Given a compact K?hler manifold (X, ω0) let H0 be the set of K?hler forms cohomologous to ω0. 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 H, the set of K?hler potentials of ω0. 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] → H, connecting u_0 ∈ H with u_1 ∈ H, we establish a Morse theoretic result relating the critical points of u_1 - u_0 to the critical points of ?_ 0 = du/dt|_(t=0). As an application of this result, we prove that on all K?hler manifolds, connecting K?hler potentials with smooth geodesics is not possible in general. In particular, in the case X ≠= CP~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.
展开▼
