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


Authors: Xiuxiong Chen and Jingrui Cheng
Journal: J. Amer. Math. Soc. 34 (2021), 937-1009
MSC (2020): Primary 53C55, 53C21; Secondary 35J30, 35J60, 35J96
DOI: https://doi.org/10.1090/jams/966
Published electronically: June 7, 2021
MathSciNet review: 4301558
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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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Additional Information

Xiuxiong Chen
Affiliation: Institute of Geometry and Physics, University of Science and Technology of China, No. 96 Jinzhai Road, Hefei, Anhui, 230026, China AND Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794-3651, USA
MR Author ID: 632654
Email: xiu@math.sunysb.edu

Jingrui Cheng
Affiliation: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI, 53706, USA AND Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794-3651, USA
MR Author ID: 1185151
Email: jingrui.cheng@stonybrook.edu

Received by editor(s): February 17, 2018
Received by editor(s) in revised form: February 24, 2020, and September 24, 2020
Published electronically: June 7, 2021
Additional Notes: The first author was partially supported by NSF grant DMS-1515795 and Simons Foundation grant 605796
Article copyright: © Copyright 2021 American Mathematical Society