Related papers: The K\"ahler-Ricci flow and the $\bar\partial$ ope…

In this work, we obtain some existence results of Chern-Ricci Flows and the corresponding Potential Flows on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as $t\rightarrow 0$. These…

We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the…

The J-flow is a parabolic flow on Kahler manifolds. It was defined by Donaldson in the setting of moment maps and by Chen as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. It is shown here that under…

We consider the K\"ahler-Ricci flow $(X, \omega(t))_{t \in [0,T)}$ on a compact manifold where the time of singularity, $T$, is finite. We assume the existence of a holomorphic map from the K\"ahler manifold $X$ to some analytic variety $Y$…

We consider the K\"ahler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the K\"ahler-Ricci flow on total…

For a Fano manifold, We consider the geometric quantization of the K\"ahler-Ricci flow and the associated entropy functional. Convergence to the original flow and entropy is established. It is also possible to formulate the…

S. K. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence the normalized Donaldson-Futaki invariants. We answer the question for the Ricci curvature formalism, in place of the scalar curvature. The…

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…

In this paper, we give some convergence results of Lagrangian mean curvature flow under some stability conditions in a general K\"ahler-Einstein manifold. In particular, we prove that the flow will converge if the initial data is some small…

We consider the formation of singularities along the Calabi flow with the assumption of the uniform Sobolev constant. In particular, on K\"ahler surface we show that any "maximal bubble" has to be a scalar flat ALE K\"ahler metric. In some…

We show that on Kahler manifolds M with c_1(M)=0 the Calabi flow converges to a constant scalar curvature metric if the initial Calabi energy is sufficiently small. We prove a similar result on manifolds with c_1(M)<0 if the Kahler class is…

We develop interconnections between the complex normalizing flow for data drawn from Borel probability measures on the twofold realification of the complex manifold and a nonlinear flow nearly K\"ahler-Ricci. The complex normalizing flow…

In this paper, we study the uniformly strong convergence of K\"ahler-Ricci flow on a Fano manifold with varied initial metrics and smooth deformation complex structures. As an application, we prove the uniqueness of K\"ahler-Ricci solitons…

We give a criterion under which a solution g(t) of the Kahler-Ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. As t tends to the singular time T from each direction, we prove…

For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.

B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$, which are nonnegative in a suitable sense, to every $Ad_{SO(n,\C)}$ invariant subset $S \subset {\bf so}(n,\C)$. For curvature…

We study stability of non-compact gradient Kaehler-Ricci flow solitons with positive holomorphic bisectional curvature. Our main result is that any compactly supported perturbation and appropriately decaying perturbations of the Kaehler…

In an earlier work joint with X. X. Chen and G. Tian, we introduced the weak K\"ahler-Ricci flow for various geometric motivations. In the current work, we take further consideration on setting up the weak flow. Namely, the initial class is…

We prove a compactness theorem for K\"ahler metrics with various bounds on Ricci curvature and the $\mathcal I$ functional. We explore applications of our result to the continuity method and the Calabi flow.

In this paper, we prove that any solution of K\"ahler-Ricci flow on a Fano compactification $M$ of semisimple complex Lie group, is of type II, if $M$ admits no K\"ahler-Einstein metrics. As an application, we found two Fano…