Remarks on Kähler Ricci Flow

Non-kähler Ricci Flow Singularities That Converge to Kähler–ricci Solitons

We investigate Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities with the property that parabolic rescalings at the singularities converge to singularity models taking the form of shrinking Kähler–Ricci solitons. More specifically, the singularity models for these solutions are given by the “blowdown soliton” discovered in [FIK03]. Our results support th...

Ricci Lower Bound for Kähler–ricci Flow

Kähler-Ricci flow, Kähler-Einstein metric, and K-stability

We prove the existence of Kähler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kähler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kähler-Ricci flow on Fano manifolds. This is in turn based on a general finite dimensional discussion, which is interesting in its own and could potentially apply to other prob...

Kähler-ricci Flow on Stable Fano Manifolds

We study the Kähler-Ricci flow on Fano manifolds. We show that if the curvature is bounded along the flow and if the manifold is K-polystable and asymptotically Chow semistable, then the flow converges exponentially fast to a Kähler-Einstein metric.

On the Convergence of a Modified Kähler-Ricci Flow

We study the convergence of a modified Kähler-Ricci flow defined by Zhou Zhang. We show that the flow converges to a singular metric when the limit class is degenerate. This proves a conjecture of Zhang.