The Kähler-Ricci flow and K-polystability

By G'abor Sz'ekelyhidi. 2008

105 citations recorded by the source corpus.

We consider the K\"ahler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a K\"ahler-Einstein metric. The main ingredient is a result that says that a sufficiently small perturbation of a cscK manifold admits a cscK metric if it is K-polystable.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5007ddd3feb615118baf342351144cf20ce99b23

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Greatest lower bounds on the Ricci curvature of Fano manifolds (2009, background): “The fact that ψ′(2) =−10/7 means that the metric has conical singularities with angle 2 sin−1 √ 5/7 along a line not meeting the exceptional divisor (i.e. along the divisor at infinity in P(O(−1)⊕O)).”
Space of Ricci Flows I (2009, background): “For other important progress in Kähler Ricci flow, we refer interested readers to the following papers(far away from being complete): [72], [110], [109], [101], [111], [79], [101], [78], [68], [104], [88], as well as references listed therein.”
Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, background): “This yields a purely analytical proof of an extension of a theorem due to Chen [10] and Székelyhidi [44].”
On the evolution of a Hermitian metric by its Chern-Ricci form (2011, background): “It gives a flow of Kähler metrics whose behavior is deeply intertwined with the complex and algebro-geometric properties of M (see [8, 9, 15, 31, 32, 34, 35, 36, 37, 38, 40, 41, 45, 50, 51, 52, 56], for example).”
Kähler–Ricci flow, Kähler–Einstein metric, and K–stability (2015, result): “About the second ingredient, very recently Datar and Székelyhidi [11] have adapted the results of [4] to the Aubin-Yau continuity path, which gives a new proof of the theorem of Chen-Donaldson-Sun.”
Contracting exceptional divisors by the Kähler–Ricci flow II (2010, background): “Convergence is also known under certain weaker assumptions (see [6, 14, 17, 18, 27, 31, 37] and the discussions therein).”