Olivier Biquard, Thibaut Delcroix
We obtain Ricci flat Kähler metrics on complex symmetric spaces of rank two by using an explicit asymptotic model whose geometry at infinity is interpreted in the wonderful compactification of the symmetric space. We recover the metrics of Biquard-Gauduchon in the Hermitian case and obtain in addition several new metrics.
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Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A crucial step in the proof is to show a vanishing theorem of Dolbeault-Morse-Novikov cohomology.
On the Ricci curvature of Kahler-Ricci Flow
本文研究Kähler-Ricci流中Kodaira维数为一的流形的Ricci曲率收敛性，揭示其在局部拓扑下的收敛行为。
We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log terminal singularities.
We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.
By establishing two general quadratic inequalities, we obtain some inequalities related to Ricci curvatures for Lagrangian submanifolds of K$\ddot{\mathrm{a}}$hler QCH-manifolds, which generalize some results for Lagrangian submanifolds of complex space forms.
Non-ancient solution of the Ricci flow
对任意具有非负有界全纯双截曲率的完备非紧$\ddot{a}$流形，本文给出了里奇流非古解的充要条件.
We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact K{ä}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential K{ä}hler-Ricci flows on compact K{ä}hler manifolds of general type as well as on stable varieties.
We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact type.
One-sided type-D Ricci-flat multi-centre metrics
本文基于LeBrun–Tod ansatz的Tod ansatz简化形式构造单侧D型里奇平坦多中心度量，系统研究其结构并结合多孤子解方案得出相关孤子数结论，推测结论对一般中心数成立。
We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K $\in$ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2$π$ and its dimension is at most equal to N. This gives a new wide class of geometric examples of metric measure spaces satisfying the RCD(K, N) curvature-dimension condition, including for instance spherical suspensions, orbifolds, K{ä}hler-Einstein manifolds with a divisor, Einstein manifolds with conical singularities along a curv…
We investigate the existence of a holomorphic and isometric immersion in the complex projective space for the complete Ricci-flat Kaehler metrics constructed by M. B. Stenzel on the cotangent bundle of a compact, rank one, globally symmetric space.
On asymptotics of complete Ricci-flat Kähler metrics on open manifolds
Tian和Yau构造了一个完备的Ricci平坦Kähler度量，它是关于一个充分光滑的反正则因子的补的。我们探讨这个度量对边界因子的行为，并证明了一个缓慢的衰减率的差异，一个适当的explicitly给定的参考度量。
We show that the singularities of the twisted Kähler--Einstein metric arising as the long-time solution of the Kähler--Ricci flow or in the collapsed limit of Ricci-flat Kähler metrics is intimately related to the holomorphic sectional curvature of the reference conical geometry. This provides an alternative proof of the second-order estimate obtained by Gross--Tosatti--Zhang with explicit constants appearing in the divisorial pole.
We show that the Ricci flat Calabi's metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [arXiv:1705.03908v2 [math.DG]] by proving that any multiple of the Eguchi-Hanson metric on the blow-up of C^2 at the origin is not projectively induced.
Chern-Ricci flow and t-Gauduchon Ricci-flat condition
本文研究Chern-Ricci流下的t-Gauduchon Ricci平坦条件，提供了不保持该条件的例子、非平凡新解及几何流的Gromov-Hausdorff极限行为，明确描述了特定主丛上pluriclosed流的极限。
Ricci-flat metrics on Calabi-Yau manifolds
研究 Calabi-Yau 流形上 Ricci-平坦 Kahler 度量的性质及其退化问题，并总结最新研究成果。
Let $X$ be a Kähler manifold which is fibered over a complex manifold $Y$ such that every fiber is a Calabi-Yau manifold. Let $ω$ be a fixed Kähler form on $X$. By Yau's theorem, there exists a unique Ricci-flat Kähler form $ρ\vert_{X_y}$ for each fiber, which is cohomologous to $ω\vert_{X_y}$. This family of Ricci-flat Kähler forms $ρ\vert_{X_y}$ induces a smooth $(1,1)$-form $ρ$ on $X$ with a normalization condition. In this paper, we prove that the direct image of $ρ^{n+1}$ is positive on the base $Y$. We also discuss several byproducts, among them the local triviality of families of Calabi…
Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{ä}hler-Einstein equation $e^{-Φ} = \det D^2 Φ$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 Φ$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2 Φ$ for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case. The general case remains open.
On pseudo-Hermitian Einstein spaces
我们在这里描述和构造的伪厄米结构$θ$没有扭转（即与横向对称）的Webster-里奇曲率张量是一个常数倍的外微分$dθ$。我们称这些结构为伪厄米爱因斯坦，我们的结果表明，它们都可以从局部K{ä}hler-Einstein度量导出。此外，我们还讨论了伪厄米爱因斯坦结构的相应的费曼度规。这些费曼度规永远不是爱因斯坦，但它们总是局部共形的爱因斯坦。
Ricci Flow on 3-dimensional Lie groups and 4-dimensional Ricci-flat manifolds
研究了三维李群上的里奇流与四维里奇平坦流形的关系。特别地，我们构造了关于三维李群的Ricci平坦上齐性1度量。
We give an overview of some recent interactions between the geometry of K3 surfaces and their Ricci-flat Kahler metrics and the dynamical study of K3 automorphisms with positive entropy.
Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.
A four dimensional example of Ricci-flat metric admitting almost-Kähler non-Káhler structure
我们构造了一个四维Ricci平坦几乎Kähler非Kähler结构的例子。
We construct examples of Bach-flat gradient Ricci solitons which are neither half conformally flat nor conformally Einstein.
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.
We study the stability and instability of ALE Ricci-flat metrics around which a Lojasiewicz inequality is satisfied for a variation of Perelman's $λ$-functional adapted to the ALE situation and denoted $λ_{\operatorname{ALE}}$. This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Lojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5.
Dynamical stability and instability of Ricci-flat metrics
本文改进了Sesum和Haslhofer在里奇流下证明的Ricci-平坦度量的动力学稳定性和不稳定性的结果，去掉了可积性假设.
Weighted Sobolev inequalities and Ricci flat manifolds
证明了具有非负里奇曲率且满足逆倍体积条件的流形上的一个加权Sobolev不等式和一个Hardy不等式.它使我们能够得到刚性的结果，Ricci平坦流形，推广早期的工作Bando，Kasue和Nakajima。
Conformally flat pseudo-Riemannian manifolds with generalized Ricci recurrent, $(GR)_n$ structure are completely classified in this short report. A conformally flat generalized Ricci recurrent pseudo-Riemannian manifold is shown to be either a de Sitter space or an anti-de Sitter space. In particular, a conformally flat generalized Ricci recurrent spacetime must be either a de Sitter spacetime or an anti-de Sitter spacetime.
We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat homogeneous Ricci solitons.
Existence of complete Kahler Ricci-flat metrics on crepant resolutions
本文得到了完备Ricci平坦Kahler度量关于Calabi-Yau簇奇点的可解的存在性结果。此外，对于某些渐近平坦的Calabi-Yau簇，我们证明了解析流形上的Ricci平坦度量与初始簇具有相同的渐近行为。
We give an alternative proof of a result of Cantat and Dupont, showing that any automorphism of a K3 surface with measure of maximal entropy in the Lebesgue class must be a Kummer example. Our method exploits the existence of Ricci-flat metrics on K3s and also covers the non-projective case.
In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Kahler Calabi-Yau surfaces or Hopf surfaces.
On Locally Conformally Flat Gradient Shrinking Ricci Solitons
本文首先利用Ricci孤子的一个积分恒等式证明了闭的局部共形平坦梯度Ricci孤子具有常截面曲率。在里奇曲率有界和黎曼曲率张量至多指数增长性的条件下，我们将这个积分恒等式推广到完全非紧梯度收缩Ricci孤子.作为这个身份的结果，我们分类完全的局部共形平坦梯度收缩Ricci孤子与里奇曲率有界从下面。
The stability inequality for Ricci-flat cones
本文深入研究了Ricci-平坦锥的稳定性不等式。也许最重要的是，我们证明了CP ^2上的Ricci-平坦锥是稳定的，表明第一个稳定的非平坦Ricci-平坦锥出现在最小的可能维度上。另一方面，我们证明了4-流形上的Ricci平坦锥的许多其他例子是不稳定的，并且爱因斯坦流形的积和h^（1，1）>1的卡勒-爱因斯坦流形上的Ricci平坦锥在维度小于10时是不稳定的。作为独立有趣的结果，我们的计算表明Page度规和Chen-LeBrun-Weber度规是不稳定的Ricci收缩。作为最后的奖励，我们给出了大量的动机，并部分证实了Tom Ilmanen的一个猜想，该猜想涉及到了锥型初始数据下里奇流的非唯一性、正质量定理和泛函。
Stability of piecewise flat Ricci flow
研究分段平坦里奇流的稳定性，通过线性分析和数值模拟发现并修正数值不稳定性，并用于证明其收敛性。
We give topological conditions to ensure that a noncollapsed almost Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is that the manifold is spin and has a nonzero A-hat genus. Another condition is that the fundamental group is infinite or, more generally, of sufficiently large cardinality.
Some Ricci Flat (pseudo-) Riemannian Geometries
定义了一类与平面三维几何中的极小曲面保形相关的二维曲面。利用此类曲面的度量，我们给出了$2 N$维Ricci平坦（伪）Riemannian几何的度量的构造。
ALE Ricci-flat Kahler metrics and deformations of quotient surface singularities
设N_0 = C^2/H是一个孤立的商奇点，H在U（2）的有限子群中.我们证明了对于N_0的任何Q-Gorenstein光滑，在任何Kahler类中，附近的纤维都允许ALE Ricci平坦Kahler度量.此外，我们推广了Kronheimer关于超Kahler四维流形的结果，给出了ALE Ricci平坦Kahler曲面的一个明确分类。我们在上述非单连通流形上构造了ALF Ricci平坦Kahler度量。这些结果提供了新的ALF Ricci平坦Kahler 4-流形的例子，在无穷远处具有立方体积增长和循环基本群。
We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such that the $a_{n+1}$ coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conje…
A Family of Steady Ricci Solitons and Ricci-flat Metrics
我们产生了新的非Kähler完全稳定梯度Ricci孤子，其渐近性结合了Bryant孤子和Hamilton雪茄。我们还得到了一族具有渐近局部锥渐近性的完备Ricci平坦度量。最后，我们得到了四元数射影空间上距离球面丛分别为扭量和${\rm Sp}(1)$丛的向量丛上的完全稳定孤子结构的数值证据。
Ricci-flat Kaehler metrics on canonical bundles
证明了实解析Kaehler流形X的典范丛的零截面邻域上存在一个（唯一的）S^1不变Ricci平坦Kaehler度量，推广了X上的度量.
We define a system of ODE that gives Einstein 4-dimensional metrics. We found new Ricci-flat incomplete metric of cohomogeneity 1 in explicit formulas and study its characteristics.
Harmonic forms on ALE Ricci-flat 4-manifolds
本文研究了ALE Ricci-flat 4-manifolds上调和函数和1形式的展开问题。
On the Perelman's reduced entropy and Ricci flat manifolds with maximal volume growth
本文利用Perelman的里奇流的约化体积研究了具有最大体积增长的Ricci平坦流形。证明了若$(M^n,g)$是一个非紧完备Ricci平坦流形，其最大体积增长满足$|Rm|(x)\to 0$为$d(x)=d_g(x,p)\to \infty$，则$M^n$具有二次曲率衰减.最后给出了这一结果的一些应用。
We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.
Bismut Ricci flat manifolds with symmetries
本文构造了具有不变Bismut Ricci平坦联络的紧齐性黎曼流形例子，证明广义Alekseevsky-Kimelfeld定理不成立，并给出5维分类及稳定性讨论。
Locally Conformally Flat Lorentzian Gradient Ricci Solitons
研究表明，当势函数的梯度为非零时，局部共形平坦的洛伦兹梯度Ricci孤子与Robertson-Walker翘曲积局部等距;当势函数的梯度为零时，局部共形平坦的洛伦兹梯度Ricci孤子与平面波局部等距.后一种梯度Ricci孤子必然是稳定的。
Given a complete, Ricci-flat 4-manifold with a Killing field, we give an estimate on the manifold's energy in terms of a certain asymptotic quantity of the Killing field. If the Killing field has no zeros and satisfies a certain asymptotic condition, the manifold is flat.
Kaimakamis and Panagiotidou in \cite{KP} introduced the notion of $^*$-Ricci soliton and studied the real hypersurfaces of a non-flat complex space form admitting a $^*$-Ricci soliton whose potential vector field is the structure vector field. In this article, we consider that a real hypersurface of a non-flat complex space form admits a $^*$-Ricci soliton whose potential vector field belongs to the principal curvature space and the holomorphic distribution.
This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose $^{*}$-Ricci tensor satisfies conditions of parallelism. More precisely, extension of existing results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel $^{*}$-Ricci tensor in case of ambient space being the complex hyperbolic space are provided. Furthermore, new results concerning $ξ$-parallelism of $^{*}$-Ricci tensor of real hypersurfaces in non-flat complex space forms are presented.
On Locally Conformally Flat Gradient Steady Ricci Solitons
本文对n维（n>2）完全非紧局部共形平坦梯度稳定孤子进行了分类。特别地，我们证明了一个完全的非紧非平坦共形平坦梯度稳态Ricci孤子，在标度下，是Bryant孤子。
Bach-flat gradient steady Ricci solitons
本文研究梯度稳态Ricci孤立子的分类问题，证明了高维Bach平坦且Ricci曲率为正的情形及三维Bach张量散度为零的情形均为平坦或与Bryant孤立子等距，改进了已有局部共形平坦情形的分类定理。
Ricci-flat deformations of asymptotically cylindrical Calabi--Yau manifolds
研究了一类产生于拟投射流形上的渐近圆柱Ricci平坦Kähler度量.利用Calabi-Yau几何和分析以及Kodaira-Kuranishi-Spencer理论，并在N.Koiso关于紧流形的结果的基础上，我们证明了在相当一般的假设下，渐近柱面Ricci平坦Kähler度量的任何“小”渐近柱面Ricci平坦变形都是Kähler的，可能是关于扰动的复结构的。我们还发现这些小的变形的模空间的尺寸。在$2n$ -流形上的渐近圆柱Ricci平坦度量类中，完整约化为SU（n）是一个开条件.
Linear and dynamical stability of Ricci flat metrics
我们可以讨论里奇流在里奇平坦度量下的两种稳定性。其中之一是线性稳定性，定义相对于Perelman的功能$\mathcal{F}$。另一个是动力学稳定性，它指的是一个里奇流的收敛开始于任何度量在一个考虑的里奇平坦度量的邻域。我们证明了动力学稳定性蕴含着线性稳定性。我们还表明，线性稳定性与可积性假设意味着动态稳定。作为推论，我们得到了$K3$曲面的稳定性结果，其中一部分已在\cite{dan2002}中完成。
On four-dimensional anti-self-dual gradient Ricci solitons
本文证明了任何四维半共形平坦梯度稳定Ricci孤子必是Bryant孤子或Ricci平坦的。我们还对四维半共形平坦的有界曲率梯度收缩Ricci孤子进行了分类。
We compute the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and give a detailed description of the corresponding radially homogeneous tensor fields in its kernel. For a Ricci-flat conifold $(M,g)$ which may have asymptotically conical as well as conically singular ends, we compute at each end a lower bound for the order with which the metric converges to the tangent cone. As a special subcase of our result, we show that any Ricci-flat ALE manifold $(M^n,g)$ is of order $n$ and thereby close a small gap in a paper by Cheeger and Tian.
We show that the moduli space of Ricci flat metrics of unit volume (including orbifold metrics) on a K3 surface is simply connected and that it has the same rational cohomology as the automorphism group of the K3 lattice $(-E_8)^{\oplus 2}\oplus U^{\oplus 3}$.
On an $n$-dimensional complete manifold $M$, consider an $h$-almost gradient Ricci soliton, which is a generalization of a gradient Ricci soliton. We prove that if the manifold is Bach-flat and $dh/du>0$, then the manifold $M$ is either Einstein or rigid. In particular, such a manifold has harmonic Weyl curvature. Moreover, if the dimension of $M$ is four, the metric $g$ is conformally flat.
We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.
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