Feng Wang, Xiaohua Zhu
In this paper, we prove the Hamilton-Tian conjecture for Kähler-Ricci flow based on a recent work of Liu-Székelyhidi on Tian's partical $C^0$-estimate for poralized Kähler metrics with Ricci bounded below. The Yau-Tian-Donaldson conjecture for the existence of Kähler-Einstein metrics on Fano manifolds will be also discussed.
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On the Hamilton-Tian Conjecture in a compact transverse Fano Sasakian $5$-manifold
本文确认了在紧致横贯Fano准正则萨克森5流形中Sasaki-里奇流的Hamilton-天猜想，并推导了横贯Fano准正则萨克森5流形上Sasaki-里奇溶洞的紧致性定理，进一步通过第二萨克森结构定理确认了紧致横贯Fano萨克森5流形中的Hamilton-天猜想，并展示了梯度Sasaki-里奇溶洞轨道度量在紧致萨克森5流形上为Sasaki-爱因斯坦的条件。
Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we prove the Hamilton-Tian conjecture and the partial-$C^0$-conjecture of Tian.
Relative uniform Yau--Tian--Donaldson correspondence for projective bundles over a curve
本文研究了曲面投影包的相对均匀 Yau-Tian-Donaldson 对应关系，通过构造兼容测试配置，证明了相对均匀稳定性与极值度量存在性的等价性。
In this short Note, we establish that the constant $C_1$ in Lemma $0.4$ of the correction (Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture, arXiv/math/DG:1512.00699 (2015)) by John Morgan and Gang Tian to their Clay Institute Monograph (Ricci Flow and the Poincare Conjecture, vol. 3, Clay Mathematics Monograph, AMS and Clay Institute, (2007)) can be taken to be zero. This implies that the counter-example which we provided recently in ArXiv/math/DG:1422760 still stands after this correction. Let us observe that the unjustified division by $k$ in the original monograph-the o…
Arcs, stability of pairs and the Mabuchi functional
本文在K-稳定性理论中引入弧的概念，刻画了Mabuchi泛函的强制性，证明Tian猜想的一个版本，并给出Yau-Tian-Donaldson猜想的新证明及自同构群有限性的推广。
In this paper we prove a new matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in \cite{Ch1}. This new inequality is shown to be connected with Perelman's entropy formula through a family of differential equalities. In the rest of the paper, We show several applications of this new estimate and its linear version proved earlier in \cite{CN}. These include a sharp heat kernel comparison theorem, generalizing the earlier result of Li and Tian, a manifold version of Stoll's theorem on the …
We formulate a notion of K-stability for Kähler manifolds, and prove one direction of the Yau-Tian-Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. coercive) implies K-semistability (resp. uniformly K-stable). In particular this shows that the existence of a constant scalar curvature Kähler metric implies K-semistability, and K-stability if one assumes the automorphism group is discrete. We also show how Stoppa's argument holds in the Kähler case, giving a simpler proof of this K-stability statement.
Laplace comparison on Kähler Ricci flow and convergence
该研究证明Fano流形上Kähler Ricci流的一致积分Laplace比较结果，结合相关理论与已有成果，给出Hamilton-Tian猜想的直接独立证明，并梳理相关文献。
It is conjectured that the existence of constant scalar curvature Kähler metrics will be equivalent to K-stability, or K-polystability depending on terminology (Yau-Tian-Donaldson conjecture). There is another GIT stability condition, called the asymptotic Chow polystability. This condition implies the existence of balanced metrics for polarized manifolds $(M, L^k)$ for all large $k$. It is expected that the balanced metrics converge to a constant scalar curvature metric as $k$ tends to infinity under further suitable stability conditions. In this survey article I will report on recent results…
Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of an expression in the curvature and its first two covariant derivatives, of the bundle of 2-forms Whitney sum 1-forms is nonnegative. The idea is to consider the 2-form which minimizes the associated quadratic form to obtain a symmetric 2-tensor. A long but straightforward computation implies this 2-tensor is a subsolution to heat-type equation. A standard a…
We prove that if an ALE Ricci-flat manifold $(M,g)$ is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to $g$. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable.
A relative Yau-Tian-Donaldson conjecture and stability thresholds
本文在光滑 Fano 簇的广义 b-除子上定义 $\tildeδ$，建立带给定奇异性 KE 度量与一致 $\mathbf{D}$-对数 K/Ding 稳定性的等价关系，并将 δ-赋值准则推广至 $\mathbf{D}$-对数情形。
The existence of the Kähler-Ricci soliton degeneration
本研究证明了所有对数法诺对的Hamilton-Tian猜想代数版本，证实其存在典范两步退化至约化一致Ding稳定三元组，且基域为复数域时该三元组有Kähler-Ricci孤立子。
In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension $\geq 4$. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away fro…
The existence of Kähler-Einstein metrics on a compact Kähler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has become one of the most powerful tools in geometric analysis. We study the Kähler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical is a generalized Kähler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metri…
Weighted cscK metric (II): the continuity method
本文研究了带权常标量曲率Kähler度量的存在性，并通过带权Mabuchi函数的 coercivity 证明了其存在性，同时探讨了其在Kähler和toric几何中的应用。
In this paper we prove a compactness result for Ricci flows with bounded scalar curvature and entropy. It states that given any sequence of such Ricci flows, we can pass to a subsequence that converges to a metric space which is smooth away from a set of codimension $\geq 4$. The result has two main consequences: First, it implies that singularities in Ricci flows with bounded scalar curvature have codimension $\geq 4$ and, second, it establishes a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. In the course of the proof, we will also establish the fol…
Ding stability and Kähler-Einstein metrics on manifolds with big anticanonical class
该研究针对具有大丰沛反典范类的射影流形提出一致Ding稳定性概念，通过建立奇异度量的Deligne泛函理论与Ding泛函斜率公式，证明Kähler-Einstein度量存在性蕴含该稳定性，推广了Fano情形的相关结果并探讨其与模空间的关联。
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