Volker Branding, Klaus Kroencke
The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also holds for connections with metric torsion in the case of non-positive Euler characteristic. Our main tool is an adapted form of the Ricci flow.
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Torsional Rigidity on Compact Riemannian Manifolds with lower Ricci Curvature Bounds
本文证明了紧致黎曼流形中具有里奇曲率下界的区域上狄利克雷问题的基本特征函数的逆赫尔德不等式.我们还证明了这类域的扭转刚度的一个等周不等式。
We examine the moduli spaces of Type~A connections on oriented and unoriented surfaces both with and without torsion in relation to the signature of the associated symmetric Ricci tensor. If the signature of the symmetric Ricci tensor is (1,1) or (0,2), the moduli spaces are smooth. If the signature is (2,0), there is an orbifold singularity.
Torsion of SU(2)-structures and Ricci curvature in dimension 5
在类比R. Bryant关于G-$_2$ -结构的Ricci张量的方法，我们研究了5维流形上SU-$(2)$ -结构的内禀挠率，导出了Ricci和数量曲率的挠率形式及其导数的显式表达式.作为这个公式的结果，我们证明了$α$ -Einstein条件迫使某些特殊的SU（2）-结构成为Sasaki-Einstein。
Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions
本文证明了可解李代数上代数广义Ricci孤立子的调和性，并用广义nilsoliton的正规扩张刻画扩张孤立子，得到尖锐的秩界。
Combinatorial differential geometry and ideal Bianchi-Ricci identities II - the torsion case
本文是arXiv：0809.1158的继续，讨论一般的，不一定是无挠的联络。它的特点是所有可能的系统的发电机的向量场值的运营商，自然依赖于一组向量场和线性连接，描述了这种运营商的空间的大小，并证明了存在的“理想”的基础上的运营商与给定的首项，满足（广义）Bianchi-Ricci身份没有纠正。
Ruled Ricci surfaces and curves of constant torsion
本文证明三维欧氏空间中赋予 Ricci 度量的非可展直纹曲面可由常挠率曲线及其副法线构造，并据此刻画正螺面为唯一具平面腰线参数化及常平均曲率的此类曲面。
The Riemannian curvature identities on almost Calabi-Yau with torsion 6-manifold and generalized Ricci solitons
该文研究紧致带挠率近复Calabi-Yau 6维流形的曲率性质，证明了Nijenhuis张量的平行性、挠率闭时的广义梯度Ricci孤立子特性，以及挠率联络Ricci平坦和曲率满足黎曼第一Bianchi恒等式的等价条件。
Connections with skew-symmetric Ricci tensor on surfaces
将曲面上具有反对称Ricci张量的无挠联络的一些已知结果推广到有挠联络，并简化了这类联络的Wong正则坐标形式。
The Riemannian curvature identities for the torsion connection on $Spin(7)$-manifold and generalized Ricci solitons
本文研究紧$Spin(7)$流形上挠率联络的曲率性质，推导了曲率满足相关恒等式的等价条件，给出闭挠率的精确判据，并证明具有闭挠率的紧$Spin(7)$流形是广义梯度Ricci孤立子。
On a smooth closed Riemannian manifold, we show short time existence of smooth solutions to the $(α,β)$-Ricci-Yamabe flow, which is a natural generalization of the Ricci flow and the Yamabe flow. We also establish some long time existence theorems for the closed $(α,β)$-Ricci-Yamabe flow by estimating its curvatures.
Ricci flow with metric torsion on surfaces of positive Euler characteristic
本文研究正欧拉示性数曲面上带度量挠率的适配Ricci流，证明球面上无非平凡孤子，给出不收敛例子，并在实射影平面上证明收敛，进而得到球面在反对称条件下的收敛性，并利用Łojasiewicz--Simon不等式证明接近稳定点时的收敛。
The Riemannian Bianchi identities of metric connections with skew torsion and generalized Ricci solitons
本文研究带完全反对称挠率的度量联络的曲率性质，推导了$\nabla$-爱因斯坦流形标量曲率与挠率的关系，给出了紧广义梯度Ricci孤立子为Ricci平坦的充要条件，还得到了这类联络满足两类黎曼比安基恒等式的条件及相关平坦性结论。
A Lower Bound of The First Eigenvalue of a Closed Manifold with Positive Ricci Curvature
给出了具有正Ricci曲率的闭黎曼流形的Laplacian第一非零特征值的下界的一个估计，该估计用内径和里奇曲率的下界表示.
The Ricci tensor of SU(3)-manifolds
我们遵循Bryant的方法，研究了SU（3）-流形的内禀挠率，导出了关于挠率形式的Ricci曲率和数量曲率的一些公式。作为结果，我们证明了在某些特殊情况下，爱因斯坦条件的力量消失的内在扭转。
Non-negative Ricci curvature on closed manifolds under Ricci flow
在这个简短的说明中，我们表明，非负里奇曲率不保存下的闭流形的4维及以上的里奇流，加强了以前的结果，克诺夫在\cite{K}的完全非紧流形的有界曲率。这就把Böhm和Wilking对12维及以上维的结果\cite{BW}降到了4维。此外，这里构造的流形是\Kahler流形，并且涉及到陈秀雄在\cite{XC}，\cite{XCL}中提出的一个问题。
Lower Bounds of the First Closed and Neumann Eigenvalues of Compact Manifolds with Positive Ricci Curvature
利用里奇曲率的直径和下界给出了具有正里奇曲率的紧致流形的第一闭特征值或Neumann特征值的下界的新估计.结果改进了以前的估计。
A Lower Bound of the First Eigenvalue of a Closed Manifold with Negative Lower Bound of the Ricci Curvature
沿着杨猜想的思路，利用内径和里奇曲率的下界，给出了具有负Ricci曲率下界的闭黎曼流形的第一非零特征值的下界的一个新估计.
As is known from studies of gravity models in the Palatini formalism, there exist two inequivalent definitions of the generalized Ricci tensor in terms of the generalized curvature namely, $\widetilde{R}_{μν}=R^ρ_{μρν}$ and $R^μ_ν=g^{αβ}R^μ_{ανβ}$. A deep formal investigation of theories with lagrangians of the form $L=L(\widetilde{R}_{(μν)})$ was initiated in [4]. In that work, the authors leave the connection free, and find out that the torsion only appears as a projective mode. This agrees with the widely employed condition of vanishing torsion in these theories as a simple gauge choice. In…
Singularity Types for Long-Time Chern-Ricci Flow
本文将Kähler-Ricci流中关于长时间解奇点类型独立性的结果推广至Chern-Ricci流，证明了在一致有界挠率和曲率条件下，相同$\partial \bar{\partial}$类的初始度量导出的解同样具有一致有界性。
Analytic torsion, vortices and positive Ricci curvature
本文刻画了Kahler流形X上的充裕线丛L上定义在所有正曲度量空间上的某个非局部泛函的全局最大值。该泛函是由Berglessson引入的Donaldson L-泛函的伴随形式，推广了临界点为正里奇曲率的Kahler-Einstein度量的Ding-Tian泛函.给出了（1）Fano流形上的解析挠率，（2）环面上的Chern-Simons-Higgs涡旋和（3）Kahler几何的应用.特别地，证明了（1）Gillet-Soulé和Fang（关于双球面上Dolbeault Laplacian的正则化行列式）（2）Tarantello和（3）Aubin（关于Moser-Trudinger型不等式）在这三种情形下的证明.对Kahler几何中的一些结果也作了新的证明，包括Mabuchi K-能量的一个下界和Bando-Mabuchi的Fano流形上Kahler-Einstein度量的唯一性结果.本文是它所取代的预印本arXiv：0905.4263的实质性扩展版本。
The Riemannian curvature identities of a $G_2$ connection with skew-symmetric torsion and generalized Ricci solitons
本文研究常型可积$G_2$流形上特征联络的曲率性质，推导了挠率调和、曲率对称Ricci平坦等的等价条件，明确了Hull联络为$G_2$-瞬子的判据，并证明紧带闭挠率可积$G_2$流形是广义梯度Ricci孤立子。
Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers
该研究证明在闭合流形上存在一个里齐流解，其在有限时间形成基于渐近圆锥收缩子流形的奇异性和模型。
Convergence of the Chern-Ricci flow on complex minimal surfaces of general type
本文研究了复极小型表面的Chern-Ricci流的收敛性，通过统一的直径估计、体积非坍缩估计和Gromov-Hausdorff收敛性，去除了之前工作中对null locus附近的局部Kahler假设，验证了Tosatti-Weinkove猜想。
We examine moduli spaces of locally homogeneous surfaces of Type~$\mathcal{B}$ with torsion where the symmetric Ricci tensor is non-degenerate. We also determine the space of affine Killing vector fields in this context.
Ricci measure for some singular Riemannian metrics
定义了流形的切丛上某些奇异无挠联络的里奇曲率作为测度。该定义使用积分公式和向量值半密度。我们给出了相关的例子，其中可以计算的Ricci措施。在不定常情形下，给出了流形上里奇流解的一个弱概念.
On the blow-up of four dimensional Ricci flow singularities
本文证明了Feldman-Ilmanen-Knopf在\cite{FIK}中提出的一个猜想，即他们在$\CP^1$上重言式线丛上构造的梯度收缩孤子度量是闭流形上I型里奇流奇点爆破的一致极限.我们利用这个结果证明了四维闭流形上里奇流奇点的爆破极限不一定具有非负的里奇曲率。
In this paper, we establish the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in $\mathbb{R}^3$ as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex surfaces whose metrics converge uniformly to the metric of the initial surface in intrinsic sense.
Generalized Bochner formulas and Ricci lower bounds for sub-Riemannian manifolds of rank two
研究一类新的秩为2的次黎曼流形，它包括黎曼流形、Webster-Tanaka挠为零的CR流形、黎曼流形上的正交丛和阶为2的分次幂零李群.这些流形允许一个正则水平联络和一个正则次拉普拉斯算子。我们在这些流形上构造了一个类似的黎曼Ricci张量，并证明了次拉普拉斯算子的Bochner型公式。因此，有可能在这些空间上形成所谓曲率维数不等式的次黎曼模拟。满足这个不等式的次黎曼流形与里奇曲率是下界的黎曼流形具有许多共同的性质
The local description of the Ricci and Bianchi identities for an h-normal N-linear connection on the dual 1-jet space J^{1*}(T,M)
本文描述了对偶1-jet空间J^{1*}（T，M）上的h-正规N-线性联络DΓ（N）的局部Ricci和Bianchi恒等式.为此，我们首先给出了DΓ（N）产生的挠率张量和曲率张量的局部可分辨（d-）适应分量的表达式，然后分析了它们所附加的局部Ricci恒等式.导出的偏转d-张量恒等式。最后，我们给出了局部Bianchi恒等式（在Cartan型h-正规N-线性联络的特殊情况下），它几何地连接了线性联络DΓ（N）的局部挠和曲率d-张量.
Torsion and curvature of ACYT and AHKT 8-manifold
研究ACYT和AHKT 8维流形的挠率与曲率，通过分析几乎厄米特8维流形上尼延胡伊斯张量、里奇张量与挠率三形式的关系，得出相关结论，还给出几乎超厄米特8维流形成为AHKT流形的条件及与HKT流形的关系。
The Ricci Curvature of Half-flat Manifolds
利用半平坦流形和非紧$G_2$完整流形之间的关系，得到了半平坦流形的Ricci曲率张量和标量的内禀挠类表达式.我们的表达式测试岩泽和更一般的幂零流形。我们还推导出表达式，在语言的Calabi-Yau模空间，扭转类和Ricci曲率的\emph{particular}半平坦的流形，自然产生通过镜像对称的通量紧化。利用这些表达式，我们得到了这些半平坦流形上的第二型弦论的凯勒模空间的一个约束。
In this paper, we bend a closed Riemannian manifold in the conformal class, through solving a fully nonlinear equation. As a result, we prove that each metric of quasi-negative Ricci curvature is conformal to a metric with negative Ricci curvature.
Flows of geometric structures II
该论文推进张量\(\mathrm{H}\)-结构流理论，比较两种演化，证明多种情况下流的短时存在性和唯一性，处理\(\mathrm{SU}(m)\)等情况，得到修正里奇 - 调和流的相关方程和估计，在六维转化为\(\mathrm{SU}(3)\)-结构标准形式描述流族。
In this paper, motivated by the work of Qi S. Zhang in [28], we derive Li-Yau gradient bounds for positive solutions of the f-heat equation on closed manifolds with Bakry-Emery Ricci curvature bounded below.
Ricci-harmonic flow of $\mathrm{G}_2$ and Spin(7)-structures
本文研究了$\mathrm{G}_2$和Spin(7)结构的Ricci-谐波流，证明了其存在性、估计和孤子性质。
We give a simple proof of a recent result due to Agostiniani, Fogagnolo and Mazzieri.
In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial Ω$ in $M$, with equality holding true if and only if $(M{\setminus}Ω, g)$ is isometric to a truncated cone over $\partialΩ$. An optimal version of Huisken's Isoperimetric Inequality for $3$-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version …
In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler [Bam20c]. The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. Among other things, we prove: (1) that in these cases the tangent flow must be unique, (2) that if a Ricci flow with finite-time singularity has a closed singularity model, then the singularity is of Type I and the singularity model is the tangent flow at the singular point; this answers a quest…
We show that a complete gradient Ricci soliton $(M^n,\,g)$ with constant scalar curvature and a non-parallel closed conformal vector field is isometric to either the Euclidean space, or an Euclidean sphere, or negatively Einstein warped product of the real line with a complete non-positively Einstein manifold. Moreover, we show that a Kähler gradient Ricci soliton of real dimension $\ge 4$, with a non-parallel closed conformal real vector field is Ricci-flat (Calabi-Yau) and is flat in dimension 4. Finally, we show that a Ricci soliton whose associated 1-form is harmonic, has constant scalar c…
The paper is devoted to Hardy type inequalities on closed manifolds. By means of various weighted Ricci curvatures, we establish several sharp Hardy type inequalities on closed weighted Riemannian manifolds. Our results complement in several aspects those obtained recently in the noncompact Riemannian setting.
In this article, we prove Talenti's comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $L^1$- and $L^\infty$-moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.
Ricci-Flat Holonomy: a Classification
分析了无挠仿射联络的约化完整代数，目的是确定哪些约化完整代数可以对应于具有相同性质的Ricci平坦联络。各种家庭的holonomies消除通过不同的代数手段，并建立了例子（在本文和'射影几何II：Holonomy分类'，由同一作者）在其余的情况下，从而解决了这个问题完全，还原holonomy。
Cocalibrated G_2-manifolds with Ricci flat characteristic connection
任意7维协校准G_2流形都存在唯一的具有斜对称挠率的联络$\nabla$.我们在$\nabla$ -Ricci张量为零的条件下研究了这类流形.特别是，我们描述了它们的几何形状的情况下，最大数量的$\nabla$ -平行向量场。
Derivative estimates of pluriclosed flow
本文为pluriclosed流提供导数估计，控制高阶导数，并在二维及Hermitian-symplectic情形下导出挠率估计和单调量，证明所有Hermitian-symplectic孤子为Kähler Ricci孤子。
We study the quasi-convergence equivalence of some families of metrics on locally homogeneous closed 4-manifolds with trivial isotropy group, and identify the dimension of each equivalence class under certain conditions.
Intrinsic torsion in quaternionic contact geometry
从内蕴挠率的观点研究四元数切触流形。我们论证了这种几何的自然结构群是一个包含Sp（n）H^* 的非紧李群K，并证明了任何qc结构都产生一个具有常数内挠率的正则K-结构，除了在七维空间中，当这个条件等价于Duchemin意义下的可积性时。我们证明了对Sp（n）H^*（或等价地，qc分布的补）的约化的选择产生唯一的满足挠率和曲率的自然条件的K-联络。我们证明了在qc分布上相容度量的选择决定了一个到Sp（n）Sp（1）的正则约化和一个其曲率几乎完全由挠率决定的正则Sp（n）Sp（1）-联络。我们表明，它的Ricci张量，以及Ricci张量的Biquard连接，有一个解释的内在扭转。
In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.
Eigenvalues of the Laplacian under the Ricci Flow
在一个技术性假设下，导出了由里奇流演化的闭流形上拉普拉斯算子的特征值的一阶变分公式，并给出了一些应用.
How to produce a Ricci Flow via Cheeger-Gromoll exhaustion
证明了具有非负复截面曲率的开流形上的里奇流的短时存在性。我们不需要曲率上界。通过考虑Cheeger-Gromoll凸穷竭中包含的凸集的加倍，并通过求解这些闭流形上的里奇流的奇异初值问题，得到了具有非负复截面曲率的里奇流的一系列闭解，这些解下收敛于开流形上的里奇流的一个解.此外，我们找到了一个最优的体积增长条件，保证长期存在的，我们给出了里奇流的长期行为的分析。最后，我们构造了一个明确的例子，不朽的非负曲率的里奇流的所有时间的解决方案。
Generalized Ricci flow I: Higher derivatives estimates for compact manifolds
考虑了具有给定（不一定是闭的）三形式的广义里奇流，建立了紧致流形的高阶导数估计.作为应用，我们证明了这种广义里奇流的紧性定理.对于更广义的里奇流，类似的结果仍然成立.
Ricci flow on locally homogeneous closed 4-manifolds
讨论了齐次四维流形上的里奇流。在对这些流形进行分类之后，我们注意到存在初始度量族，使得我们可以将它们对角化，并且里奇流保持对角化。我们分析了这些家庭的长期行为。我们发现，如果一个解决方案存在的所有时间，那么流动表现出第三型奇异性的意义上的Hamilton。
This article is a continuation of [CMZ21b], where we proved that a Ricci flow with a closed and smooth tangent flow has unique tangent flow, and its corresponding forward or backward modified Ricci flow converges in the rate of $t^{-β}$ for some $β>0$. In this article, we calculate the corresponding $\mathbb{F}$-convergence rate: after being scaled by a factor $λ>0$, a Ricci flow with closed and smooth tangent flow is $|\log λ|^{-θ}$ close to its tangent flow in the $\mathbb{F}$-sense, where $θ$ is a positive number, $λ\gg 1$ in the blow-up case, and $λ\ll 1$ in the blow-down case.
Stability of solutions of certain extended Ricci flow systems
考虑了四个推广的里奇流系统--即与其它几何流耦合的里奇流--并证明了这些流的某些定常解的动力稳定性。这些系统包括与调和映射流耦合的里奇流（在翘曲积上的里奇流的背景下抽象地研究）、与调和映射流和Yang-Mills流耦合的里奇流以及与度量相容连接的挠率的热流耦合的里奇流。用于证明稳定性的方法遵循Guenther、Isenberg和Knopf概述的程序，该程序使用拟线性抛物系统的最大正则性理论和Simonett的结果。
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.
Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature
本文推导了具有Bakry-Emery里奇曲率从下有界的黎曼流形的Cheng-Yau、Li-Yau、Hamilton估计，以及闭黎曼流形上加权热方程正有界解的Hessian的整体和局部上界。
Smoothing metrics on closed Riemannian manifolds through the Ricci flow
在一致局部Sobolev不等式的假设下，证明了具有绝对里奇曲率界和小黎曼曲率积分界的黎曼度量可以光滑化为具有截面曲率界.这在一定程度上扩展了以前的先验估计叶莉（J. Geom. Anal. 17（2007）495-511; Advances in Mathematics 223（2010）1924-1957）。
In this paper, we study the torsion flow which is served as the CR analogue of the Ricci flow in a closed pseudohermitian manifold. We show that there exists a unique smooth solution to the CR torsion flow in a small time interval with the CR pluriharmonic function as an initial data. In spirit, it is the CR analogue of the Cauchy-Kovalevskaya local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems
Closed $G_2$-Structures with Negative Ricci Curvature
本文研究负 Ricci 曲率闭 $G_2$-结构的存在性，证明非紧情形的 $G_2$-Goldberg 猜想，并给出完备与不完备 Einstein 情形的无挠性及测地线长度限制。
Let M be a Riemannian n-manifold with n greater than or equal to 3. For k between 1 and n, we say M has k-positive Ricci curvature if at every point of M the sum of any k eigenvalues of the Ricci curvature is strictly positive. In particular, one positive Ricci curvature is equivalent to positive Ricci curvature and n-positive Ricci curvature is equivalent to positive scalar curvature. Let G be the fundamental group of the closed manifold M. We say that G is virtually free if G contains a free subgroup of finite index, or equivalently, if some finite cover of M has a fundamental group that is …
Degeneration of shrinking Ricci solitons
设$(Y,d)$是具有一致上界直径和下界体积的闭收缩Ricci孤子的Gromov-Hausdorff极限。证明了在余维至少为2的闭子集上，$Y$是满足收缩Ricci孤子方程的光滑流形.
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