Morse theory and geodesics in the space of Kähler metrics

Abstract:

Given a compact Kähler manifold $(X,\omega _0)$ let $\mathcal H_{0}$ be the set of Kähler forms cohomologous to $\omega _0$. As observed by Mabuchi, 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 Kähler 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 Kähler manifolds, connecting Kähler potentials with smooth geodesics is not possible in general. In particular, in the case $X \neq \mathbb {C} P^1$, we will also prove that the set of pairs of potentials that cannot be connected with smooth geodesics has nonempty interior. This is an improvement upon the findings of Lempert and Vivas and of the author and Lempert.