Convergence of Kähler-Ricci flow

By G. Tian, Xiaohua Zhu. 2007

224 citations recorded by the source corpus.

In this paper, we prove a theorem on convergence of Kahler-Ricci flow on a compact Kahler manifold which admits a Kahler-Ricci soliton

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/22e13c38871fc4e4b7458b337c58b0f8187d8e79

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Contracting exceptional divisors by the Kähler–Ricci flow II (2010, background): “It was shown in [16, 19, 30] that on a manifold with positive first Chern class the flow converges to a Kähler– Einstein metric when one exists.”
Space of Ricci Flows I (2009, methodology): “The first written proof was given by Tian-Zhu in [100].”
Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons (2014, methodology): “In the case when X is smooth the previous existence result was shown in [20], using a variant of Aubin’s continuity method and the convergence result was shown in [49], using Perelman’s deep estimates [38].”
A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics (2010, unknown): “This corollary generalizes a recent result of Chen-Tian-Zhou [30], saying that any maximizer u such that ωu has locally bounded coefficients is necessarily smooth and Kähler-Einstein.”
The Kähler–Ricci flow through singularities (2009, methodology): “We refer the readers to an incomplete list of literatures [24,25,27,28,37,43,44] for some recent development.”
Multiplier Ideal Sheaves and the K\"ahler-Ricci Flow (2006, unknown): “R IC C I FLO W 1 D.H.Phong ,Natasa Sesum ,and Jacob Sturm y A bstract M ultiplieridealsheavesareconstructed asobstructionsto theconvergenceofthe K ahler-Ricci ow on Fano m anifolds,following earlier constructions ofK ohn,Siu, and Nadel,and using therecentestim atesofK olodziejand Perelm an.”
The Kähler–Ricci flow and optimal degenerations (2016, methodology): “When ωKRS is actually Kähler-Einstein, this was proven by Tian-Zhu [37, 38] using Perelman’s estimates [27, 34] (see also [13]).”
The Kähler-Ricci flow and K-polystability (2008, result): “Note that once the existence of a Kähler-Einstein metric is established, convergence of the flow follows from the work of Tian and Zhu [21].”
Lectures on Stability and Constant Scalar Curvature (2008, methodology): “When restricted to the Kähler-Einstein case, [147] also yields a proof of Theorem 15.”
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, methodology): “In the case when ∆ = 0 and X is a Fano manifold the functional M(X,∆,V ) essentially coincides with the “modified Mabuchi functional” appearing in [57].”
A new approach to static numerical relativity and its application to Kaluza–Klein black holes (2009, methodology): “There the flow was performed in the space of Kähler metrics, and a theorem of Tian-Zhu guaranteed that any initial Kähler metric (in the correct class) would flow to the Kähler-Einstein metric [77].”
The Kähler–Ricci flow on Hirzebruch surfaces (2009, background): “It is shown in [9] that M admits a Kähler–Ricci soliton and we refer the reader to the results of [35] and also [24].”