Kahler-Ricci solitons on toric manifolds with positive first Chern class

By Xu-jia Wang, Xiaohua Zhu. 2004

408 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/58cdc72fb162336fd5b942508a5e347853289ca4

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Rigidity of Quasi-Einstein Metrics (2008, background): “Kähler Quasi-Einstein Metrics There are many nontrivial examples of shrinking Kähler-Ricci solitons [2, 17].”
Rigid subsets of symplectic manifolds (2007, result): “Moreover, if M is toric the opposite is also true: if the Futaki invariant vanishes for any V generating a subgroup of the torus T acting onM then M admits a Kähler-Einstein metric – this follows from a theorem by Wang and Zhu [67], combined with a previous result of Mabuchi [38].”
Hamilton-Perelman's Proof of the Poincar\'e Conjecture and the Geometrization Conjecture (2006, background): “By contrast, there do exist nontrivial compact gradient shrinking Ricci solitons (see Koiso [83], Cao [14] and Wang-Zhu [131] ).”
On Four-Dimensional Anti-self-dual Gradient Ricci Solitons (2011, background): “The second author also benefits a lot from communication with Prof Xiaohua Zhu.”
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, background): “In the case when X is smooth and ∆ is the trivial divisor the previous theorem was first shown in the seminal work [58] by Zhou-Wang, except for the last point, proven in [59].”
On second variation of Perelman’s Ricci shrinker entropy (2010, methodology): “Specifically in dimension n = 4, Koiso [13] and the first author [1] independently constructed a gradient Kähler-Ricci shrinking soliton on CP (2)#(−CP (2)), and Wang-Zhu [16] on CP (2)#(−2CP (2)), while in the noncompact case Feldman-Ilmanen-Knopf [8] constructed the U(2)-invariant gradient shrinki”
Kähler–Einstein metrics along the smooth continuity method (2015, result): “In particular this recovers the result of Wang-Zhu [50] showing that all toric Fano manifolds admit a Kähler-Ricci soliton.”
Geometry of Complete Gradient Shrinking Ricci Solitons (2009, background): “Compact toric Kähler shrinkers In [39], Wang-Zhu found a gradient Kähler-Ricci soliton on CP 2#2(−CP (2)) which has U(1) × U(1) symmetry.”
Ricci-flat Kähler metrics on crepant resolutions of Kähler cones (2008, result): “The analogous result for toric Fano manifolds was proved in [41].”
Symmetries, quotients and Kähler-Einstein metrics (2004, background): “Although by now there is a good deal of examples, the only broad class of manifolds for which the problem is solved is the one of toric Fano manifolds, thanks to a recent theorem of Xujia Wang and Xiaohua Zhu ([27], see also Donaldson’s work [12] for related results for extremal metrics).”