Vitali Kapovitch, Alexander Lytchak
We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.
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In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [24] to prove to prove a weak chord arc type result for these disks. In Section 4 we apply this weak chord arc result to obtain an intrinsic version of the one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature. In a natural sense, these one-sided curvature estimates generalize respectively, the extrinsic …
We prove several sharp one-sided pinching estimates for immersed and embedded hypersurfaces evolving by various fully nonlinear, one-homogeneous curvature flows by the method of Stampacchia iteration. These include sharp estimates for the largest principal curvature and the inscribed curvature ('cylindrical estimates') for flows by concave speeds and a sharp estimate for the exscribed curvature for flows by convex speeds. Making use of a recent idea of Huisken and Sinestrari, we then obtain corresponding estimates for ancient solutions. In particular, this leads to various characterisations of…
On the dimension of the singular set of perimeter minimizers in spaces with a two-sided Ricci curvature bound
本文研究非塌缩 Ricci 极限空间中周长极小元的奇异集，证明其 Hausdorff 维数不超过环境维数减五，并表明该界尖锐。
We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of Huisken--Sinestrari and the cylindrical estimate of Huisken--Sinestrari. Namely, we show that the curvature of the solution pinches onto the convex cone generated by the curvatures of any shrinking cylinder solutions admitted by the initial data. For example, if the initial data is $(m+1)$-convex, then the curvature of the solution pinches onto the convex hull of the…
Two-sided non-collapsing curvature flows
最近证明了由Weingarten映射的凹（凸）度一次齐次函数给出速度的欧氏超曲面流的嵌入解是内（外）非坍缩的.这些结果随后被推广到球面和双曲空间中的超曲面流。在本文的第一部分中，我们证明了局部凸的解决方案是外部非崩溃的速度函数比以前考虑的更大的一类，更准确地说，我们表明，以前的结果保持时，速度函数的凸性放宽到反相。我们注意到，一个大类的凹速度函数，满足逆的。因此，我们得到了一个大类的双边非塌陷流，而以前的双边非塌陷只知道的中曲率流。在本文的第二部分中，我们证明了双边非塌陷的实用性与一个简单的证明收敛的紧凑，凸超曲面的圆形点。非崩溃估计的证明与前面提到的结果类似，因为我们证明了外球曲率是线性化流的粘性超解。新的成分是以下观察：由于在推导粘度不等式时提供上支持的函数是在$M\times M$上定义的（或在“边界情况”下定义为$TM$），而外部球曲率和线性化流动方程仅取决于第一个因子，因此我们有选择的自由，在计算中包括来自额外方向的二阶导数。最优选择与反凹速度函数类密切相关。
We study a notion of relative entropy motivated by self-expanders of mean curvature flow. In particular, we obtain the existence of this quantity for arbitrary hypersurfaces trapped between two disjoint self-expanders asymptotic to the same cone. This allows us to begin to develop the variational theory for the relative entropy functional for the associated obstacle problem. We also obtain a version of the forward monotonicity formula for mean curvature flow proposed by Ilmanen.
Complete stable minimal hypersurfaces in positively curved 4-manifolds
本文证明在四维流形中非负截面曲率与正标量曲率结合可迫使完备双侧稳定极小超曲面具有刚性，并给出其他曲率条件下刚性失效的反例。
Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $Σ$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $Σ$ does not separate $M$ then $Σ$ is totally geodesic and $M\setminusΣ$ is isometric to the Riemannian product $Σ\times(a,b)$, and if $Σ$ separates $M$ then the map $i_*:π_1(Σ)\rightarrow π_1(M)$ induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature $H\geq(n-1)\sqrt{k}$ in a manifold of Ricci curvature $Ric_M\geq-(n-1)k,k>0$, and for a free boundar…
Collapsing with a lower bound on the curvature operator
Cheeger和Gromov证明了F-结构与具有双侧曲率界的坍缩有关。我们定义了纤维F-结构，并将Cheeger-Gromov的一些结果推广到曲率算子具有下界的坍缩情形.
Dimension Bound of Singular Set of One-Phase Free Boundary Problems in Spaces with Two-Sided Ricci Bound
本文研究了具有双侧里奇曲率界的n维流形非塌陷极限中一相自由边界问题，证明自由边界奇异集的Hausdorff维数不超过n-5，且此界是最佳的。
On the moduli of constant mean curvature cylinders of finite type in the 3-sphere
我们证明了单侧Alexandrov嵌入常中曲率圆柱的有限型在3-球面是旋转曲面。这证实了Pinkall和Sterling的一个猜想，即在3-球面中唯一嵌入的常数中曲率环面是旋转的。
We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds on the Ricci tensor and a lower injectivity radius bound we obtain a uniform estimate on the change of the sectional curvature. Actually, our result holds for any metric with a uniform bound on the $W^{2,p}$-harmonic radius.
A remark on the symmetries of the Riemann curvature tensor
这是Riemann曲率张量的对称性可以（有效地）表示为SL（2）-不变性的基本观察。
Remark on the Calabi flow with bounded curvature
本文证明了：若Calabi流的曲率张量沿Calabi流一致有界，且Mabuchi能量适当，则Calabi流收敛于一个常数的数量曲率度量.
We prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension $n=2$. This then implies that the mean curvature flow exists for all time and converges to a translating solution.
We give a simple proof of a recent result due to Agostiniani, Fogagnolo and Mazzieri.
Remark about scalar curvature on certain noncompact manifolds
本文研究了非紧流形内部完整黎曼度量的标量曲率问题，给出了排除非负标量曲率的充分条件，并在高维情况下提供了正标量曲率度量的端点例子。
In this short note, we prove the existence of constant scalar curvature Kähler metrics on compact Kähler manifolds with semi-ample canonical bundles.
In this paper we address the relationship between Gromov-Hausdorff limits and intrinsic flat limits of complete Riemannian manifolds. In \cite{SormaniWenger2010, SormaniWenger2011}, Sormani-Wenger show that for a sequence of Riemannian manifolds with nonnegative Ricci curvature, a uniform upper bound on diameter, and non-collapsed volume, the intrinsic flat limit exists and agrees with the Gromov-Hausdorff limit. This can be viewed as a non-cancellation theorem showing that for such sequences, points don't cancel each other out in the limit. Here we prove a similar no-cancellation theorem, rep…
Remark about the spectrum of the p-form Laplacian under a collapse with curvature bounded below
给出了曲率有界的p-型Laplacian在某类坍缩中小的正特征值个数的一个下界.
Remarks on a scalar curvature rigidity theorem of Brendle and Marques
本文对\cite{BrendleMarques}中Brendle和Marques的数量曲率刚性定理给出了一些注记.主要结果是Brendle和Marques定理在大于[2]中规定的测地线球上成立。
A remark concerning universal curvature identities on 4-dimensional Riemannian manifolds
我们将使用不同于Gilkey、Park和Sekigawa \cite{GPS}所用的方法证明四维黎曼流形曲率恒等式的普适性。
Aloff-Wallach Spaces: Volumes, Curvatures, Injectivity Radii
本文给出了Aloff-Wallach空间$W(p,q),$体积的双边界，计算了空间$W(n,n+1),$的最大和最小截面曲率，并利用这些信息估计了$W(p,q)$的内射半径：我们得到了$W(p,q)$的内射半径的上界和$W(n,n+1).$的内射半径的下界
Remark about scalar curvature and Riemannian submersions
本文研究了具有光滑测度的黎曼流形的修正数量曲率函数。给定一个黎曼浸没的纤维运输是测量保持常数，我们表明，修改后的数量曲率的基础上有界的全空间和纤维的数量曲率。给出了光滑极限空间中有界曲率坍缩产生的数量曲率的一个应用。
Cross Curvature Flow on Locally Homogeneous Three-manifolds (II)
本文研究了局部齐次三维流形上的正交叉曲率流。我们描述了这些流量的长期行为。我们结合这与早期的结果有关的负交叉曲率流的渐近行为来描述局部齐次3-流形上的交叉曲率流的最大解的双边行为。我们证明了局部齐次3-流形上的正交叉曲率流产生一个Heisenberg型次黎曼几何。
A remark on the fourth order Q curvature on manifolds with dimension at least 5
本文针对维数不小于5、Yamabe不变量为正的黎曼流形，探讨其四阶Q曲率的二分性问题。
Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature
证明了Fischer-Colbrie和Schoen的一个猜想的如下版本：如果M是一个具有非负数量曲率的完备黎曼3-流形，其中包含一个在其合痕类中面积最小的双侧环面S，则M是平坦的.这是从一个本地版本中得出的文件。
Ricci curvature bounds for Statistical Submersions
本文从直接里奇曲率视角，结合统计浸没的里奇曲率关系与Hineva代数不等式，首次得到统计浸没的希内瓦型下里奇曲率估计，为统计浸没提供统一里奇曲率方法并获得双侧里奇曲率界。
Remark on semi-positive holomorphic sectional curvature and quasi-negative $k$-Ricci curvature
本文记录了两个注记：对于紧致Kähler流形，半正全纯截面曲率时MRC纤维化的有理维数等于非真正平坦方向的数量；拟负$k$-Ricci曲率或更一般的拟负混合曲率$C_{a,b}$时，典范丛是丰富的。
We give a quick direct proof of Wu's theorem on the subadditivity of Hermitian metrics of negative holomorphic sectional curvature.
We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a condition for the surface to be totally umbilic in terms of the normal curvature.
We study non-collapsed Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson-Sun, who considered non-collapsed limits of polarized Kähler manifolds with two-sided Ricci curvature bounds.
Remarks on Kähler orbifolds of non-negative Ricci curvature
本文证明了凯勒轨道折面的非负曲率版本的 Kobayashi 定理，证明了非负曲率的紧凯勒轨道折面在满足一定条件时是单连通的。
A remark on capillary surfaces in a 3-dimensional space of constant curvature
本文将J. Choe关于毛细管曲面的一个定理推广到任意三维常曲率空间。本文的主要工具是H.由于S.- S. Chern和J. Choe的两个指标引理。
Remarks on an inequality involving the normal scalar curvature
研究了实空间型中子流形的一个涉及数量曲率、法数量曲率和中曲率的点态不等式.我们把它转化为一个代数问题，使我们能够证明它的一个稍微弱的版本。我们还证明了猜想的某些类型的子流形的$\mathbb C^n$。
In this short note, using an argument by Munteanu and Wang, we provide an alternative proof of the fact first obtained by Lei Ni that shrinking gradient Kähler-Ricci solitons with positive bisectional curvature must be compact.
We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.
In [11], I. M. Gelfand, V. Retakh, and M. Shubin defined the symplectic sectional curvature of a torsion-free connection preserving a symplectic form. The present article defines the corresponding notion of constant symplectic sectional curvature and characterizes this notion in terms of the curvature tensor of the symplectic connection and its covariant derivatives. Some relations between various more general conditions on the symplectic sectional curvature and the geometry of the symplectic connection or that induced on a symplectic submanifold are explored as well.
A Remark on Compact Minimal Surfaces in $S^5$ with Non-Negative Gaussian Curvature
在本文中，我们使用接触角的概念对$S^5$中具有非负高斯曲率的紧致极小曲面进行分类。
Stable Minimal Hypersurfaces in Positively Curved $4$-Manifolds
该研究在正曲率4维流形中，证明满足特定曲率条件的稳定极小超曲面必为全测地，还构造了反例，相关刚性证明结合了谱分裂理论等方法。
We provide a general Böchner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero radial Weyl curvature and satisfies a suitable pinching condition. Moreover, we classify $V$-static spaces with non-negative sectional curvature.
We extend to higher codimension earlier characterization of the equatorial disk and the critical catenoid by a pinching condition on the length of their second fundamental form among free boundary minimal surfaces in the three dimensional Euclidean ball due to L. Ambrozio and I. Nunes.
We show a connection between the $CDE'$ inequality and the $CDψ$ inequality. In particular, we introduce a $CD_ψ^φ$ inequality as a slight generalization of $CDψ$ which turns out to be equivalent to $CDE'$ with appropriate choices of $φ$ and $ψ$. We use this to prove that the $CDE'$ inequality implies the classical $CD$ inequality on graphs, and that the $CDE'$ inequality with curvature bound zero holds on Ricci-flat graphs.
In this note we characterize compact hypersurfaces of dimension $n\geq 2$ with constant mean curvature $H$ immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when $n\geq 3$ and $H\neq 0$, they are locally contained in a rotational hypersurface. In dimension two, the integral pinching condition reduces to a topological assumption and we recover the classical Hopf-Chern result.
Some remarks on Finsler manifolds with constant flag curvature
本文是对具有常正旗曲率的Finsler流形几何的四个松散相关的评论的阐述。<p>第一个注记是在这样一个流形的测地线空间上存在一个典型的Kahler结构。<p>第二点是，有一种自然的方法可以从复射影n-空间中适当一般位置的超曲面中构造一个（不一定是完备的）具有常正旗曲率的芬斯勒n-流形。<p>第三个注记是2-球面上常曲率的Finsler度量可用其测地线空间上的黎曼度量和1-形式来描述。特别地，这允许使用2-球面上正高斯曲率的任何（黎曼）Zoll度量来构造2-球面上常正曲率的全局Finsler度量。<p>第四点是关于维数n+1>2中具有常正旗曲率的（局部）Finsler度量空间的一般性。证明了这样的度量依赖于n（n+1）个n+1元的任意函数，并且这样的度量自然地对应于2n-流形上的某些无挠S^1 x GL（n，R）-结构.作为一个副产品，它被发现，这些群体确实出现的holonomy的挠自由仿射联络在2n维，迄今未被怀疑的现象。
Remarks on scalar curvature of Yamabe solitons
本文考虑Yamabe孤子的数量曲率。特别地，我们证明了在自然条件和非正里奇曲率下，任何完备Yamabe孤子都具有常数数量曲率，即它是Yamabe度量.我们还证明了完全非紧Yamabe孤子的里奇曲率在无穷远处的二次衰减具有非负的数量曲率。给出了Kazdan-Warner条件的一个新的证明。
Remarks on the $\mathbf Q$-curvature flow
这个简短的说明的主要目的是指出，$S^n$上规定的$\mathbf Q$ -曲率问题的负梯度流可以扩展到处理$\mathbf Q$ -曲率候选者$f$可能改变符号的情况。
A remark on Kähler metrics of constant scalar curvature on ruled complex surfaces
我们指出如何在理论的常数数量曲率凯勒度量的一些最新进展，可以用来澄清这种度量的存在性问题的特殊情况下，几何规则复杂的表面。
We prove that the total CR $Q$-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the $P^\prime$-operator and the CR invariance of the total $Q^\prime$-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.
Existence of regular neighborhoods for H-surfaces
本文研究了n维流形中完备的常中曲率超曲面的整体几何。更确切地说，我们给出了暗示这类曲面的适当性的条件，并证明了在某些n-流形中，某些常数中曲率超曲面存在固定大小的单侧正则邻域。
Comparison and Rigidity Theorems in Semi-Riemannian Geometry
利用黎曼张量的单侧界，将平行超曲面的形状算子所满足的Riccati方程的比较理论推广到任意指数的半黎曼流形，在黎曼流形中，黎曼张量的单侧界对应于截面曲率的单侧界.从二维刚性结果出发，利用归纳技巧，证明了任意指标的半黎曼流形的一类新的间隙型刚性定理，推广了Gromov和Greene-Wu首先给出的结果.作为应用，我们证明了具有常曲率单连通端的半黎曼流形的刚性结果。
Moduli spaces of critical Riemannian metrics with L^{n/2} norm curvature bounds
我们考虑紧流形上极值Kähler度量的模空间.证明了在总体积有界、$\Riem$上的$L^{n\over2}$ -范数有界和Sobolev常数有界的条件下，这个模空间可以通过包含具有200个奇点的（约化）orbifolds而被紧化.我们的大多数结果通过某些其他类别的关键黎曼度量。
We exhibit a curious link between the Quadratic Orthogonal Bisectional Curvature, combinatorics, and distance geometry. The Weitzenböck curvature operator, acting on real (1,1)--forms, is realized as the Dirichlet energy of a finite graph, weighted by a matrix of the curvature. These results also illuminate the difference in the nature of the Quadratic Orthogonal Bisectional Curvature and the Real Bisectional Curvature.
In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. First, we prove that if the ratio $λ$ of the maximum and the minimum of the holomorphic sectional curvatures $< 2$, then there exists a positive constant $δ>\frac{29(λ-1)}{\sqrt{(48-24λ)^{2}+(29λ-29)^{2}}}$ such that $\cosα\geqδ$ is preserved along the flow, improving the main theorem in [LY]; Secondly, as similar as the main theorem in [HL0], we prove that when $\cosα$ is close to $1$ enough, then the symplectic mean curvature flow exists for long time and converges to a h…
We show that every one-harmonic map, in the sense of Trapani and Valli, from a Hadamard surface of pinched negative curvature to $\mathbb{H}^2$ has image the interior of the convex hull of a subset of $\partial_\infty\mathbb{H}^2$. The proof relies on Minkowski geometry, by interpreting one-harmonic maps as the Gauss maps of convex surfaces.
Remarks on prescribed scalar curvature functions
本文针对一般闭流形上带规定系数的Yamabe型半线性方程正解问题，给出最优存在性充分条件，并将其应用于标量平坦流形，解决了规定标量曲率问题中的一个长期疑问。
Stable Bernstein Problem in certain positively curved manifolds
该研究针对特定正曲率流形的稳定Bernstein问题，证明了不同曲率条件下完备双侧稳定极小浸入的全测地性与不存在性，推广了已有低维结果至高维情形。
Conformal structures and rigidity of complete stable minimal hypersurfaces
本文研究不同曲率条件下完备稳定极小超曲面的共形结构与刚性，证明非负数量曲率流形中不存在共形等价于非正数量曲率有界区域的非紧稳定极小超曲面，并给出紧致情形及扭曲积流形中的稳定性与刚性结果。
This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to provide a proof of our conjecture, mentioned in [TY], where the assumption of negative holomorphic sectional curvature is dropped to quasi-negative. We should note that a solution to our conjecture is also provided by Diverio-Trapani [DT]. Both proofs depend on our argument in [WY16]. But our argument here makes use of the argument given by the second author and…
In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry-Émery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our diameter estimate improves previous ones obtained by Wei and Wylie (J. Differential Geom. 83, 377--405, 2009) and Limoncu (Math. Z. 271, 715--722, 2012). As an application, we shall give an upper diameter bound for compact Ricci solitons in terms of the maximum value of the scalar curvature. By using such a diameter bound, we shall provide some new sufficient condit…
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