Ricci flow on Kähler-Einstein surfaces

By Xiuxiong Chen, Gang Tian. 2001

156 citations recorded by the source corpus.

Abstract.In this paper, we prove that if M is a Kähler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the Kähler-Ricci flow converges to a Kähler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general Kähler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in Kähler Ricci flow: On a compact Kähler-Einstein manifold, does the Kähler-Ricci flow converge to a Kähler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the Kähler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem.

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

View Paper: https://www.semanticscholar.org/paper/a3f5c56899c1fc991a83b4e09d08e566bdee78b4

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Citations in the corpus

On a class of fully nonlinear flows in Kähler geometry (2009, background): “Fj is shown to be independent of the choice of path [5].”
The Moser-Trudinger inequality on Kähler-Einstein manifolds (2006, unknown): “More precisely, for each t, consider the normalized Kähler-Ricci flow us,t in time s with initial data u0,t = 0, ∂us,t ∂s = log (ωφt + √ −1 2 ∂∂̄us,t) n ωnφt + us,t − hωφt .”
Global existence and convergence for a higher order flow in conformal geometry (2003, unknown): “Tian [7] to prove convergence of the Kähler-Ricci flow on Kähler-Einstein surfaces.”
Frankel conjecture and Sasaki geometry (2012, methodology): “In [35], using the maximum principle as in [16], it is proved that such manifolds must have constant transverse holomorphic bisectional curvature 1.”
On the Calabi Flow (2006, background): “[17] If Ricci curvature is bounded from below and the Kähler potential is uniformly bounded.”
Smooth and singular Kahler-Einstein metrics (2014, methodology): “The functionals E0 and En were introduced by Mabuchi [167] and Bando–Mabuchi [14], while the remaining ones by Chen–Tian [67].”
Lectures on Stability and Constant Scalar Curvature (2008, methodology): “Then X admits a G-invariant Kähler-Einstein metric if and only if there exists constants A > 0, B, so that the following inequality holds for G-invariant ω0-plurisubharmonic potentials φ, Fω0(φ) ≥ A Jω0(φ)− B (7.2) As a consequence, Tian [142] deduced the following theorem, which implies in particu”
The Kähler–Ricci flow with positive bisectional curvature (2007, methodology): “We remark that there are already proofs of Theorems 1 and 3 which first show the existence of a K¨ahler-Einstein metric and then apply the results o f [CT], [P2].”
Space of Kähler metrics III – On the lower bound of the Calabi energy and geodesic distance (2006, unknown): “[11] Let Ric(ωφ) ≥ −C1 then there is a uniform constant C such that :”
On stability and the convergence of the Kähler-Ricci flow (2006, methodology): “This inequality was instrumental in [9], where it was established under the stronger assumption of positive biholomorphic sectional curvature, using the Moser-Trudinger inequality and Liouville functionals.”
Ricci flow on locally homogeneous closed 4-manifolds (2005, background): “Note that Kähler Ricci flow with positive holomorphic bisectional curvature on CP 2 has been studied by Chen and Tian ([CT]); they prove that the (volume-normalized) Kähler Ricci flow converges exponentially fast to a Kähler metric of constant holomorphic bisectional curvature.”