On the constant scalar curvature Kähler metrics (II)—Existence results
On the constant scalar curvature Kähler metrics (II)—Existence results
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DOI：
10.1090/jams/966
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发表时间：
2020-10
影响因子：
3.9
通讯作者：
Xiuxiong Chen;Jingrui Cheng
中科院分区：
文献类型：
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作者：
Xiuxiong Chen;Jingrui Cheng
在本文中，我们应用Chen和Cheng [On the Constant Scalar Curvature Kähler Metrics（I）：a priori estimates，Preprint]中的估计来研究紧致Kähler流形上cscK度量的存在性.首先证明了在Kähler势空间中K-K-能量与L1L^1测地距离d1d_1的适当性蕴含着cscK度量的存在性.我们还证明了（E1，d1）（\mathcal {E}^1，d1）中K-K-能量的弱极小元是光滑的cscK势.最后，我们证明了cscK度量的不存在意味着存在一个不稳定的L1L^1测地线，其中K-K-能量不增，这是唐纳森猜想的一个弱版本.陈秀雄[Ann. Math. Qué. 42（2018），pp. 189.第189章是为了证明
In this paper, we apply our previous estimates in Chen and Cheng [On the constant scalar curvature Kähler metrics (I): a priori estimates, Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness of K K -energy in terms of L 1 L^1 geodesic distance d 1 d_1 in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of the K K -energy in ( E 1 , d 1 ) (\mathcal {E}^1, d_1) are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilized L 1 L^1 geodesic ray where the K K -energy is non-increasing, which is a weak version of a conjecture by Donaldson. The continuity path proposed by Xiuxiong Chen [Ann. Math. Qué. 42 (2018), pp. 69–189] is instrumental in the above proofs.