Einstein manifolds with nonnegative isotropic curvature are locally symmetric
S. Brendle
设（M，g）是n维\geq 4具有非负迷向曲率的Einstein流形.我们证明了（M，g）是局部对称的.
Let (M,g) be an Einstein manifold of dimension n \geq 4 with nonnegative isotropic curvature. We show that (M,g) is locally symmetric.
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On the smooth rigidity of almost-Einstein manifolds with nonnegative isotropic curvature
设$(M^n,g)$，$n \ge 4$是具有非负迷向曲率的紧致单连通黎曼流形.给定$0<l\le L$，证明了存在满足下列条件的$\eps = \eps (l,L,n)$：若$g$的标量曲率$s$满足$$ l \le s \le L $$，Einstein张量满足$$ | Ric - \frac {s}{n}g | \le \eps$$，则$M$同构于紧致型对称空间.这是S的结果的光滑模拟。Brendle证明了具有非负迷向曲率的紧致Einstein流形与局部对称空间等距。
Using a method introduced by R. Bamler to study the behavior of scalar curvature under continuous deformations of Riemannian metrics, we prove that if a sequence of smooth Riemannian metrics gi on a fixed compact manifold M has isotropic curvature bounded from below by a nonnegative function u, and if gi converge in C 0 norm to a smooth metric g, then g has isotropic curvature bounded from below by u. The proof also works for various other bounds from below on the curvature, such has non-negative curvature operator.
Manifolds with nonnegative isotropic curvature
本文证明了：如果$(M^n,g)$，$n \ge 4$是一个紧致的，可定向的，局部不可约的，具有非负迷向曲率的黎曼流形，则下列可能性之一成立：（i）$M$允许一个具有正迷向曲率的度量;（ii）$(M,g)$与一个局部对称空间等距;（iii）$(M,g)$是Kähler且双全纯于$\C P^\frac {n}{2}$ . (iv)$(M,g)$是四元数Kähler。这是由以下结果所暗示的：设$(M^{2n},g)$是一个紧致的，局部不可约的Kähler流形，具有非负的迷向曲率。则要么$M$双全纯到$\C P^n$，要么等距到紧致厄米对称空间。这肯定地回答了米卡列夫和王的一个问题。证明是基于S. Brendle和R. Schoen在里奇流上。
Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature
本文作为此前研究的续作，针对具有非负或半非负迷向曲率的四维梯度稳态、膨胀Ricci孤立子得到分类相关结果，并改进了梯度收缩Ricci孤立子的已有结论，给出复二维Kaehler-Ricci收缩子的刻画。
Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature
本文研究四维完备梯度收缩里奇孤立子在半正迷向曲率条件下的曲率估计与分类，给出非紧情形的二次曲率下界，并利用强极值原理完成相应分类。
Concavity for elliptic and parabolic equations in locally symmetric spaces with nonnegative curvature
本研究将Langford和Scheuer的结果拓展，建立了非负截面曲率局部对称空间上椭圆与抛物方程解的凹性原理，是首个非恒定截面曲率空间上的通用凹性原理。
Cohn-Vossen theory for locally conformally flat manifolds
通过建立非负n-超调和函数的精细奇异性估计，解析刻画了非负Ricci曲率完备非紧局部共形平坦流形的体积增长，验证了Yau关于Cohn-Vossen不等式的猜想，并证明了消除早期工作所有辅助假设的尖锐间隙定理。
Anisotropic Einstein data with isotropic nonnegative scalar curvature
我们构造了一个具有指定数量曲率的爱因斯坦方程的黑洞初始数据，或者更准确地说，是包含在黑洞内部的一段初始数据。约束转化为一个抛物型方程，半径为“时间”变量，在一个度量分量u经历爆破。度量本身是规则的，直到并包括在爆破半径处的表面，这是一个极小曲面。通过应用等变分歧理论的自相似重标度方程，我们证明了存在的爆破轮廓，不是O（3）对称-或各向异性-虽然曲率是各向同性规定。
Toric Einstein 4-manifolds with nonnegative sectional curvature
本文证明了4-流形上具有非负截面曲率的$T^2$-不变爱因斯坦度量是局部对称的。
On gradient $ρ$-Einstein solitons with Bach tensor radially nonnegative
本文研究 Bach 张量径向非负的梯度 $\rho$-Einstein 孤立子，证明其在特定条件下为局部扭曲积，并完成相关分类。
Conformally flat manifolds with nonnegative Ricci curvature
我们证明了具有非负里奇曲率的n>2维完备共形平坦流形具有很好的刚性性质：它们要么是平坦的，要么局部等距于球面和直线的乘积，要么整体共形等价于平坦空间或球面空间形式.这扩展了Q. M. Cheng，M.H. Noronha，B. L. Chen和X.- P. Zhu和S.竹
On the structure of complete 3-manifolds with nonnegative scalar curvature
本文证明了如下结果：设$\mathcal{N} $是一个完备（非紧）连通可定向黎曼三元流形，具有非负数量曲率$S \geq 0$和有界截面曲率$ K_{s} \leq K $ .设$Σ\subset \mathcal{N} $是一个完全可定向的连通面积最小柱面，使得$π_1 (Σ) \in π_1 (\mathcal{N})$ .则$\mathcal{N}$与$\mathbb{S} ^1 \times \mathbb{R} ^2 $或$\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{R}$局部等距（使用标准乘积度量）。作为推论，我们得到：设$\mathcal{N} $是一个完备（非紧）连通可定向黎曼流形，具有非负数量曲率$S \geq 0$和有界截面曲率$ K_{s} \leq K $ .设$π_1 (\mathcal{N})$包含一个与正亏格紧致曲面的基本群同构的子群。然后，$\mathcal{N}$局部等距于$\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{R}$（具有标准乘积度量）。
We consider Finsler submanifolds $M^n$ of nonnegative Ricci curvature in a Minkowski space $\mathbb{M}^{n+p}$ which contain a line or whose relative nullity index is positive. For hypersurfaces, submanifolds of codimension two or of dimension two, we prove that the submanifold is a cylinder, under a certain condition on the inertia of the pencil of the second fundamental forms. We give an example of a surface of positive flag curvature in a three-dimensional Minkowski space which is not locally convex.
Dimension of polynomial growth harmonic functions on locally conformally flat manifolds with nonnegative Ricci curvature
本文针对具非负Ricci曲率的局部共形平坦流形，证实丘成桐提出的关于多项式增长调和函数空间维数的第二个问题，即其维数不超过对应欧几里得空间的该空间维数的精确上界成立。
In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give a new proof to a rigidity result for complete manifolds with nonnegative scalar curvature admitting a proper smooth map to $T^{n-1}\times \mathbf R$ with nonzero degree. Here we introduce a trick to obtain the compactness of limit hypersurface from locally graphical convergence. Based on the same idea we also establish an optimal $2$-systole inequality for several classes of complete Riemannian man…
In this paper, we give a complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M$ having positive isotropic curvature. We prove that for a pair $(f, κ)$ of a nontrivial smooth function $f: M \to {\Bbb R}$ and a nonnegative real number $κ$, if $(M, g)$ having positive isotropic curvature satisfies $$ Ddf - (Δf)g - f{\rm Ric} = κg, $$ then $(M, g)$ is isometric to a geodesic ball in ${\Bbb S}^n$ when $κ>0$, and either $M$ isometric to ${\Bbb S}^n_+$, or the product $I \times {\Bbb S}^{n-1}$, up to finite cover when $κ=0$.
Simply connected Alexandrov 4-manifolds with positive or nonnegative curvature and torus actions
我们指出，一个4-维拓扑流形的Alexandrov度量（曲率有界以下）和一个有效的，等距作用的圆或2-环面是局部光滑的。这个观察意味着，如果我们考虑Alexandrov流形而不是黎曼流形，则具有正或非负截面曲率以及圆或2-环面的有效等距作用的紧致单连通黎曼4-流形的拓扑和等变分类也成立。
Locally Isotropic Pseudo-Riemannian Manifolds
局部迷向的伪黎曼流形是局部对称的，这个结果是由Wolf得到的。在黎曼设置的一个证明，由于绍博，使用频谱性质的所谓绍博运营商。在本文中，我们将Szabó的方法推广到伪黎曼环境，得到的结果与Wolf的结果相当。大多数关于Szabó算子的伪黎曼设置的结果是使用代数拓扑的方法。本文利用了Szabó算子的多项式性质及其在零锥上的行为，这两个性质到目前为止都没有得到太多的关注。
We prove that any complete immersed two-sided mean convex translating soliton $Σ\subset \mathbb{R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in $\mathbb{R}^3$ is the axisymmetric "bowl soliton". We also show that if the mean curvature of $Σ$ tends to zero at infinity, then $Σ$ can be represented as an entire graph and so is the "bowl soliton". Finally we classify all locally strictly convex graphical translating solitons defined over strip regions.
Let $\big(M,g^{TM}\big)$ be a noncompact complete spin Riemannian manifold of even dimension $n$, with $k^{TM}$ denote the associated scalar curvature. Let $f\colon M\rightarrow S^{n}(1)$ be a smooth area decreasing map, which is locally constant near infinity and of nonzero degree. We show that if $k^{TM}\geq n(n-1)$ on the support of ${\rm d}f$, then $ \inf \big(k^{TM}\big)< 0$. This answers a question of Gromov. We use a simple deformation of the Dirac operator to prove the result. The odd dimensional analogue is also presented.
In this paper, we prove two rigidity results for non-positively curved homogeneous Finsler metrics. Our first main result yields an extension of Hu-Deng's well-known result proven for the Randers metrics. Indeed, we prove that every connected homogeneous Finsler space with non-positive flag curvature and isotropic S-curvature is Riemannian or locally Minkowskian. We extend the Szabó's rigidity theorem for Berwald surfaces and show that homogeneous isotropic Berwald metrics with non-positive flag curvature are Riemannian or locally Minkowskian. We prove that a homogeneous $(α, β)$-metrics has i…
A generalization of Hamilton's differential Harnack inequality for the Ricci flow
在[10]中，R. Hamilton对具有非负曲率算子的里奇流的解建立了一个微分Harnack不等式.我们证明了这个不等式在MxR^2具有非负迷向曲率的较弱条件下成立。
Four-dimensional neutral signature self-dual gradient Ricci solitons
本文描述了自对偶梯度Ricci孤子在中性签名下的局域结构。如果Ricci孤子是非各向同性的，那么它是局部共形平坦的，并且局部等距于$I\times_φN(c)$形式的翘曲乘积，其中$N(c)$是常曲率空间。若Ricci孤子是各向同性的，则它局部等距于一个仿射曲面的余切丛，该曲面具有黎曼扩张的联络，并且Ricci孤子由其基础仿射结构描述.这提供了非局部共形平坦的自对偶梯度里奇孤子的例子。
Holonomy restrictions from the curvature operator of the second kind
本文研究第二类曲率算子的符号条件对完整和乐群的限制，证明 n-非负或 n-非正时流形受限完整和乐群为 SO(n) 或平坦，并加强了 Einstein、Kähler 及局部对称情形。
Some curvature properties of Cartan spaces with mth root metrics
本文证明了每一个具有迷向平均Berwald曲率的m阶根度量约化为弱Berwald度量。证明了具有迷向平均Landsberg曲率的m次根度量是弱Landsberg度量。我们找到了m次根度量的共形$β$ -变换局部对偶平坦的充要条件。最后，证明了局部射影平坦m次根度量的共形$β$ -变换是局部Minkowski变换.
We consider the class of locally boost isotropic spacetimes in arbitrary dimension. For any spacetime with boost isotropy, the corresponding curvature tensor and all of its covariant derivatives must be simultaneously of alignment type ${\bf D}$ relative to some common null frame. Such spacetimes are known as type ${\bf D}^k$ spacetimes and are contained within the subclass of degenerate Kundt spacetimes. Although, these spacetimes are $\mathcal{I}$-degenerate, it is possible to distinguish any two type ${\bf D}^k$ spacetimes, as the curvature tensor and its covariant derivatives can be charac…
Deformations of Q-curvature I
本文研究了闭黎曼流形上的$Q$ -曲率的变形问题。其中最重要的一个概念是$Q$ -奇异空间，它是由Chang-Gursky-Yang在20世纪90年代提出的。受Fischer-Marsden早期工作的启发，我们得到了与$Q$ -曲率有关的几何结果.它包括非负Einstein $Q$ -奇异空间的分类，非$Q$ -奇异空间的线性化稳定性和具有非负$Q$ -曲率的平坦流形的局部刚性结果.对于整体结果，我们证明了任何光滑函数都可以在一般的$Q$ -flat流形上实现为$Q$ -曲率，而相反，具有非负$Q$ -曲率的$n$ -tori上的局部共形平坦度量必须是平坦的.特别地，在$4$ -环面上不存在具有非负$Q$ -曲率的度量，除非它是平坦的。
On Berwald Spaces with non-Zero Flag Curvature
该研究证明非零旗帜曲率的贝瓦尔德流形是黎曼流形，并扩展了刚性定理，同时探讨了各向同性贝瓦尔德流形的性质及推广了贝瓦尔德-兰德斯度量的结果。
Smoothing $L^\infty$ Riemannian metrics with nonnegative scalar curvature outside of a singular set
该研究针对满足奇异集外光滑非负标量曲率、$L^\infty$接近欧氏度量的$L^\infty$黎曼度量，证明可用Ricci-DeTurck流时间切片逼近，还给出了流保持非负标量曲率的条件。
Exotic $\mathbb{R}^4$'s and positive isotropic curvature
我们证明了没有奇异的$\mathbb{R}^4$承认一个完整的黎曼度量一致正迷向曲率和有界几何。这基本上是一个推论的主要结果在[胡1]，并表示在[胡2]没有证据。在证明过程中，我们还证明了某些连通光滑PDMATHAMN-流形的无穷连通和（$n\geq 2$）按局部有限图的同构型不依赖于所用的胶合映射.
On the topology of locally volume collapsed Riemannian 3-orbifolds
研究了局部体积坍缩的黎曼3-orbifolds的几何和拓扑性质。我们证明了一个充分塌陷的封闭的3-orbifold没有坏的2-suborbifold要么承认一个度量的非负截面曲率或满足瑟斯顿的几何化猜想。我们还证明了一个带边界的版本。Kleiner和Lott独立地证明了类似的结果。
Cohn--Vossen-Type Inequalities for Three-Manifolds and Locally Conformally Flat Manifolds
本文在非负 Ricci 曲率的完备非紧 Riemann 流形上证明了 Cohn-Vossen 型数量曲率不等式，包括三维流形中基于渐近体积比的尖锐上界以及局部共形平坦流形上的多项式或指数增长估计。
On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions
本文证明了高维下具有正迷向曲率的收缩Ricci孤立子在特定曲率条件下必为球面或圆柱空间的商空间，并将相关分类与Ricci流古老解结果推广至更高维。
Let $(M,g^{TM})$ be a noncompact complete Riemannian manifold of dimension $n$, and let $F\subseteq TM$ be an integrable subbundle of $TM$. Let $g^F=g^{TM}|_{F}$ be the restricted metric on $F$ and let $k^F$ be the associated leafwise scalar curvature. Let $f:M\to S^n(1)$ be a smooth area decreasing map along $F$, which is locally constant near infinity and of non-zero degree. We show that if $k^F> {\rm rk}(F)({\rm rk}(F)-1)$ on the support of ${\rm d}f$, and either $TM$ or $F$ is spin, then $\inf (k^F)<0$. As a consequence, we prove Gromov's sharp foliated $\otimes_\varepsilon$-twisting conje…
Singularity removal rigidity theorems for minimal hypersurfaces in manifolds with nonnegative scalar curvature
本文证明了非负标量曲率流形中极小超曲面的奇点去除刚性定理，并利用该结果直接证明了渐近扁$8$-流形的正质量定理。
Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type~I curvature growth. We show that a $κ$-noncollapsed Type~I ancient solution which is noncompact and has nonnegative sectional curvature necessarily splits at least one Euclidean f…
Every Finsler metric naturally induces a spray but not so for the converse. The notion for sprays of scalar (resp. isotropic) curvature has been known as a generalization for Finsler metrics of scalar (resp. isotropic) flag curvature. In this paper, a new notion, sprays of constant curvature, is introduced and especially it shows that a spray of isotropic curvature is not necessarily of constant curvature even in dimension $n\ge3$. Further, complete conditions are given for sprays of isotropic (resp. constant) curvature to be Finsler-metrizabile. As applications of such a result, the local str…
Holomorphic geometric structures on Kaehler-Einstein manifolds
证明了具有第一类非负的全纯抛物几何的紧致Kaehler流形是复环面上有理齐次簇的平坦丛。我们还证明了具有负第一陈类的紧致Kaehler流形的局部Hermitian对称簇是全纯共微几何的完备的Kaehler流形。
On submanifolds in locally symmetric spaces of noncompact type
给定非紧型紧致局部对称空间X^n中的非紧型连通紧致全测地子流形Y^m，我们给出了保证[Y^m]在H_m（X^n; R）中非零的充分条件;在低维情况下，我们的条件也是必要的.本文给出了从有限覆盖（X-bar，Y-bar）到非负曲边（X_u，Y_u）的切映射存在的条件.
Screen Generic Lightlike Submanifolds of a Locally Bronze Semi-Riemannian Manifold equipped with an (l,m)-type Connection
研究赋予 (l,m) 型联络的局部青铜色半黎曼流形中屏幕一般类光子流形，建立测地性、不存在性、全脐性及极小子流形结构等结论。
Mass Lower Bounds for Asymptotically Locally Flat Manifolds
该研究针对渐近局部平坦流形建立正质量型定理，证明了特定拓扑条件下非负数量曲率对应非负质量，还得到四维带U(1)作用情形的质量下界，首次揭示渐进端非平凡拓扑对质量的影响。
Convexity in locally conformally flat manifolds with boundary
给定开单位球$B_1\subset \real^n$的一个闭子集$\La$，$n \geq 3$，我们将考虑$\bar{B_1} \setminus \La$上的一个完备黎曼度量$g$，其常数量曲率等于$n(n-1)$，并且与欧几里得度量共形相关。在本文中，我们证明了每一个闭欧几里德球$\bar{B} \subset B_1\setminus \La$是凸的关于度量$g$，假设的中曲率的边界$\partial B_1$是非负的关于内法线。
Locally Conformally Flat Manifolds with Positive Scalar Curvature: Kleinian Groups, Moduli Spaces, and Euclidean Rigidity
该研究探讨具正标量曲率的局部共形平坦流形，通过克莱因群等工具得到宏观维数、同伦群等约束，证明三维相关空间可缩性，构造了不满足欧氏刚性的反例。
Doubly Warped Product Finsler Manifolds with Some Non-Riemannian Curvature Properties
本文考虑了具有某些非黎曼曲率性质的双曲积Finsler流形。首先，我们研究了Berwald流形和迷向平均Berwald DWP-Finsler流形。然后证明了每一个真Douglas DWP-Finsler流形都是黎曼流形。我们证明了一个适当的DWP-流形是Landsbergian当且仅当它是Berwaldian。接着，我们证明了每一个相对迷向的Landsberg DWP-流形都是Landsberg流形。证明了相对迷向平均Landsberg曲积流形是弱Landsberg流形。最后证明了不存在任何局部对偶平坦的真DWP-Finsler流形。
If the flag curvature of a Finsler manifold reduces to sectional curvature, then locally either the Finsler metric is Riemannian, or the flag curvature is isotropic.
Lower bound of Ricci flow's existence time
设$(M^n, g)$是一个具有非负里奇曲率的紧致$n$ -dim（$n\geq 2$）流形，若$n\geq 3$，则假定$(M^n, g)\times \mathbb{R}$具有非负迷向曲率.证明了$(M^n, g)$上Ricci流存在时间的下界.这为一类闭Ricci流的最大存在时的一致下界提供了另一种证明，这一证明是由E. Cabezas-Rivas和B.威尔金在$Rc\geq 0$等假设下，得到了$n= 3$的内曲率估计.结合这些结果，我们证明了一大类$3$ -dim开流形上Ricci流的短时存在性，这类流形具有适当的耗尽覆盖和非负的里奇曲率.
Rigidity of Schouten Tensor under Conformal Deformation
本文研究了共形形变下 Schouten 张量有下界的度量刚性，利用该张量非负与 Ricci pinching 条件的等价性，为 Cheng 关于局部共形平坦流形的紧致性定理提供了简单证明。
We show that a closed four-manifold with $4\frac{1}{2}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both $\mathbb{CP}^2$ and $\mathbb{S}^3 \times \mathbb{S}^1$ have $4\frac{1}{2}$-nonnegative curvature operator of the second kind. In higher dimensions $n\geq 5$, we show that closed Riemannian manifolds with $4\frac{1}{2}$-positive curvature operator of the second kind are homeomorphic to spherical space forms. These results are proved by showing that $4\frac{1}{2}$-positive curvature operator of the second kind…
An almost-Schur type lemma for symmetric $(2,0)$ tensors and applications
在\cite{C}的前一篇文章中，我们将De Lellis和Topping关于具有非负Rcci曲率的闭流形的几乎Schur引理推广到任何闭流形.本文将上述结果推广到对称的$(2,0)$ -张量，并给出其应用，包括空间型中闭超曲面的$r$次中曲率和闭局部共形平坦流形的$k$数量曲率.
On Wilking's criterion for the Ricci flow
B Wilking最近证明了人们可以将一个在适当意义下是非负的曲率算子$C(S)$的里奇流不变锥与每个$Ad_{SO(n,\C)}$不变子集$S \subset {\bf so}(n,\C)$联系起来。对于复维数$n$的Kähler流形的曲率算子，考虑$Ad_{GL(n,\C)}$不变子集$S \subset {\bf gl}(n,\C)$。在这篇文章中，我们展示了：（i）若$S$是$Ad_{SO(n,\C)}$子集，则$C(S)$包含在具有非负迷向曲率的曲率算子的锥中;若$S$是$Ad_{GL(n,\C)}$子集，则$C(S)$包含在具有非负正交对分曲率的Kähler曲率算子的锥中. (ii)如果$S \subset {\bf so}(n,\C)$是闭合的$Ad_{SO(n,\C)}$不变子集，并且$C_+(S) \subset C(S)$表示在适当意义上为{ \it正}的曲率算子的锥，则两种可能性之一成立：（a）在$C_+(S)$中的任意两个曲率算子的黎曼流形的连通和也允许在$C_+(S)$中有一个曲率算子的度量（b）在$C_+(S)$中具有曲率算子的紧致黎曼流形$M$上的正规化里奇流收敛于常正截面曲率度量或常正全纯截面曲率度量，或者$M$是秩1对称空间.
Minimizers of the sharp Log entropy on manifolds with non-negative Ricci curvature and flatness
考虑标度不变的，尖锐的对数熵（泛函）介绍了Weissler \cite{W:1}非紧流形上的非负里奇曲率。它也可以被看作是一个锐化版本的Perelman的W熵\cite{P:1}在平稳的情况下。证明了它有极小元当且仅当流形与$\R^n$等距。利用这个结果，证明了一类具有非负里奇曲率的非紧流形等距于$\R^n$ .与\cite{An:1}、\cite{Ba:1}和\cite{BKN:1}中关于渐近平坦流形和渐近局部欧氏（ALE）流形的平坦性结果相比，它们关于曲率张量的衰减或积分条件被度量在无穷远处在$C^1$意义下收敛到欧氏度量的条件所取代.不需要度量的二阶条件。
On the tangent cone of Kähler manifolds with Ricci curvature lower bound
设$X$是满足$Ric(M_i)\geq -(n-1)$的点完备Kähler流形序列$(M^n_i, p_i)$的Gromov-Hausdorff极限，且其体积是非塌陷的.证明了在$X$的切锥上，存在一个同构于$\mathbb{R}$且等距作用的李群.此外，该作用在截面上局部自由。这将Cheeger-Colding的度量锥定理推广到Kähler的情形。我们还讨论了具有非负对分曲率的完备Kähler流形的一些应用。
New Curvature Operator of the Second Kind Conditions for Rigidity
本文通过第二类曲率算子建立Tachibana型刚性定理，证明维数≥5的完备Einstein流形在$C_n$-非负条件下具有常非负截面曲率，并推广至平移算子与对称空间分类。
Friedmann Cosmology and Almost Isotropy
在弗里德曼宇宙模型中，宇宙学家假设宇宙的类空切片是具有恒定截面曲率的黎曼流形。这一假设通过舒尔定理证明，类空宇宙是局部各向同性的。在这里，我们定义一个黎曼流形几乎局部各向同性的意义上，它允许弱引力透镜在所有方向和强引力透镜在局部角区域在大多数点。然后，我们证明了这样的流形是Gromov Hausdorff接近的长度空间$Y$，这是一个收集的空间型加入离散点。在本文中，我们定义了一个概念，我们称之为“指数长度空间”，并证明，如果这样的空间是局部各向同性的，那么它是一个空间型。
The Obata-Vétois argument and its applications
本文通过简化Vétois的Obata型论证，在闭共形爱因斯坦流形上确定了使$Q_4+aσ_2$为常数时流形必为爱因斯坦的区间$I_n$，并借此计算Yamabe型常数、证明尖锐Sobolev不等式，还部分回答了Branson和Ørsted关于泛函行列式极值的问题。
In this paper, we prove several structure theorems for locally conformally flat, positive Yamabe orbifolds and nonnegative scalar curvature, ALE manifolds. These two kinds of spaces can be related by conformal blow-up and conformal compactification. For the orbifolds, we prove that such orbifolds admit a manifold cover. For the ALE manifolds, the homomorphism of the fundamental group for the ALE space induced by the embedding of the ALE end is always injective. Using these properties, several classifications of such ALE manifolds and orbifolds are given in low dimensions. As an application to …
Concavity of spacetimes
本文研究了Finsler时空上时间分离函数的局部凹性，通过旗曲率非负性和未来胶囊凸性给出了新的刻画方法。
For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^{k}$ for any nonnegative integer $k$ away from a finite set of points, and the limit is Hamiltonian stationary in ${\mathbb{C}}^{n}$. We also obtain a theorem on extending Hamiltonian stationary Lagrangian submanifolds $L$ across a compact set $N$ of Hausdorff codimension at least …
A note on the boundedness of Riesz transform for some subelliptic operators
设$\M$是光滑连通非紧流形，具有光滑测度$μ$和满足$L1=0$的光滑局部次椭圆扩散算子$L$，且关于$μ$对称.我们证明了，如果$L$满足，与非负的曲率参数$ρ_1$，广义曲率不等式在\eqref{CD}下面，那么里斯变换是有界的$L^p (\bM)$的每一个$p>1$，即\[\| \sqrt{Γ((-L)^{-1/2}f)}\|_p \le C_p \| f \|_p, \quad f \in C^\infty_0(\bM), \]，其中$Γ$是\textit{carré du champ}与$L$。我们的结果特别适用于所有的Sasakian流形的水平Tanaka-Webster Ricci曲率是非负的，所有的卡诺集团与步骤2，广泛的子类的主丛黎曼流形的Ricci曲率是非负的。
$L^p_{loc}$ positivity preservation and Liouville-type theorems
本文在完备黎曼流形上研究了微分不等式 $-Δu + λu \geq 0$ 的 $L^{p}_{loc}$ 分布解，证明了在适当增长条件下解必非负，并探讨了其在完备极小子流形外在距离 $L^{p}$ 增长估计中的应用。
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