On Modified Mabuchi Functional and Mabuchi Moduli Space of Kähler Metrics on Toric Bundles
On Modified Mabuchi Functional and Mabuchi Moduli Space of Kähler Metrics on Toric Bundles
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DOI：
10.4310/mrl.1999.v6.n5.a7
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发表时间：
1999
影响因子：
1
通讯作者：
D. Guan
中科院分区：
文献类型：
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作者：
D. Guan
Mabuchi在[Mb1]中引入了Mabuchi泛函，证明了它对于处理紧致流形上具有常数量曲率的Kahler度量是非常有用的(见[BM]等)。我们还可以预期，具有常数量曲率的Kahler度量的存在几乎等同于Mabuchi泛函的下界的存在(例如，参见[Ti])。但对于极值度量，Mabuchi泛函不适用。因此，对于在Aut(M)的极大紧连通子群K下不变的度量，我们需要一个新的(或修正的)泛函。我们直到[FM]的出现才获得这一功能(当我们在1995年审查[FM]时)。Mabuchi也独立地发现了这一功能[Mb3](另见[Sm])。[GC]中给出了该泛函的定义。我们将在本文中给出一些结果和应用。证明了修正的Mabuchi泛函M(ω1，ω2)具有这样的性质：对于任意gω∗ω(K)，M(ω1，gω2)=M(∈1，CK2)，其中CKC(K)是K在K的复化K中的中心子，并且极值度量正好是该泛函的局部极小点.因此，我们期望极值度量的存在几乎等同于该泛函的下界的存在。令人惊讶的是，这个泛函的第一个应用不是在光滑环面簇上的极值度量的存在而是唯一，即具有开(C∗)n轨道的光滑Kahler流形。因此，例如，在任何Kahler流形类中，通过爆破二维复射影空间的两点或三点而得到的流形中至多存在一个极值度量。为了具有唯一性，我们考虑环簇上Kahler度量的Mabuchi模空间(见[MB2]，这是由Semmes[SE1]和Donaldson[CH]重新发现的)。在这种情况下，模空间是平坦的(另见[Se1，2])。此外，对于任意两个Kahler度量，都存在唯一的测地线
Mabuchi introduced the Mabuchi functional in [Mb1], and it turns out that it is very useful for dealing with Kahler metrics with constant scalar curvatures on compact manifolds (see [BM], etc.). One can also expect that the existence of Kahler 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 Kahler manifolds with an open (C∗)n-orbit. Therefore, there is for example at most one extremal metric in any Kahler 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 Kahler 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 Kahler metrics there is a unique geodesic