Abstract: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 $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of the $K$-energy in $(\mathcal {E}^1, d_1)$ are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilized $L^1$ geodesic ray where the $K$-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.

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中文翻译：

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关于恒定标量曲率 Kähler 度量 (II)—存在结果

摘要：在本文中，我们在 Chen 和 Cheng 中应用我们之前的估计 [ On the constant scalarCurve Kähler metrics (I): a priori估计, 预印本] 研究紧凑 Kähler 流形上 cscK 度量的存在。首先，我们证明在 Kähler 势空间中，$K$-energy 在 $L^1$ 测地距离 $d_1$ 方面的适当性意味着 cscK 度量的存在。我们还表明 $(\mathcal {E}^1, d_1)$ 中 $K$-energy 的弱极小值是平滑的 cscK 势。最后，我们证明不存在 cscK 度量意味着存在不稳定的 $L^1$ 测地线，其中 $K$-energy 不增加，这是 Donaldson 猜想的弱版本。陈秀雄提出的连续性路径[Ann. 数学。阙。42 (2018), pp. 69–189] 有助于上述证明。

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