Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties

By Robert J. Berman, S. Boucksom, Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi. Journal für die Reine und Angewandte Mathematik, 2011

225 citations recorded by the source corpus.

We prove the existence and uniqueness of Kähler–Einstein metrics on {{\mathbb{Q}}}-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e07271cd63c3bd75c4f9cc4610e59e5696aac7d6

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In Applied Mathematics

1. Chi Li
2. Yuchen Liu
3. Jingrui Cheng
4. H. Blum
5. Chenyang Xu
6. Kento Fujita
7. Xiuxiong Chen
8. Mattias Jonsson

In Geometry and Topology

1. Chi Li
2. Yuchen Liu
3. Jingrui Cheng
4. Mattias Jonsson
5. H. Blum
6. Chenyang Xu
7. Robert J. Berman
8. Kento Fujita

Citations in the corpus

• On the constant scalar curvature Kähler metrics (II)—Existence results (2020, background): “In the above, the space E1 is the abstract metric completion of the space H under the Finsler metric d1 in H (see section 2 for more details) and the notion of finite energy geodesic segment was introduced in [4] (c.”
• Kähler–Einstein metrics and volume minimization (2016, background): “We recall some notions in the logarithmic setting following [5].”