The Space of Kähler Metrics

Abstract

Donaldson conjectured [14] that the space of Kähler metrics is geodesic convex by smooth geodesic and that it is a metric space. Following Donaldson's program, we verify the second part of Donaldson's conjecture completely and verify his first part partially. We also prove that the constant scalar curvature metric is unique in each Kähler class if the first Chern class is either strictly negative or 0. Furthermore, if C1 ≤ 0, the constant scalar curvature metric realizes the global minimum of Mabuchi energy functional; thus it provides a new obstruction for the existence of constant curvature metric: if the infimum of Mabuchi energy (taken over all metrics in a fixed Kähler class) isn't bounded from below, then there doesn't exist a constant curvature metric. This extends the work of Mabuchi and Bando[3]: they showed that Mabuchi energy bounded from below is a necessary condition for the existence of Kähler-Einstein metrics in the first Chern class. * Research was supported partially by NSF postdoctoral fellowship. * Around the same time with Mabuchi's work, Bourguignon J. P. has worked on something similar in a related subject [6].† Here we mean the mixed second derivatives is uniformly bounded. See theorem 3 in section 3 for details. ‡ The sufficient part of this result was proved in [12]. § Tian inform us that he [39] has conjectured that constant scalar curvature metrics exist if and only if Mabuchi functional is proper. * * In [14], Donaldson provided a formal proof to this proposition after assuming the existence of a smooth geodesic between any two metrics. Our proof follows his idea closely.

The Monge-Ampère operator and geodesics in the space of Kähler potentials

“…Now let h 0 , h 1 ∈ H. It is known by the work of Chen [5] (see also more recent progress in Donaldson [13] and Chen-Tian [6]) that there is a unique C 1,1 geodesic h : [0, 1] → H joining h 0 to h 1 . As discussed in Donaldson [10] the optimal regularity properties of this geodesic are of considerable interest.…”

The $\mathcal C^{2,\alpha}$ estimate of complex Monge-Ampere equation

“…Kolodziej ([29,30]) proved the existence and Hölder estimate of solution to the complex Monge-Ampère equation when the right hand side is a nonnegative L p function for p > 1. There are further existence and regularity results on the complex Monge-Ampère equation with right hand side less regular or degenerate, see references [3,13,26,4,52,43,27,20,19,22] for details.…”

On the weak Kähler-Ricci flow

“…However, even if we obtained a C 1,1 minimizer of the K-energy functional, we would still face the regularity problem. In [8], the first named author made the following conjecture: Any C 1,1 minimizers of the K-energy in a given Kähler class must be smooth. As a consequence of the above theorem, we can solve this conjecture in canonically polarized cases.…”

“…It was known that any Kähler metric with constant scalar curvature is the absolute minimizer of the K-energy on the space of all Kähler metrics with a fixed Kähler class ( [8], [18], [24] and [9]) 1 . From an analytic point of view, it is an extremely difficult problem to prove the existence of a Kähler metric with constant scalar curvature in general cases.…”