THE INVERSE MONGE-AMPÈRE FLOW AND APPLICATIONS TO KÄHLER-EINSTEIN METRICS

Tristan C. Collins, Tomoyuki Hisamoto, Ryosuke Takahashi

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)51-95
Number of pages45
JournalJournal of Differential Geometry
Volume120
Issue number1
DOIs
Publication statusPublished - 2022
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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