The Conical Kähler–Ricci Flow with Weak Initial Data on Fano Manifolds

2016 ◽  
pp. rnw171
Author(s):  
Jiawei Liu ◽  
Xi Zhang
Keyword(s):  
Initial Data ◽  
Ricci Flow ◽  
Fano Manifolds

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

2019 ◽  
Vol 2019 (751) ◽  
pp. 27-89 ◽  
Author(s):  
Robert J. Berman ◽  
Sebastien Boucksom ◽  
Philippe Eyssidieux ◽  
Vincent Guedj ◽  
Ahmed Zeriahi
Keyword(s):  
Ricci Flow ◽  
Existence And Uniqueness ◽  
Additional Condition ◽  
Einstein Metrics ◽  
Discrete Version ◽  
Fano Varieties ◽  
Fano Manifolds ◽  
Smooth Convergence ◽  
Kähler Einstein Metrics ◽  
Special Case

AbstractWe 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.

scholarly journals Bergman kernel along the Kähler–Ricci flow and Tian’s conjecture

2016 ◽  
Vol 2016 (717) ◽  
Author(s):  
Wenshuai Jiang
Keyword(s):  
Recent Work ◽  
Ricci Curvature ◽  
Ricci Flow ◽  
Bergman Kernel ◽  
Fano Manifolds ◽  
Bergman Kernels ◽  
Complex Dimension ◽  
Short Time

AbstractIn this paper, we study the behavior of Bergman kernels along the Kähler–Ricci flow on Fano manifolds. We show that the Bergman kernels are equivalent along the Kähler–Ricci flow for short time under certain condition on the Ricci curvature of the initial metric. Then, using a recent work of Tian and Zhang, we can solve a conjecture of Tian for Fano manifolds of complex dimension at most 3.

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