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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On the constant scalar curvature Kähler metrics (II)—Existence results
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by Xiuxiong Chen and Jingrui Cheng HTML | PDF
J. Amer. Math. Soc. 34 (2021), 937-1009 Request permission

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
  • © Copyright 2021 American Mathematical Society
  • 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
  • MathSciNet review: 4301558