On the convergence and singularities of the J‐Flow with applications to the Mabuchi energy
On the convergence and singularities of the J‐Flow with applications to the Mabuchi energy
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DOI：
10.1002/cpa.20182
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发表时间：
2004-10
影响因子：
3
通讯作者：
Jian Song;B. Weinkove
中科院分区：
文献类型：
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作者：
Jian Song;B. Weinkove
S.K.Donaldson和X.X.Chen的J流是具有两个Kähler度量的Kähler流形上的抛物型流。这是J泛函的梯度流，出现在陈的马布奇能量公式中。我们找到了关于这两个度量的一个正性条件，它是J-流收敛到一个临界度量的充要条件。利用这一结果，我们证明了在具有充分正规丛的流形上，Mabuchi能量在由正性条件定义的正则类的开邻域内的所有Kähler类上是真的。这改进了陈和第二作者以前的结果。我们讨论了这一点对常数量曲率Kähler度量存在性问题的影响。
The J‐flow of S. K. Donaldson and X. X. Chen is a parabolic flow on Kähler manifolds with two Kähler metrics. It is the gradient flow of the J‐functional that appears in Chen's formula for the Mabuchi energy. We find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the J‐flow to a critical metric. We use this result to show that on manifolds with ample canonical bundle, the Mabuchi energy is proper on all Kähler classes in an open neighborhood of the canonical class defined by a positivity condition. This improves previous results of Chen and of the second author. We discuss the implications of this for the problem of the existence of constant‐scalar‐curvature Kähler metrics.