An end-to-end construction for compact constant mean curvature surfaces
M. Jleli, F. Pacard
我们解释了如何利用现有的知识，一套完整的非紧凑的常数中曲率表面可以产生新的例子紧凑的常数中曲率表面的属大于或等于3。
We explain how the current knowledge on the set of complete noncompact constant mean curvature surfaces can be exploited to produce new examples of compact constant mean curvature surfaces of genus greater than or equal to 3.
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In this note we classify compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature
Compact Mean Curvature Flows from Non-Compact Initial Data
该研究在ℝⁿ⁺¹中构造了源自非紧初始数据、存在有限时间的紧嵌入平均曲率流解。
Compact four-manifolds with pinched self-dual Weyl curvature
本文研究了具有调和自双曲Weyl曲率和挤压条件的紧致四流形的性质。
Eternal forced mean curvature flows I - A compactness result
为了构造CMC超曲面的Morse/弗洛尔同调理论，我们证明了紧致双曲流形中带强迫项的常中曲率流模断裂轨线的一个紧性结果.
We make rigorous an old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.
Shrinkers with curvature pinching conditions are compact
本文通过建立Ricci曲率正性与渐近非负截面曲率两类曲率夹挤条件，证明了相应收缩孤立子的紧致性，并给出相关分类结果。
In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqueness result on Ricci flows. A backward uniqueness theorem is also proved for mean curvature flow with bounded curvatures.
On the Gauss curvature of compact surfaces in homogeneous 3-manifolds
对具有4维等距群的齐次黎曼3-流形的紧平坦曲面进行了分类。建立了这类三维流形中紧致常高斯曲率曲面的不存在性结果。
In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean curvature surfaces in three manifold with positive Ricci curvature.
Let $M^n(n\geq3)$ be an $n$-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $M^n$ satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds with harmonic curvature and positive scalar curvature. In particular, Theorem 1.4, Corollary 1.6 and Theorem 1.9 are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of equality.
4-dimensional compact manifolds with nonnegative biorthogonal curvature
本文的目的是研究S. T. Yau在1990年用一个关于双正交曲率的较弱条件代替截面曲率。此外，我们还对具有非负双正交曲率的四维紧致定向黎曼流形进行了分类。特别地，我们得到了关于四维紧致流形的拼挤定理的丘猜想的部分答案。
We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $σ_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant $σ_2$ is isometric to a quotient of the round $\mathbb{S}^4$.
In this paper, we characterize round spheres in the Euclidean space under some suitable conditions on the r-mean curvature.
An example of compact Kähler manifold with nonnegative quadratic bisectional curvature
我们构造了一个非负二次对分曲率的紧致Kähler流形，它不允许任何非负正交对分曲率的Kähler度量.该流形是一个第二贝蒂数等于1的7维Kähler C-空间，其正则度量是一个正数量曲率的Kähler-Einstein度量
The space of Constant Mean Curvature surfaces in compact Riemannian Manifolds
本文的主要观点是，在适当的中曲率和Ricci曲率的条件下，我们可以将Choi-Schoen紧性定理推广到具有正里奇曲率（中曲率小，取决于里奇曲率）的黎曼流形中的紧嵌入极小曲面到简单浸入紧H-曲面。证明了单连通1/4-pinched流形中的凸嵌入（固定）常中曲率超曲面空间是紧的。
Compactness and rigidity of Kähler surfaces with constant scalar curvature
证明了一类具有常数数量曲率、体积有界、直径有界、里奇曲率有界、签名有界的Kähler曲面族的紧性定理.进一步证明了Einstein-Maxwell系统的分裂定理和刚性定理。
We propose a new approach to the study of compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary or positive Ricci curvature and convex boundary. Several conjectures are formulated. Some partial results that support these conjectures are established.
Compact Gradient Shrinking Ricci Solitons with Positive Curvature Operator
本文研究了Hamilton猜想：任何具有正曲率算子的紧梯度收缩Ricci孤子必是Einstein孤子。我们首先推导出几个恒等式。然后在一个附加条件下证明了该猜想成立。此外，这样的孤子必须是常曲率的。
Open problems on compact constant mean curvature surfaces with boundary
本文收集了带边紧致常平均曲率曲面理论中若干易于表述的公开问题，并综述了这些问题的现有解答进展。
Uniform Estimates for the Constant Mean Curvature Solutions of the Vacuum Einstein Constraint Equations on Compact Manifolds
给出了紧致流形上共形真空Einstein约束方程的常中曲率解的一致估计。这些解的存在性在J. Isenberg的一篇论文中给出。
Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature
本文给出了具有非负里奇曲率的完全非紧扩张（或定常）梯度Ricci孤子的数量曲率的时滞估计。作为应用，我们证明了任何具有正压缩里奇曲率的完全非紧膨胀（或定常）梯度Kähler-Ricci孤子都是Ricci平坦的.这一结果回答了Chow，Lu和Ni在一本书中提出的Kähler-Ricci孤子的问题。
On Compact Finsler Spaces of Positive Constant Curvature
在欧几里得空间$\R^{n+1}$中，一个$n$维（$n\geq 2$）单连通的、紧致的、无边界的、具有正常截面曲率的Finsler空间共形同胚于一个n-球面.
Geometric realizations of curvature models by manifolds with constant scalar curvature
我们证明了任何黎曼曲率模型都可以由具有常数数量曲率的流形几何地实现。我们还证明了任意的伪厄米特曲率模型、仿厄米特曲率模型、超伪厄米特曲率模型和超仿厄米特曲率模型都可以由具有常标量和 *-数量曲率的流形来实现.
In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.
Density estimates for compact surfaces with total boundary curvature less than 4pi
本文得到了R^n中全边界曲率小于4 π且中曲率的L^p范数足够小的紧致曲面的密度估计，p>2.所得结果推广了文[2]中的主要结果。然后，我们应用我们的估计来讨论这样的表面的几何和拓扑。
Prescribing scalar curvatures: non compactness versus critical points at infinity
研究通过例子展示在黎曼流形上，标量曲率的规范问题中，非紧性与无穷远处临界点的转变现象。
A 2-systolic inequality on non-rational compact Kähler surfaces with positive scalar curvature
本文研究了具有非常数全纯映射到正亏曲线的紧正标量曲率凯勒表面的2-系统不等式，证明了此类表面必为纤维化 over 复曲线的有理面。
Compactness of capillary hypersurfaces with mean curvature prescribed by ambient functions
本文证明了由环境函数规定平均曲率的毛细超曲面的紧致性定理，推广了Schätzle和Bellettini的结果，并扩展了曲率varifold定义至有向积分varifold，同时讨论了边界平均曲率规定情形。
We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.
Singular minimal surfaces with constant curvature
本文证明了常高斯曲率的奇异极小曲面仅为平面、球面和圆柱面，并进一步分类了常主曲率和常平均曲率的奇异极小曲面。
Compact Kähler manifolds with nonpositive bisectional curvature
设$(M^n, g)$是具有非正对分曲率的紧致Kähler流形.本文证明了紧致Kähler流形$N^k$上的有限覆盖是双全纯的，等距于具有$c_1 < 0$的平坦环面丛。这证实了丘的一个猜想。作为推论，对于任何具有非正对分曲率的紧致Kähler流形，Kodaira维数等于Ricci张量的最大秩。我们还证明了一个整体分裂的结果，在一定的浸入复子流形的假设。
Mean curvature and compactification of surfaces in a negatively curved Cartan-Hadamard manifold
本文证明了一个Chern-Osserman型不等式，它涉及到完全曲面的体积增长，该曲面具有受控的中曲率，且适当地浸入Cartan-Hadamard流形$N$中，该流形的截面曲率从上到下由一个负量$K_{N}\leq b<0$限定
We study a version of the Hermitian curvature flow on compact homogeneous complex manifolds. We prove that the solution has a finite exstinction time $T>0$ and we analyze its behaviour when $t\to T$. We also determine the invariant static metrics and we study the convergence of the normalized flow to one of them.
Compactness results for immersions of prescribed Gaussian curvature I - analytic aspects
我们推广了关和Spruck的结果，证明了Hadamard流形中常高斯曲率超曲面的存在性.
Conformal product structures on compact manifolds with constant sectional curvature
本文证明了常数截面曲率且非平坦的紧致流形不具有共形积结构，并进一步展示了这些方法可以自然推广到非正曲率的不可约紧致局部对称空间。
Constant Mean Curvature n-noids with Platonic Symmetries
具有Platonic对称性的常中曲率n-类星体
In this short note, using Siu-Yau's method [14], we give a new proof that any n-dimensional compact Kahler manifold with positive orthogonal bisectional curvature must be biholomorphic to $\mathbb{P}^n$.
Rigidity Theorems for Compact Manifolds with Boundary and Positive Ricci Curvature
我们证明了一些边界刚性的结果半球下的下限里奇曲率。主要结果可以看作是一个Min-Oo猜想的Ricci版本。
Minimal hypersurfaces in $\mathbb{S}^5$ with constant scalar curvature and zero Gauss curvature are totally geodesic
本文证明五维球面中常数量曲率且零高斯曲率的闭极小超曲面必为全测地超曲面。
Spacelike surfaces in Generalized Robertson-Walker spacetimes whose mean curvature function satisfies a natural nonlinear inequality are analyzed. Several uniqueness and nonexistence results for such compact spacelike surfaces are proved. In the nonparametric case, new Calabi-Bernstein type problems are solved as a consequence.
In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of self-adjoint operators, we can describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.
On the Curvature and Topology of Compact Stationary Spacetimes
本文基于 Petersen 与 Wink 2021 年的结果，研究紧致洛伦兹流形在具有单位长度类时 Killing 向量场时其曲率与拓扑所受到的阻碍。
Let (M,g) be a complete noncompact riemannian manifold with bounded geometry and parallel Ricci curvature. We show that some operators, "affine" relatively to the Ricci curvature, are locally invertible, in some classical Sobolev spaces, near the metric g.
Fundamental group of compact decent spacetimes with lightlike hypersurface curvature
本文研究带有类光超曲面曲率的紧致正则4维时空的基本群，证明其叶状结构的叶紧致时基本群为多循环群，非全恶性的4维紧致pp波时空也具有多循环基本群。
Volume comparison of conformally compact manifolds with scalar curvature $R\geq -n\left(n-1\right)$
本文利用规范化Ricci-DeTurk流证明了严格稳定共形紧Einstein流形的一个稳定性结果。作为应用，我们给出了具有数量曲率$R\geq -n\left(n-1\right)$的共形紧致流形的局部体积比较，以及当重整化体积为零时的刚性结果。
Prescribed Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree
本文研究具有负Gauduchon度的紧Hermitian流形上预定Chern数量曲率问题，利用几何流收敛性在非正及变号曲率函数情形下建立了存在性与不存在性结果。
Compact almost Ricci solitons with constant scalar curvature are gradient
本文证明了任何具有常数量曲率的紧致非平凡几乎Ricci孤子$\big(M^n,\,g,\,X,\,λ\big)$等距于欧氏球面$\Bbb{S}^{n}$ .由此得到了每一个具有常数量曲率的紧致非平凡几乎Ricci孤子都是梯度的。此外，向量场$X$分解为Killing向量场$Y$与适当函数的梯度的和。
We show that under Ricci curvature integral assumptions the dimension of the first cohomology group can be estimated in terms of the Kato constant of the negative part of the Ricci curvature. Moreover, this provides quantitative statements about the cohomology group, contrary to results by Elworthy and Rosenberg.
Algebraic curvature tensors whose skew-symmetric curvature operator has constant rank 2
设R是签名（p，q）的非退化内积的代数曲率张量，其中q>4.若$π$是类空2平面，则设$R(π)$是相关的反对称曲率算子。我们对代数曲率张量进行了分类，使R（-）具有常秩2，并证明了这些张量在平坦空间中可由超曲面几何实现.我们还分类的Ivanov-Petrova代数曲率张量秩2;这些是代数曲率张量常秩2，使复杂的约旦标准形的R（-）是常数。
Hypersurfaces of Constant Curvature in Hyperbolic Space II
在双曲空间中，对于一类包含高阶中曲率及其曲率导数的（椭圆）曲率函数，在无穷远处具有预定渐近边界的常曲率完备超曲面.
Compactness for immersions of prescribed Gaussian curvature II - geometric aspects
我们发展了局部凸浸入族在边界附近的一个紧性结果。我们还发展了一个模2度理论的浸入常数（和规定）的高斯曲率与规定的边界。然后，这些被用来解决高原问题的浸入常数（和规定）高斯曲率一般阿达玛流形。
Collapse of the mean curvature flow for isoparametric submanifolds in non-compact symmetric spaces
已知非紧型对称空间上的Hermann作用的主轨道是曲率适应的等参子流形，它在周围对称空间的理想边界上没有非欧几里得型的焦点。本文研究了曲率适应等参子流形及其焦点子流形的中曲率流。具体地说，研究了子流形通过伪黎曼浸没提升到无限维伪希尔伯特空间的中曲率流.
Almost Kähler manifolds of constant antiholomorphic sectional curvature
证明了如果一个维数大于或等于6的AK 2-流形具有逐点常反全纯截面曲率，则它是一个具有常负截面曲率的6维流形或一个具有常全纯截面曲率的Kähler流形.
Kahler surfaces with quasi-constant holomorphic curvature
本文的目的是描述具有拟常全纯曲率的Kahler曲面
Algebraic curvature tensors for indefinite metrics whose skew-symmetric curvature operator has constant Jordan normal form
我们用$q\ge5$对符号为$(p,q)$的连通伪黎曼流形进行了分类，使得在$M$的每一点处，反对称曲率算子在类空2平面集上具有常秩2和常Jordan标准形，并且反对称曲率算子对$M$的至少一点不是幂零的.
Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature
本文找到了Minkowski空间中所有具有常中曲率且其母曲线为多项式或洛伦兹圆的螺旋曲面。在第一种情况下，我们证明了多项式的次数是$0$或$1$，曲面是直纹的。如果母曲线是洛伦兹圆，我们证明了唯一的可能性是轴是类空的，圆心位于轴上。
Extending Aubin's construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|, t\in\mathbb{R}$. In particular, there are no topological obstructions for metrics with $\varepsilon$-pinched Weyl curvature and negative scalar curvature.
In this paper, we study the relationship between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces. We give some examples of isoparametric hypersurfaces with (non)constant principal curvatures on Randers manifolds with nonconstant flag curvatures. Furthermore, we construct an example of a conformally flat Randers manifold which admits a family of nonisoparametric hyperplanes with constant principal curvatures.
ParaSasakian manifolds with a constant paraholomorphic section curvature
本文考虑具有常仿全纯截面曲率的仿Sasakian流形。
Constant scalar curvature metrics on toric surfaces
本文完成了一个程序，以确定哪些复曲面允许常数数量曲率的Kahler度量。
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