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

Abstract

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 $\mathcal H$, the set of K\"ahler potentials of $\omega_0$. 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 $u_1 \in \mathcal H$, we establish a Morse theoretic result relating the critical points of $u_1-u_0$ to the critical points of $\dot u_0 = du/dt|_{t=0}$. 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 $X \neq \Bbb C P^1$, 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}.