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 \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 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 $X\ne \mathbb{C}{P}^1$, 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}.

Final version. To appear in PAMS

References in corpus (1)

• The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan

Cited by in corpus (3)

• Moment maps, nonlinear PDE, and stability in mirror symmetry
• Geodesic Rays and Kähler-Ricci Trajectories on Fano Manifolds
• Harmonic Discs of Solutions to the Complex Homogeneous Monge-Ampère Equation