The Kähler–Ricci flow on surfaces of positive Kodaira dimension

By Jian Song, G. Tian. 2006

268 citations recorded by the source corpus.

The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has become one of the most powerful tools in geometric analysis. We study the K\"ahler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical is a generalized K\"ahler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric classification for K\"aher surfaces with a numerical effective canonical line bundle by the K\"ahler-Ricci flow. In general, we propose a program of finding canonical metrics on canonical models of projective varieties of positive Kodaira dimension.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c934ec485f924794e6efc754595211b39bc3b079

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Infinite time singularities of the Kähler-Ricci flow (2014, methodology): “The second setup is as follows (see Song and Tian [20; 22; 21], Fong and Zhang [5] and Tosatti, Weinkove and Yang [27]).”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, background): “Song-Tian [39, 40] answered this question affirmatively by showing that each variety with semi-ample canonical class admits a twisted Kähler-Einstein metric of the form Ricω − α = −ω, (6) where roughly α comes from a Weil-Petersson type metric on the moduli space of Calabi-Yau manifolds.”
Collapsing of abelian fibred Calabi-Yau manifolds (2011, background): “The general question of understanding the geometric behaviour of these metrics was raised by Yau [46, 47], Wilson [44] and others, and much work has been devoted to it, see for example [18, 29, 30, 34, 37, 38] and references therein.”
Dimensional reduction and the long-time behavior of Ricci flow (2007, methodology): “A similar phenomenon occurs in the work of Song and Tian concerning collapsing in the Kähler-Ricci flow on elliptic fibrations [52].”
The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits (2014, methodology): “This was conjectured by Song-Tian [39] in the case of elliptic surfaces.”
Lectures on Stability and Constant Scalar Curvature (2008, background): “They are all consequences of Perelman’s crucial monotonicity formulas [101, 36, 129], with the first three statements directly due to him, and the last one to Ye [161] and Zhang [165]:”
Gromov-Hausdorff collapsing of Calabi-Yau manifolds (2013, background): “By [35] and [12], ω̃t converges smoothly to f ∗ω on f−1(K) for any compact K ⊂ N0 as t → 0, and on N0, where ω is the Kähler metric on N0 with Ric(ω) = ωWP obtained in [35] and [29] (see also [28]), and ωWP is a Weil-Petersson semipositive form on N0 coming from the variation of the complex structur”