Casey Lynn Kelleher, Gang Tian
We introduce a new curvature flow which matches with the Ricci flow on metrics and preserves the almost Hermitian condition. This enables us to use Ricci flow to study almost Hermitian manifolds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this work, we first establish short time existence and Shi's type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of Kaehler-Einstein metric on complete noncompact Hermitian manifolds.
Invariant complex structures on the homogeneous manifold $U(n+1)/U(n)\times U(p+1)/U(p)$ are reseached. The critical point of the functional of the scalar curvature is found.
In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-Kähler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $S^{2n-…
On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (Riemannain real vector bundle) with an arbitrary metric connection over a compact Hermitian manifold, we can derive various vanishing theorems for Hermitian manifolds and complex vector bundles by the second Ricci curvature tensors. We will also introduce a natural geometric flow on Hermitian manifolds by using the second Ricci curvature tensor.
We consider the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci form. This is an evolution equation first studied by M. Gill, and coincides with the Kahler-Ricci flow if the initial metric is Kahler. We find the maximal existence time for the flow in terms of the initial data. We investigate the behavior of the flow on complex surfaces when the initial metric is Gauduchon, on complex manifolds with negative first Chern class, and on some Hopf manifolds. Finally, we discuss a new estimate for the complex Monge-Ampere equation on Hermitian manifolds.
Using a stability criterion due to Kröncke, we show, providing ${n\neq 2k}$, the Kähler--Einstein metric on the Grassmannian $Gr_{k}(\mathbb{C}^{n})$ of complex $k$-planes in an $n$-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kröncke and Knopf--Sesum on the instability of the Fubini--Study metric on $\mathbb{CP}^{n}$ for $n>1$. The key to the proof is using the description of Grassmannians as certain coadjoint orbits of $SU(n)$. We are also able to prove that Kröncke's method will not work on any of the oth…
Hermitian metrics with vanishing second Chern Ricci curvature
研究埃尔米特度量的第二陈型瑞奇曲率的刚性现象，揭示了几类流形上第二陈型瑞奇平坦度量的特征及此类度量不存在性的结果。
In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao and Feldman-Ilmanen-Knopf.
We show that on a smooth Hermitian minimal model of general type the Chern-Ricci flow converges to a closed positive current on M. Moreover, the flow converges smoothly to a Kahler-Einstein metric on compact sets away from the null locus of K_M. This generalizes work of Tsuji and Tian-Zhang to Hermitian manifolds, providing further evidence that the Chern-Ricci flow is a natural generalization of the Kahler-Ricci flow.
This paper is concerned with Chern-Ricci flow evolution of left-invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger-Gromov) sense to a Chern-Ricci soliton. We give some results on the Chern-Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgr…
In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of Chern-Ricci curvature will be preserved if the initial metric has non-positive bisectional curvature. As an application, we show that the canonical line bundle of a compact Hermitian manifold with nonpositive bisectional curvature and quasi-negative Chern-Ricci curvature is ample.
The behavior of the second Ricci flow on complex parallelizable manifolds
本文研究第二Chern-Ricci流在紧致复平行流形上的正则性，并证明Chern平坦度量是动态稳定的。
$t$-Gauduchon Ricci-flat metrics on non-Kähler Calabi-Yau manifolds
该研究在Picard数大于1的有理齐次簇上的紧非Kähler Calabi-Yau主环面丛中，对所有t<1构造了新的t-Gauduchon Ricci平坦度量，并给出了特定Fano三维簇上主T²-丛的多种Ricci平坦及平衡Hermitian度量新实例。
We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature, and complete non-Kähler metric with zero holomorphic sectional curvature and nonvanishing curvature tensor.
Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds. It is proved that the Ricci tensor of such a soliton is the vertical component of both B-metrics multiplied by a constant. It is established that gradient almost Ricci-like solitons have constant soliton coefficients. Explicit examples of Lie groups as manifolds of dimensions 3 and 5 equipped with the structures studied are provided.
Kählerness of compact Hermitian surfaces under semi-definite Strominger-Bismut-Ricci curvatures
通过Strominger-Bismut联络的Ricci曲率与挠率的关系以及Chern数恒等式，证明了紧Hermitian曲面在半定Ricci曲率条件下的Kählerness准则。
We study the decomposition of the Riemannian curvature R tensor of an almost quaternion-Hermitian manifold under the action of its structure group Sp(n)Sp(1). Using the minimal connection, we show that most components are determined by the intrinsic torsion ξand its covariant derivative \widetilde\nablaξand determine relations between the decompositions of ξ\otimesξ, \widetilde\nablaξand R. We pay particular attention to the behaviour of the Ricci curvature and the q-Ricci curvature.
This paper studies gradient almost Ricci-harmonic soliton with respect to a fixed metric. We rely on analytic techniques to estabilish some basic elliptic and integral equations for the structure of almost Ricci-harmonic soliton which generalizes that of Ricci-hamonic solitons on one hand and that of almost Ricci soliton on the other hand. The consequences of these formulas in relation to classification and rigidity results for gradient almost Ricci-harmonic solitons are also discussed.
On Ricci forms of canonical metrics over noncompact complex manifolds
研究非紧复流形上里奇曲率相关的几何问题，包括负里奇曲率下紧致凯勒流形上凯勒-爱因斯坦度量的存在性，以及完备赫尔米特流形上给定里奇曲率的规范赫尔米特度量的存在性，提出构造阿费-爱因斯坦度量的方法。
Back to almost Ricci solitons
本文利用几何动力学与几何分析方法研究完备的几乎 Ricci 孤立子，在同类中刻画了 Einstein 流形，并借助 Ricci 张量的正交展开研究紧致情形以证实该概念。
We introduce the concept {\it $h$-almost Ricci soliton} which extends naturally the {\it almost Ricci soliton} by Pigola-Rigoli-Rimoldi-Setti and show that a compact nontrivial $h$-almost Ricci soliton of dimension no less than three with $h$ having defined signal and constant scalar curvature is isometric to a standard sphere with the potential function well determined. We also consider the {\it $h$-Ricci soliton} which is a particular case of the $h$-almost Ricci soliton and a generalization of the {\it Ricci soliton} and give characterizations for a special class of gradient $h$-Ricci solit…
Hermitian, Ricci-flat toric metrics on non-compact surfaces à la Biquard-Gauduchon
本文反转 Biquard-Gauduchon 的统一构造，证明在矩多面体温和假设下，环面除子余集上所有共形 Kähler、Ricci 平坦的环面度量均属于已知族且均为 ALF。
Manifolds with non-positive second Chern-Ricci curvature
本文研究具有非正第二陈-里奇曲率的紧致埃尔米特曲面，建立了判断其凯勒性或射影性的新准则。
We construct an example of Ricci-flat almost-Kähler non-Kähler structure in four dimensions.
In this article we study an almost $f$-cosymplectic manifold admitting a Ricci soliton. We first prove that there do not exist Ricci solitons on an almost cosymplectic $(κ,μ)$-manifold. Further, we consider an almost $f$-cosymplectic manifold admitting a Ricci soliton whose potential vector field is the Reeb vector field and show that a three dimesional almost $f$-cosymplectic is a cosymplectic manifold. Finally we classify a three dimensional $η$-Einstein almost $f$-cosymplectic manifold admitting a Ricci soliton.
The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton then it is isometric to a unit sphere S2n+1. Next, we extend this result on a compact k-almost Ricci soliton when the flow vector field X is contact. Finally, we study some special types of k-almost Ricci soliton where the potential vector field X is point wise collinear with the Reeb vector field ξ of the contact metric structure.
In this paper we show that all conformal metrics to a pseudo-euclidean space invariant under the translation group, and all the conformal metrics product manifold also invariant by translation where F m it is Ricci flat semi-Riemannian manifold, are gradient Ricci almost soliton. We also proved that all conformal metrics to a euclidean space and invariant by rotation are gradient Ricci almost soliton.
The aim of this paper is to determine left-invariant strictly almost Kähler structures on 4-dimensional Lie groups $(g, J, Ω)$ such that the Ricci tensor is $J$-invariant.
We discuss a technique to construct Ricci-flat hermitian metrics on complements of (some) anticanonical divisors of almost homogeneous manifolds and discuss when this metric is complete and Kähler. This construction has a strong interplay with invariance groups of the same dimension as the manifold acting with an open orbit. Lie groups of this type we call divisorial. As an application we can describe compact manifolds admitting a divisorially invariant Kähler metric on an open subset. Finally, we see a connection between the reducibility of the anticanonical divisor and the non-triviality of …
In this short note it is established that there does not exist Ricci soliton with the Reeb potential vector field in an almost Kenmotsu manifold (briefly, $\mathcal{AKM}$).
Some results on Ricci-Bourguignon solitons and almost solitons
本文研究Ricci-Bourguignon流孤子，引入几乎孤子概念，推导积分公式，并证明具有常数量曲率或共形向量场的紧致梯度几乎孤子等距于欧几里得球面。
H-almost Ricci-Yamabe solitons in paracontact geometry
本文在仿切触几何框架下分类了 h-几乎 Ricci-Yamabe 孤立子，系统刻画了 para-Kenmotsu、para-Sasakian 和 para-cosymplectic 流形上的相关结构，并给出两个实例加以说明。
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 paper, we give a complete classification of $κ$-solutions of Kähaler-Ricci flow on compact complex manifolds. Namely, they must be quotients of products of irreducible compact Hermitian symmetric manifolds.
Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line bundle for compact Kähler manifolds with nonpositive $k$-Ricci curvature, we consider a natural notion of {\em almost nonpositive $k$-Ricci curvature}, which is weaker than the existence of a Kähler metric with nonpositive $k$-Ricci curvature. When $k=1$, this is just the {\em almost nonpositive holomorphic sectional curvature} introduced by Zhang. We firstly give a lower bound for the existence time of the twisted Kähler-Ricci flow when there exists a Kähler metric with $k$-Ricci curvature bounded from above by a pos…
It is shown that a locally homogeneous proper Ricci almost soliton is either of constant sectional curvature or a product $\mathbb{R}\times N(c)$, where $N(c)$ is a space of constant curvature.
Almost splitting and quantitative stratification for super Ricci flow
本文研究了 Muller 量非负的超 Ricci 流的几乎刚性，得出了几乎分裂与定量分层定理，并发现关于标量曲率积分量的几乎常数性新结果。
We will show the Cheeger-Colding segment inequality for manifolds with integral Ricci curvature bound. By using this segment inequality, the almost rigidity structure results for integral Ricci curvature will be derived by a similar method as in \cite{CC1}. And the sharp Hölder continuity result of \cite{CoN} holds in the limit space of manifolds with integral Ricci curvature bound.
A nongradient Ricci almost soliton
本研究给出扭曲积中非梯度型Ricci几乎孤立子的构造方法与显式实例，并证明高斯孤立子的刚性结果，为微分几何中Ricci孤立子相关研究提供新案例与结论。
We derive several vanishing theorems for genera under almost nonnegative Ricci curvature and infinite fundamental group, which includes Todd genus, $\widehat{A}$-genus, elliptic genera and Witten genus. A vanishing theorem of Euler characteristic number for almost nonnegatively curved Alexandrov spaces is also proved.
Almost Gradient Ricci Solitons on Static Spacetime
本文研究静态时空上几乎梯度里奇孤子的几何性质，确定其存在条件，证明在真空静态时空下退化为稳态孤子，并探讨膨胀孤子满足薛定谔方程、能量条件及底流形为爱因斯坦流形等结果。
Kählerity of complete almost-Kähler gradient shrinking Ricci solitons
本文证明任意偶维下完备（紧致或非紧致）殆凯勒梯度收缩Ricci孤立子均为凯勒的，并结合凯勒-Ricci收缩曲面分类得到实四维该类孤立子的完整分类。
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.
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 results can be viewed as generalization of an existence result by Giesen and Topping for surfaces of hyperbolic type of Ricci flow to higher dimensions in certain sense. On the other hand, we also discuss the long time behaviour of the solution and obtain some sufficient conditions for the existence of Kähler-Einstein metric on complete noncompact Hermitian manifo…
Almost Ricci solitons on Finsler spaces
本文在Finsler测度空间上引入并研究（梯度）几乎Ricci孤立子，给出其等价刻画，并分类Randers情形，构造了Finsler几何中首批非平凡的梯度Ricci孤立子例子。
In this paper the notion of Ricci $ρ$-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3- dimensional almost Kenmotsu Einstein manifold $M$ be a $ρ$-soliton, then $M$ is a Kenmotsu manifold of constant sectional curvature $-1$ and the $ρ$-soliton is expanding, with $λ=2$.
Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kaehler-Weyl structure; this structure is locally conformally Kaehler if and only if the alternating Ricci tensor vanishes. The alternating Ricci tensor takes values in a certain representation space. In this paper, we show that any algebraic possibility in this representation space can in fact be geometrically realized by a left-invariant Kaehler-Weyl structure on a 4-dimensional Lie group in either the Hermitian or the para-Hermitian setting.
On 4-dimensional Ricci-flat ALE manifolds
本文建立了Hermitian非Kähler ALE引力瞬子与带有一个orbifold点的Bach-flat Kähler orbifold之间的一一对应，并证明除反向定向的Eguchi-Hanson度量外，不存在结构群含于SU(2)的此类瞬子。
In this paper, we consider $*$-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if the metric of a Kenmotsu manifold $M$ is a $*$-Ricci soliton, then soliton constant $λ$ is zero. For 3-dimensional case, if $M$ admits a $*$-Ricci soliton, then we show that $M$ is of constant sectional curvature -1. Next, we show that if $M$ admits a $*$-Ricci soliton whose potential vector field is collinear with the characteristic vector field $ξ$, then $M$ is Einstein and soliton vector field is equal to $ξ$. Finally, we prove that if $g$ is a gradient almost $*$-Ricci soliton, the…
Holonomy and the Ricci curvature of complex Hermitian manifolds
本文研究复埃尔米特流形的限制和乐群表示的几何效应，给出了埃尔米特流形为凯勒流形的条件、广义Calabi-Yau的判据，以及紧凯勒流形为射影流形的相关结论。
Ricci almost solitons: complete examples
本文通过扭曲积构造了多种Ricci几乎溶iton和Ricci-Bourguignon溶iton的新例子，并用初等函数明确描述了它们的结构。
The aim of this paper is to classify bi-Hermitian compact surfaces $(M,g)$ whose Ricci tensor $ρ$ satisfies the relation $\nabla_Xρ(X,X) =\frac13Xτg(X,X)$.
In this paper we characterize the Einstein metrics in such broader classes of metrics as almost $η$-Ricci solitons and $η$-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an $η$-Ricci soliton is Einstein metric if either it is $η$-Einstein or the potential vector field $V$ is an infinitesimal contact transformation or $V$ is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost $η$-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then …
The covariant derivative of the Kähler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can be realized geometrically.
The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a $(k,μ)'$-almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$. For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a $(k,μ)$-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property $Q \cdot P = 0$ is locally isometric to the hyperbolic space…
This manuscript examines almost Kenmotsu manifolds (briefly, AKMs) endowed with the almost Ricci-Yamabe solitons (ARYSs) and gradient ARYSs. The condition for an AKM with ARYS to be $η$-Einstein is established. We also show that an ARYS on Kenmotsu manifold becomes Ricci-Yamabe soliton under certain restrictions. In this series, it is proven that a $(2n+1)$-dimensional $(κ, μ)'$-AKM equipped with a gradient ARYS is either locally isometric to $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$ or the Reeb vector field and the soliton vector field are codirectional. The properties of $3$-dimensional non-K…
We introduce transverse Chern-Ricci flow for transversely Hermitian foliations, which is analogous to the Chern-Ricci flow. We show that when $\mathcal{F}$ is homologically orientable and the basic first Bott-Chern class is zero, starting at any transversely Hermitian metric the flow exists for all time and as $t\rightarrow \infty$ converges smoothly to a transversely Hermitian metric $ω_\infty$ with the transverse Chern-Ricci form $ρ^T(ω_\infty)=0$. We also determine the maximal existence time of the flow in the general case. These are foliated version of results of Gill and Tosatti-Weinkove,…
Integral formulas in $h$-Almost Ricci-Bourguignon solitons
本文研究紧致梯度 h-几乎 Ricci-Bourguignon 孤立子的积分公式，推广了 Ricci 几乎孤立子的已有结果，并证明在势向量场共形或标量曲率常数条件下，维数不小于 3 的紧致非平凡孤立子等距于欧氏球面。
Let M be a smooth compact manifold without boundary. We consider two smooth Sub-Semi-Riemannian metrics on M. Under suitable conditions, we show that they are almost conformally isometric in an Lp sense. Assume also that M carries a Riemannian metric with parallel Ricci curvature. Then an equation of Ricci type, is in some sense solvable, without assuming any closeness near a special metric.
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.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。