Valentino Tosatti
We study the Kahler-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 Kahler-Einstein metric.
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Extremal Kähler--Ricci solitons on Fano manifolds are Kähler--Einstein
本文证明Fano流形上所有极值Kähler--Ricci孤子均为Kähler--Einstein，完全解决了Calamai和Petrecca提出的问题。
A short proof of the convergence of the Kahler-Ricci flow on Fano manifolds admitting a Kahler-Einstein metric or a Kahler-Ricci soliton is given, using a variety of recent techniques
We prove the existence of Kähler-Ricci solitons on toric Fano orbifolds, hence extend the the theorem of Wang and Zhu [WZ] to the orbifold case.
In this paper, we establish several sufficient and necessary conditions for the convergence of a Kähler-Ricci flow, on a Kähler manifold with positive first Chern class, to a Kähler-Einstein metric (or a shrinking Kähler-Ricci soliton).
In this short note, based on the work of Wang-Zhu, we determine the greatest lower bounds on Ricci curvature for all toric Fano manifolds.
In this article we prove the existence of Kahler-Ricci solitons on smoothable, K-stable Q-Fano varieties. We also investigate the behavior of twisted Kahler-Ricci solitons in the Gromov-Hausdorff topology under this smoothing family.
We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.
In this paper, we give an alternative proof for the convergence of Kähler-Ricci flow on a Fano mnaifold $(M,J)$. This proof differs from that in [TZ3]. Moreover, we generalize the main theorem of [TZ3] to the case that $(M,J)$ may not admit any Kähler-Einstein metrics.
In this short paper, we improve the result of Phong-Song-Sturm on degeneration of Fano Kähler-Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let $\mathcal{KR}(n)$ be the space of Kähler-Ricci solitons on $n$-dimensional Fano manifolds. We show that after passing to a subsequence, any sequence in $\mathcal{KR}(n)$ converge in the Gromov-Hausdorff topology to a Kähler-Ricci soliton on an $n$-dimensional $\mathbb{Q}$-Fano variety with log terminal singularities.
In this paper, we extend Tosatti's method to study the lower boundedness of modified $K$-energy on a Fano manifold and apply this result to study the relative $K$-stability of the deformation space of a Kähler Ricci soliton.
In this paper, we study the long-term behavior of the conical Kähler-Ricci flow on Fano manifold $M$. First, based on our work of locally uniform regularity for the twisted Kähler-Ricci flows, we obtain a long-time solution to the conical Kähler-Ricci flow by limiting a sequence of these twisted flows. Second, we study the uniform Perelman's estimates of the twisted Kähler-Ricci flows. After that, we prove that the conical Kähler-Ricci flow must converge to a conical Kähler-Einstein metric if there exists one.
In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded below, there exists a sequence of weak almost Kähler-Ricci solitons which converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology.
In this note, we provide some general discussion on the Ricci lower bound along Kähler-Ricci flow with singularity over closed manifold.
In this paper, we study the uniformly strong convergence of Kähler-Ricci flow on a Fano manifold with varied initial metrics and smooth deformation complex structures. As an application, we prove the uniqueness of Kähler-Ricci solitons in sense of diffeomorphism orbits. The result generalizes Tian-Zhu's theorem for the uniqueness of of Kähler-Ricci solitons on a compact complex manifold, and it is also a generalization of Chen-Sun's result of for the uniqueness of of Kähler-Einstein metric orbits.
In this paper, we prove that any solution of Kähler-Ricci flow on a Fano compactification $M$ of semisimple complex Lie group, is of type II, if $M$ admits no Kähler-Einstein metrics. As an application, we found two Fano compactifications of $\mathrm{SO}_4(\mathbb{C})$ and one Fano compactification of $\mathrm{Sp}_4(\mathbb{C})$, on which the Kähler-Ricci flow will develop singularities of type II. To the authors' knowledge, these are the first examples of Ricci flow with singularities of type II on Fano manifolds in the literature.
We study the behavior of the Kähler-Ricci flow on compact manifolds developing finite-time singularities, in particular, when the flow contracts exceptional divisors or collapses the Fano fibers of a holomorphic fiber bundle. We present a technique using holomorphic vector fields to prove estimates related to the work of Song-Weinkove and Fu-Zhang.
We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are shrinking Kahler-Ricci solitons. Specifically, the singularity model for these solutions is expected to be the "blowdown soliton" discovered in [FIK03]. Our partial results support the conjecture that the blowdown soliton is stable under Ricci flow, as well as the conjectured stability of the subspace of Kahler metrics under Ricci flow.
Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold
本文研究Fano流形上Kähler-Ricci流的几何量子化及相关熵泛函，证明了其向原始流和熵的收敛性，并提出了反典范极化下最优退化的有限维类似物。
We study the blowup behavior at infinity of the normalized Kahler-Ricci flow on a Fano manifold which does not admit Kahler-Einstein metrics. We prove an estimate for the Kahler potential away from a multiplier ideal subscheme, which implies that the volume forms along the flow converge to zero locally uniformly away from the same set. Similar results are also proved for Aubin's continuity method.
In this paper, we will establish a regularity theory for the Kähler-Ricci flow on Fano $n$-manifolds with Ricci curvature bounded in $L^p$-norm for some $p > n$. Using this regularity theory, we will also solve a long-standing conjecture for dimension 3. As an application, we give a new proof of the Yau-Tian-Donaldson conjecture for Fano 3-manifolds. The results have been announced in \cite{TiZh12b}.
In this paper, we prove the existence of a Kahler Ricci soliton on any smooth Fano horospherical manifold by a study of the Kahler-Ricci flow. Indeed, we prove that the renormalized Kahler Ricci flow converges in the sense of Cheeger Gromov and that this limit is a Kahler-Ricci soliton.
$L^\infty$ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation
本文在 Perelman 估计假设下，利用 Chen-Cheng 辅助方程和 ABP 最大值原理，为 Fano 流形上归一化 Kähler-Ricci 流给出不依赖多势理论的 $L^\infty$ 估计新证明。
We prove that on one Kähler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical Kähler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which generalizes Li-Rubinstein's estimate and derive high order estimates from this estimate.
In this note, a modified Kähler-Ricci flow is introduced and studied. The main point is to show the flexibility of Kähler-Ricci flow and summarize some useful techniques.
In this paper, we extend the method in [TZhu5] to study the energy level $L(\cdot)$ of Perelman's entropy $λ(\cdot)$ for Kähler-Ricci flow on a Fano manifold. Consequently, we first compute the supremum of $λ(\cdot)$ in Kähler class $2πc_1(M)$ under an assumption that the modified Mabuchi's K-energy $μ(\cdot)$ defined in [TZhu2] is bounded from below. Secondly, we give an alternative proof to the main theorem about the convergence of Kähler-Ricci flow in [TZhu3].
We investigate the limiting behavior of the unnormalized Kahler-Ricci flow on a Kahler manifold with a polarized initial Kahler metric. We prove that the Kahler-Ricci flow becomes extinct in finite time if and only if the manifold has positive first Chern class and the initial Kahler class is proportional to the first Chern class of the manifold. This proves a conjecture of Tian for the smooth solutions of the Kahler-Ricci flow.
In this short note we determine the greatest lower bounds on Ricci curvature for all Fano $T$-manifolds of complexity one, generalizing the result of Chi Li. Our method of proof is based on the work of Datar and Székelyhidi, using the description of complexity one special configurations given by Ilten and Süß.
In this short note we announce a regularity theorem for Kähler-Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of Kähler-Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate to the Kähler-Ricci flow under the regularity assumption, which extends previous works on Kähler-Einstein metrics and shrinking Kähler-Ricci solitons (cf. \cite{Ti90}, \cite{DoSu12}, \cite{Ti12}, \cite{PSS12}). The detailed proof will appear in \cite{TiZh13}.
We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.
In this paper, we consider the twisted Kähler-Ricci soliton, and show that the existence of twisted Kähler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical Kähler-Ricci soliton, and obtain some existence results. In particular, under some assumptions on the divisor and $α$-invariant, we get the properness of the modified log K-energy and the existence of conical Kähler-Ricci soliton with suitable cone angle.
A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds
本文为Fano流形上Kähler-Ricci流的Perelman标量曲率和直径估计提供新证明，基于时空函数族的新Harnack估计，并推广至一般有限时间解。
We provide a direct proof of time-slice weak compactness along the Kähler Ricci flow on Fano manifolds.
In this paper we prove that a nonflat Kähler-Ricci soliton of the Ricci flow on a complex two-dimensional Kähler manifold with nonnegative holomorphic bisectional curvature can not be of maximal volume growth.
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.
We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kahler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kahler metrics.
In this paper we show that on a Fano manifold the convergence of the Kähler-Ricci flow to a Kähler-Einstein metric follows from the integrability of the $L^2$ norm of the Ricci potential for positive time.
In this article we study the limiting behavior of the Kähler Ricci flow on complete non-compact Kähler manifolds. We provide sufficient conditions under which a complete non-compact gradient Kähler-Ricci soliton is biholomorphic to $\ce^n$. We also discuss the uniformization conjecture by Yau \cite{Y} for complete non-compact Kähler manifolds with positive holomorphic bisectional curvature.
Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold
研究紧Kähler流形上长时间Kähler-Ricci流，证明沿规范化Kähler-Ricci流的一致μ-熵或一致Sobolev不等式，并推出标量曲率收敛到负Kodaira维数。
In this paper, we prove that on a Fano manifold $M$ which admits a Kähler-Ricci soliton $(\om,X)$, if the initial Kähler metric $\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak Kähler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if $\vphi_0$ is also $K_X$-invariant, then the weak modified Kähler-Ricci flow converges exponentially to a unique Kähler-Ricci soliton nearby. Especially, if the Futaki invariant vanishes, we may delete the $K_X$-invariant assumption. The methods based on the metric geometry of the space of the Kähler metrics are poten…
In this paper we survey the recent developments of the Ricci flows on complete noncompact Kähler manifolds and their applications in geometry.
Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I
本文证明紧致Kähler曲面上有限时间体积坍缩的Kähler–Ricci流满足Type I曲率界，无需对称性或有理类假设，方法结合辛拓扑与孤子分类，并利用Bamler紧致性理论。
We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kähler-Ricci flow on a minimal elliptic Kähler surface converges in the sense of currents to a generalized conical Kähler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor.
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 exponentially. This generalizes a result of Perelman on the convergence of the Kahler-Ricci flow, and it builds on work of Tian-Zhu.
These notes are based on a lecture series given at the Park City Math Institute in the summer of 2013. The notes are intended as a leisurely introduction to the Kähler-Ricci flow on compact Kähler manifolds, aimed at graduate students with some background in differential geometry.
Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize Kronheimer's results on hyperkahler 4-manifolds, by giving an explicit classification of the ALE Ricci-flat Kahler surfaces. We construct ALF Ricci-flat Kahler metrics on the above non-simply connected manifolds. These provide new examples of ALF Ricci-flat Kahler 4-manifolds, with cubic volume growth and cyclic fundamental group at infinity.
We prove a uniform isoperimetric inequality for all time along the twisted Kähler-Ricci flow on Fano manifolds.
We show that every entire self-shrinking solution on $\mathbb{C}^1$ to the Kähler-Ricci flow must be generated from a quadratic potential.
In this short note, we show that given a special Kähler-Einstein degeneration with bounded geometry, for any noncentral fiber, there exists a Kähler-Ricci flow which converges to the Kähler-Einstein metric of the central fiber. As an example, Tian's deformations \cite{Tian97} of the Mukai 3-fold admit Kähler-Ricci flows which converge to Donaldson's Kähler-Einstein metric \cite{Don08} over the Mukai 3-fold.
We study the behaviour of the normalized Kähler-Ricci flow on complete Kähler manifolds of negative holomorphic sectional curvature. We show that the flow exists for all time and converges to a Kähler-Einstein metric of negative scalar curvature, recovering a result of Wu and Yau.
In this paper, we extend the result about the existence of Kähler-Ricci soliton on toric manifold (proved by Wang and Zhy) by proving this existence on horospherical varieties using the continuity method.
In this short note, we use classic computations for Kähler-Ricci flow to achieve scalar curvature bound for minimal manifold of general type.
We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kahler surfaces, projective manifolds of complex dimension three, and for projective manifolds of all dimensions if assuming the abundance conjecture.
We study the non Ricci flat gradient steady Kähler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely $\underline{\lim}_{r\to \infty} r^{-1}\int_{B_r} S=0$, and show that it is a quotient of $Σ\times \mathbb{C}^{n-1-k}\times N^k$, where $Σ$ and $N$ denote the Hamilton's cigar soliton and some compact Kähler Ricci flat manifold respectively. As an application, we prove that any non Ricci flat gradient steady Kähler Ricci soliton with $Ric\geq 0$, together with subquadratic volume growth or $\limsup_{r\to \infty} rS<1$ must have u…
In this paper, we consider Kahler-Ricci flow on n-dimensional Kahler manifold with semi-ample canonical line bundle and 0< m:= Kod(X)<n. Such manifolds admit a Calabi-Yau fibration over its canonical model. We prove that the scalar curvature of the Kahler metric along the normalized Kahler-Ricci flow converge to -m outside the singular set of this fibration.
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 Kähler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reduction theorem for complete ancient solutions of the Kähler-Ricci flow with nonnegative bisectional curvature on noncompact complex manifolds under a finiteness assumption on the Chern number $c^n_1$.
We establish a parabolic version of Tian's $C^{2,α}$-estimate for conical complex Monge-Ampere equations, which includes conical Kähler-Einstein metrics. Our estimate will complete the proof of the existence of unnormalized conical Kähler-Ricci flow in arXiv:1411.7284.
We generalize the maximal time existence of Kähler-Ricci flow in Tian-Zhang and Song-Tian to conical case. Furthermore, if the twisted canonical bundle $K_{M}+(1-β)[D]$ is big or big and nef, we can expect more on the limit behaviors of such conical Kähler-Ricci flow. Moreover, the results still hold for simple normal crossing divisor.
We give a maximum principle proof of interior derivative estimates for the Kähler-Ricci flow, assuming local uniform bounds on the metric.
Let $(M,J_0)$ be a Fano manifold which admits a Kähler-Ricci soliton, we analyze the behavior of the Kähler-Ricci flow near this soliton as we deform the complex structure $J_0$. First, we will establish an inequality of Lojasiewicz's type for Perelman's entropy along the Kähler-Ricci flow. Then we prove the convergence of Kähler-Ricci flow when the complex structure associated to the initial value lies in the kernel $Z$ or negative part of the second variation operator of Perelman's entropy. As applications, we solve the Yau-Tian-Donaldson conjecture for the existence of Kähler-Ricci solitons…
We show that the Kähler-Ricci flow on a manifold with positive first Chern class converges to a Kähler-Einstein metric assuming positive bisectional curvature and certain stability conditions.
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