On the constant scalar curvature Kähler metrics (II)—Existence results

Chen, Xiuxiong; Cheng, Jingrui

doi:10.1090/jams/966

On the constant scalar curvature Kähler metrics (II)—Existence results
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by Xiuxiong Chen and Jingrui Cheng;

J. Amer. Math. Soc. 34 (2021), 937-1009

DOI: https://doi.org/10.1090/jams/966

Published electronically: June 7, 2021

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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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