space of K\"ahler metrics

Given a compact K"ahler manifold $(X,{\omega }_0)$ let ${\mathcal{H}}_0$ be the set of K"ahler forms cohomologous to ${\omega }_0$. As observed by Mabuchi \cite{m}, this space has the structure of an infinite dimensional Riemannian manifold, if one identifies it with a totally geodesic subspace of $\mathcal H$,thesetofK\ddot{a}hlerpotentialsof$\omega_0$. Following Donaldson’s research program, existence and regularity of geodesics in this space is of fundamental interest. In this paper, supposing enough regularity of a geodesic $u:[0,1]\to \mathcal{H}$, connecting $u_0\in \mathcal{H}$ with $u_1 \in \mathcal H$,weestablishaMorsetheoreticresultrelatingthecriticalpointsof$u_1-u_0$tothecriticalpointsof$\dot u_0 = du/dt|_{t=0}$. As an application of this result, we prove that on all K"ahler manifolds, connecting K"ahler potentials with smooth geodesics is not possible in general. In particular, in the case $X\mathrm{\ne }\mathbb{C}{P}^1$, we will also prove that the set of pairs of potentials that can not be connected with smooth geodesics has nonempty interior. This is an improvement upon the findings of \cite{lv} and \cite{dl}.