Ancient Solutions to Kähler-ricci Flow

On dimension reduction in the Kähler-Ricci flow

We consider dimension reduction for solutions of the Kähler-Ricci flow with nonegative bisectional curvature. When the complex dimension n = 2, we prove an optimal dimension reduction theorem for complete translating KählerRicci solitons with nonnegative bisectional curvature. We also prove a general dimension reduction theorem for complete ancient solutions of the Kähler-Ricci flow with nonneg...

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...

On quasi-Einstein Finsler spaces‎

‎The notion of quasi-Einstein metric in physics is equivalent to the notion of Ricci soliton in Riemannian spaces‎. ‎Quasi-Einstein metrics serve also as solution to the Ricci flow equation‎. ‎Here‎, ‎the Riemannian metric is replaced by a Hessian matrix derived from a Finsler structure and a quasi-Einstein Finsler metric is defined‎. ‎In compact case‎, ‎it is proved that the quasi-Einstein met...

Ricci Lower Bound for Kähler–ricci Flow

Ricci Flow Unstable Cell Centered at an Einstein Metric on the Twistor Space of Positive Quaternion Kähler Manifolds of Dimension

We construct a 2-parameter family FZ of Riemannian metrics on the twistor space Z of a positive quaternion Kähler manifold M satisfying the following properties : (1) the family FZ contains an Einstein metric gZ and its scalings, (2) the family FZ is closed under the operation of making the convex sums, (3) the Ricci map g 7→ Ric(g) defines a dynamical system on the family FZ, (4) the Ricci flo...