Related papers: On the K\"ahler-Ricci flow near a K\"ahler-Einstei…

By means of a Kaluza-Klein type argument we show that the Perelman's F-functional is the Einstein-Hilbert action in a space with extra ``phantom'' dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and…

We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound $R(X)$ to the existence of conical Kahler-Einstein metrics on a Fano manifold $X$. In particular, if $D\in |-K_X|$ is a smooth simple divisor and the…

In this note, we construct families of functionals of the type of $\mathcal{F}$-functional and $\mathcal{W}$-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a…

We provide a direct proof of time-slice weak compactness along the K\"ahler Ricci flow on Fano manifolds.

We establish the existence of K\"ahler-Ricci flow on pseudoconvex domains with general initial metric without curvature bounds. Moreover we prove that this flow is simultaneously complete, and its normalized version converge to the complete…

We consider the K\"ahler-Ricci flow $\frac{\partial}{\partial t}g_{i\bar{j}} = g_{i\bar{j}} - R_{i\bar{j}}$ on a compact K\"ahler manifold $M$ with $c_1(M) > 0$, of complex dimension $k$. We prove the $\epsilon$-regularity lemma for the…

In this paper we study the behavior of the Ricci flow at infinity for the full flag manifold $SU(3)/T$ using techniques of the qualitative theory of differential equations, in special the Poincar\'e Compactification and Lyapunov exponents.…

We first define Pseudo-Calabi flow, as {equation*} {{aligned}{{\partial \varphi}\over {\partial t}}&= -f(\varphi), \triangle_varphi f(\varphi) &= S(\varphi) - \ul S.{aligned}. \end{equation*} Then we prove the well-posedness of this flow…

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 paper, we prove that the $L^4$-norm of Ricci curvature is uniformly bounded along a K\"ahler-Ricci flow on any minimal algebraic manifold. As an application, we show that on any minimal algebraic manifold $M$ of general type and…

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 prove a uniform isoperimetric inequality for all time along the twisted K\"{a}hler-Ricci flow on Fano manifolds.

We show that a K\"ahler-Ricci soliton on a Fano manifold can always be smoothly approximated by a sequence of relative anticanonically balanced metrics, also called quantized K\"ahler-Ricci solitons. The proof uses a semiclassical estimate…

We show that the normalized K\"ahler-Ricci flow on a compact K\"ahler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted K\"ahler-Einstein metric on the canonical…

We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded…

We show that on a Kahler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kahler class. On toric manifolds we give a numerical characterization of when the J-flow converges, verifying a…

We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the…

In this paper, we prove the Hamilton-Tian conjecture for K\"ahler-Ricci flow based on a recent work of Liu-Sz\'ekelyhidi on Tian's partical $C^0$-estimate for poralized K\"ahler metrics with Ricci bounded below. The Yau-Tian-Donaldson…

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 $Y$…

In this note, we show that the solution of K\"ahler-Ricci flow on every Fano threefold from the family No.2.23 in the Mori-Mukai's list develops type II singularity. In fact, we show that no Fano threefold from the family No.2.23 admits…