Related papers: Numerical Kaehler-Ricci soliton on the second del …

We numerically calculate Perelman's entropy for a variety of canonical metrics on $\mathbb{CP}^{1}$-bundles over products of Fano K\"ahler-Einstein manifolds. The metrics investigated are Einstein metrics, K\"ahler-Ricci solitons and…

In this expository note, we study the second variation of Perelman's entropy on the space of Kahler metrics at a K\"ahler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelman's entropy is…

In this announcement, we exhibit the second variation of Perelman's $\lambda$ and $\nu$ functionals for the Ricci flow, and investigate the linear stability of examples. We also define the "central density" of a shrinking Ricci soliton and…

The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a Kahler-Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly.…

The third del Pezzo surface admits a unique Kaehler-Einstein metric, which is not known in closed form. The manifold's toric structure reduces the Einstein equation to a single Monge-Ampere equation in two real dimensions. We numerically…

We consider two classes of non-toric log del Pezzo $\mathbb{C}^*$-surfaces: on the one side the 1/3-log canonical ones and on the other side those of Picard number one and Gorenstein index at most 65. In each of the two classes we figure…

On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the K\"ahler-Ricci flow with a certain initial value converges in $C^\infty$-norm exponentially fast to a K\"ahler-Einstein metric. The proof is…

We explain a characterization of Einstein-Fano manifolds in terms of the lower bound of the density of the volume of the K\"ahler-Ricci Flow. This is a direct consequence of Perelman's uniform estimate for the K\"ahler-Ricci Flow and a…

We show that the solution constructed in an earlier work of Y-G. Shi and the authors can be used to obtain sharp gradient estimates for the Kaehler-Ricci flow which achieves equality on a steady soliton. The estimate can be applied to…

In this paper, we prove that K\"ahler-Ricci flow converges to a K\"ahler-Einstein metric (or a K\"ahler-Ricci soliton) in the sense of Cheeger-Gromov as long as an initial K\"ahler metric is very closed to $g_{KE}$ (or $g_{KS}$) if a…

We study a Boltzmann's type entropy functional (which appeared in existing literature) defined on K\"ahler metrics of a fixed K\"ahler class. The critical points of this functional are gradient K\"ahler-Ricci solitons, and the functional…

We introduce a flow of Riemannian metrics and positive volume forms over compact oriented manifolds whose formal limit is a shrinking Ricci soliton. The case of a fixed volume form has been considered in our previous work. We still call…

We consider the Kaehler-Ricci flow on complete finite-volume metrics that live on the complement of a divisor in a compact Kaehler manifold X. Assuming certain spatial asymptotics on the initial metric, we compute the singularity time in…

In this work, we first establish short time existence and Shi's type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of Kaehler-Einstein…

In this paper we study a generalization of the Kahler-Ricci flow, in which the Ricci form is twisted by a closed, non-negative (1,1)-form. We show that when a twisted Kahler-Einstein metric exists, then this twisted flow converges…

Let $(M,J_0)$ be a Fano manifold which admits a K\"ahler-Ricci soliton, we analyze the behavior of the K\"ahler-Ricci flow near this soliton as we deform the complex structure $J_0$. First, we will establish an inequality of Lojasiewicz's…

In this paper, we shall use the K\"ahler geometry formulation to study the global behavior of the Ricci flow on $R^2$. The geometric feature of our Ricci flow is that it has finite width. Our aim is to determine the limiting metric (which…

We present a general numerical method for investigating prescribed Ricci curvature problems on toric K\"ahler manifolds. This method is applied to two generalisations of Einstein metrics, namely Ricci solitons and quasi-Einstein metrics. We…

In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler,…

We consider a normalization of the Ricci flow on a closed Riemannian manifold given by the evolution equation $\partial_{t}g(t)=-2(Ric(g(t))-\frac{1}{2\tau}g(t))$ where $\tau$ is a fixed positive number. Assuming that a solution for this…