TY - JOUR
T1 - THE INVERSE MONGE-AMPÈRE FLOW AND APPLICATIONS TO KÄHLER-EINSTEIN METRICS
AU - Collins, Tristan C.
AU - Hisamoto, Tomoyuki
AU - Takahashi, Ryosuke
N1 - Funding Information:
Bando, Robert Berman, Sébastien Boucksom, Ryoichi Kobayashi and Yuji Odaka for helpful conversations. T.C.C. and T. H. would like to thank Chalmers University for their hospitality during a visit in May, 2017, where this project started. T.C.C. was supported in part by National Science Foundation grant DMS-1506652, the European Research Council and the Knut and Alice Wallenberg Foundation. T.H. was supported by JSPS KAKENHI Grant Number 15H06262 and 17K14185. R.T. was supported by Grant-in-Aid for JSPS Fellows Number 16J01211. Finally, the authors are grateful to the referees for many helpful comments.
Publisher Copyright:
© 2022 International Press of Boston, Inc.. All rights reserved.
PY - 2022
Y1 - 2022
N2 - We introduce the inverse Monge-Ampère flow as the gradient flow of the Ding energy functional on the space of Kähler metrics in 2πλc1(X) for λ = ±1. We prove the long-time existence of the flow. In the canonically polarized case, we show that the flow converges smoothly to the unique Kähler-Einstein metric with negative Ricci curvature. In the Fano case, assuming the X admits a Kähler-Einstein metric, we prove the weak convergence of the flow to the Kähler-Einstein metric. In general, we expect that the limit of the flow is related with the optimally destabilizing test configuration for the L2-normalized non-Archimedean Ding functional. We confirm this expectation in the case of toric Fano manifolds.
AB - We introduce the inverse Monge-Ampère flow as the gradient flow of the Ding energy functional on the space of Kähler metrics in 2πλc1(X) for λ = ±1. We prove the long-time existence of the flow. In the canonically polarized case, we show that the flow converges smoothly to the unique Kähler-Einstein metric with negative Ricci curvature. In the Fano case, assuming the X admits a Kähler-Einstein metric, we prove the weak convergence of the flow to the Kähler-Einstein metric. In general, we expect that the limit of the flow is related with the optimally destabilizing test configuration for the L2-normalized non-Archimedean Ding functional. We confirm this expectation in the case of toric Fano manifolds.
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U2 - 10.4310/JDG/1641413788
DO - 10.4310/JDG/1641413788
M3 - Article
AN - SCOPUS:85123772125
VL - 120
SP - 51
EP - 95
JO - Journal of Differential Geometry
JF - Journal of Differential Geometry
SN - 0022-040X
IS - 1
ER -