On Modified Mabuchi Functional and Mabuchi Moduli Space of Kähler Metrics on Toric Bundles

Mabuchi introduced the Mabuchi functional in [Mb1], and it turns out that it is very useful for dealing with Kähler metrics with constant scalar curvatures on compact manifolds (see [BM], etc.). One can also expect that the existence of Kähler metric with constant scalar curvature is almost equivalent to the existence of a lower bound of the Mabuchi functional (see, e.g., [Ti]). But for the case of extremal metrics, the Mabuchi functional is not applicable. Therefore, we need a new (or a modified) functional for metrics which are invariant under a maximal compact connected subgroup K of Aut(M). We did not obtain this functional until the appearing of [FM] (while we were reviewing [FM] in 1995). Mabuchi also found this functional independently [Mb3] (see also [Sm]). A definition of this functional was given in [GC]. We shall give some results and applications in this paper. It turns out that our modified Mabuchi functional M(ω1, ω2) has the property that M(ω1, g∗ω2) = M(ω1, ω2), for any g ∈ CKC(K), where CKC(K) is the centralizer of K in the complexification K of K. Moreover, the extremal metrics are exactly the local minimal points of this functional. Therefore, we expect that the existence of an extremal metric is almost equivalent to the existence of a lower bound of this functional. Surprising enough that the first application of this functional is not the existence but the uniqueness of extremal metrics on smooth toric varieties, i.e., smooth Kähler manifolds with an open (C∗)n-orbit. Therefore, there is for example at most one extremal metric in any Kähler class of the manifold obtained by blowing up two points or three points of a two dimensional complex projective space. To have the uniqueness, we consider the Mabuchi moduli space of the Kähler metrics on the toric varieties (see [Mb2], which was rediscovered by Semmes [Se1] and Donaldson [Ch]). It turns out that the moduli space is flat in this situation (see also [Se1,2]). Moreover, for any two Kähler metrics there is a unique geodesic