Closed Constant Curvature Space Curves

局部共形平坦黎曼流形中具有预定中曲率的闭超曲面

Closed hypersurfaces of prescribed mean curvature in locally conformally flat Riemannian manifolds

我们证明了存在光滑的封闭超曲面的规定中曲率同胚$S^n$小$n, n\le6$，提供了障碍。

Entropy in A Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces

Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifo…

Uniqueness of closed self-similar solutions to the Gauss curvature flow

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$ shrinks a strictly convex closed smooth hypersurface to a round sphere.

Li-Yau gradient estimates on closed manifolds under bakry-emery ricci curvature conditions

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.

The equi-affine and Frenet curvatures of curves in pseudo-Riemannian 2-manifolds

For an arbitrary nondegenerate curve in a pseudo-Riemann\-ian (including Riemannian) 2-manifold, we express the equi-affine curvature with the help of the Frenet (geodesic) curvature of this curve.

关于闭凸超曲面的H^k中曲率流的一个推广

On an extention of the H^k mean curvature flow of closed convex hypersurfaces

本文证明了闭凸超曲面的H^k（k为奇数且大于2）中曲率流可以在极大时间上扩张，只要对某些p，中曲率的总L^p积分是有限的

欧几里得平面单位切 bundle 上 Legendre 曲线的曲率等价性

Curvature equivalence for Legendre curves in the unit tangent bundle over Euclidean plane

本文研究了欧几里得平面单位切 bundle 上 Legendre 曲线的曲率等价性，探讨其性质并给出局部和全局分类。

Remark on an inequality for closed hypersurfaces in complete manifolds with nonnegative Ricci curvature

We give a simple proof of a recent result due to Agostiniani, Fogagnolo and Mazzieri.

Rigidity of Einstein metrics as critical points of quadratic curvature functionals on closed manifolds

In this paper, we prove some rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals on closed manifolds, characterized by some point-wise inequalities. Moreover, we also provide a few rigidity results that involve the Weyl curvature, the trace-less Ricci curvature and the Yamabe invariant, accordingly.

非正曲率对称空间中闭超曲面的全曲率估计

A total curvature estimate of closed hypersurfaces in non-positively curved symmetric spaces

本文证明了单连通非正曲率对称空间中闭超曲面的全曲率估计，并以此推得此类流形的等周不等式。

关于脐点曲线的主曲率线模式

On the Patterns of Principal Curvature Lines around a Curve of Umbilic Points

本文研究光滑曲面脐点曲线附近的主曲率线的性质。

平面曲线非局部曲率的逼近

Approximating the Nonlocal Curvature of Planar Curves

本文研究平面曲线非局部曲率的分解与逼近，通过将定义扩展到曲线外点并利用不完全 beta 函数给出线段公式，证明了 Hölder 正则曲线的非局部曲率可由线性插值样条的非局部曲率逼近。

Closed minimal surfaces of high Morse index in manifolds of negative curvature

We show that compact Riemannian three-manifolds with negative sectional curvature possess closed minimal surfaces of arbitrarily high Morse index.

The geodesic total curvature of spherical curves

The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of some new integral-geometric formulas concerning both the Euclidean and geodesic total curvature of spherical curves.

Ancient Solutions to Curve Shortening with Finite Total Curvature

We construct ancient solutions to Curve Shortening in the plane whose total curvature is uniformly bounded by gluing together an arbitrary chain of given Grim Reapers along their common asymptotes.

$\mathbb{R}^{2}$ 中配备典范线性联络的曲线的曲率

Curvatures of curves in $\mathbb{R}^{2}$ endowed with a canonical linear connection

研究配备典范线性联络的 $\mathbb{R}^2$ 中平面曲线的几何，引入切向和测地曲率，证明给定曲率曲线的存在唯一性定理，并分类常曲率曲线及推广曲率为弧长线性函数的曲线。

Hardy type inequalities on closed manifolds via Ricci curvature

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.

负曲率幂次流的闭自相似解

Closed self-similar solutions to flows by negative powers of curvature

本文研究翘曲乘积流形中负曲率幂次流的闭自相似解，证明其必为切片，关键引入新辅助函数。

The equi-affine curvatures of curves in 3-dimensional pseudo-Riemannian manifolds

In this paper, the Cartan frames and the equi-affine curvatures are described with the help of the Frenet frames and the Frenet curvatures of a non-null and non-degenerate curve in a 3-dimensional pseudo-Riemannian manifold. The constancy of the Frenet curvatures of such a curve always implies the constancy of the equi-affine curvatures. We show that the converse statement does not hold in general. Finally, we study the equi-affine curvatures of null curves in 3-dimensional Lorentzian manifolds, and prove that they are related to their pseudo-torsion.

On Closed Manifolds with Harmonic Weyl Curvature

We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key ingredients are new Bochner-Weitzenböck-Lichnerowicz type formulas for the Weyl tensor, which are generalizations of identities in dimension four.

常曲率比曲线

Curves with constant curvature ratios

${\mathbb R}^n$中的曲线，其两个连续曲率之间的比率是常数，其特征在于它们的切线指标是平坦环面中的测地线。对于$n= 3,4$，也研究了这种球面曲线，并与球面中的内禀螺旋线进行了比较。

正则曲线的曲率及其导数的欧氏不变量

The Curvatures of Regular Curves and Euclidean Invariants of their Derivatives

众所周知的公式表示的曲率和挠曲线在$R^3$在其导数的欧几里德不变量。对于$R^n$中的所有曲线的曲率，我们都得到了这种表达式。因此，$R^n$中的曲线由其n阶导数的范数确定直到等距。我们将这些观察推广到任意黎曼流形中的曲线。

到具有常数sectional curvature的黎曼流形中的k-调和曲线

k-harmonic curves into a Riemannian manifold with constant sectional curvature

J.Eells和L. Lemaire引入了k-调和映射，T. Ichiyama，J. Inoguchi和H.Urakawa给出了一阶变分公式。本文将$3$ -调和曲线的常微分方程描述到具有常截面曲率的黎曼流形中，证明了双调和曲线是k-调和曲线$(k\geq 2)$ .

Whitney-Graustein Homotopy of Locally Convex Curves via a Curvature Flow

Let $X_0, \widetilde{X}$ be two smooth, closed and locally convex curves in the plane with same winding number. A curvature flow with a nonlocal term is constructed to evolve $X_0$ into $\widetilde{X}$. It is proved that this flow exits globally, preserves both the local convexity and the elastic energy of the evolving curve. If the two curves have same elastic energy then the curvature flow deforms the evolving curve into the target curve $\widetilde{X}$ as time tends to infinity.

Realizing metrics of curvature $\leq -1$ on closed surfaces in Fuchsian anti-de Sitter manifolds

We prove that any metric with curvature $\leq -1$ (in the sense of A. D. Alexandrov) on a closed surface of genus $>1$ is isometric to the induced intrinsic metric on a space-like convex surface in a Lorentzian manifold of dimension $(2+1)$ with sectional curvature $-1$. The proof is done by approximation, using a result about isometric immersion of smooth metrics by Labourie--Schlenker.

具有常非负数量曲率的5维欧氏球面中闭极小超曲面的一个内蕴刚性定理

An intrinsic rigidity theorem for closed minimal hypersurfaces in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature

设M是5维欧氏球面中具有常非负数量曲率的闭极小超曲面。证明了若所有主曲率的立方之和与不同主曲率的个数为常数，则M是等参的.此外，还给出了M的第二基本型的平方长度的所有可能值.这一结果为陈省身猜想提供了另一个支持证据。

Existence of curves with constant geodesic curvature in a Riemannian 2-sphere

We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.

曲线曲面的中曲率演化

Evolution of curves and surfaces by mean curvature

本文介绍了中曲率流，一些关于它的发现，以及一些尚未解决的问题。

位势为球面曲线曲率的周期薛定谔算子的几何估计

A geometric estimate for a periodic Schrödinger operator whose potential is the curvature of a spherical curve

我们从下面的几何数据估计的周期Sturm-Liouville算子$- 4 d^2/ds^2 + κ^2 (s)$的特征值与潜在的闭曲线的曲率。

A-简单奇点R,0→R²,0的曲率准则及其平行曲线

Curvature criteria of A-simple singularities R,0-> R^2,0 and their parallel curves

本文提出了一种确定A-简单奇点R,0→R²,0曲率准则的方法，并探讨了其平行曲线的奇点类型。

具有指定曲率的平面闭曲线

Planar closed curves with prescribed curvature

本文利用变分方法并结合Struwe的单调性技巧，通过引入参数获得有界Palais-Smale序列，从而证明某些曲率函数类下具有指定曲率的平面闭曲线的存在性。

Remarks on the boundary curve of a constant mean curvature topological disc

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.

On bounds of entropy and total curvature for ancient curve shortening flows

Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are one-dimensional mean curvature flows. As an application, we give a short proof of the uniqueness of tangent flow at infinity of an ancient smooth complete non-compact curve shortening flow with finite entropy embedded in $\mathbb{R}^2$.

Gaustral-Kronecker曲率为零的$\Bbb{S^4}(1)$的非闭极小超曲面

Non-closed minimal hypersurfaces of $\Bbb{S^4}(1)$ with identically zero Gauß-Kronecker curvature

在不对$M^3$的紧性作任何假设的情况下，给出了单位4-球面$\Bbb{S}^4(1)$中Gaussian-Kronecker曲率函数为零的极小超曲面$M^3$的部分局部刻划.

Curvatures of moduli space of curves and applications

In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g, ω_{WP})$ of curves with genus $g>1$ has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positive Riemannain curvature operator and also non-positive complex sectional curvature. As applications, we prove that any submanifold in $M_g$ which is totally geodesic in $A_g$ with finite volume must be a ball quotient.

Rigidity of closed metric measure spaces with nonnegative curvature

We show that one-dimensional circle is the only case for closed smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature whose spectrum of the weighted Laplacian has an optimal positive upper bound. This result extends the work of Hang-Wang in the manifold case (Int. Math. Res. Not. 18 (2007), Art. ID rnm064, 9pp).

渠道曲面上参数曲线与曲率线的关系

The relation between parameter curves and lines of curvature on canal surfaces

耳道表面是具有变化半径的移动球体的包络，由其中心的轨迹C（t）（脊曲线）和半径函数r（t）定义。在本文中，我们调查时，参数曲线的渠道表面也是曲率线。最后，对于特殊的脊柱曲线，我们得到了椎管曲面的半径函数。

Submersions and curves of constant geodesic curvature

Considering Riemannian submersions, we find necessary and sufficient conditions for when sub-Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvatures. We describe a canonical extension of the sub-Riemannian metric and study geometric properties of the obtained Riemannian manifold. This work contains several examples illustrating the results.

曲率为零的环面曲线

Torus Curves With Vanishing Curvature

设T是R^3中的标准旋转环面，半径为b和1，0<b<1。设α是T上的（p，q）环面曲线.我们证明了对于T的可变半径b=p^2/（p^2+q^2），α上存在曲率为零的点。曲线α对于所有其他b值都具有非零曲率。此外，对于这个b值，α上恰好有q个曲率为零的点。

On the construction of closed nonconvex nonsoliton ancient mean curvature flows

We construct closed, embedded, ancient mean curvature flows in each dimension $n\ge 2$ with the topology of $S^1 \times S^{n-1}$. These examples are not mean convex and not solitons. They are constructed by analyzing perturbations of the self-shrinking doughnuts constructed by Drugan and Nguyen (or, alternatively, Angenent's self shrinking torus when $n =2$)

Conformal Killing $L^{2}-$forms on complete Riemannian manifolds with nonpositive curvature operator

We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^{2}-$form. Moreover, we prove vanishing theorems for closed and co-closed conformal Killing $L^{2}-$forms on some complete Riemannian manifolds.

用闭曲线的曲率逼近R^n的周期函数

Approximations of periodic functions to R^n by curvatures of closed curves

我们证明了对于任意n个实周期函数f_1，.，f_n具有相同的周期，使得对于i<n，f_i>0，且实数e >0，则在R^{n+1}中存在曲率为k_1，.的闭曲线，k_n，使得|k_i（t）-f_i（t）|对于所有的i和t。这既不适用于双曲空间H^{n+1}中的闭曲线，也不适用于R^{n+1}中的闭曲线的参数族。

具有闭曲率线的极小曲面

Minimal surfaces with closed curvature lines

研究有限总曲率的非定向极小曲面，其端面由闭曲率线覆盖，证明此类曲面的刚性，排除单端面的存在。

Curves of Constant Curvature and Torsion in the 3-Sphere

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded in the three-sphere. This behavior is very different from that of helices in three-dimensional Euclidean space, which also have constant curvature and torsion.

常中曲率曲面的达布变换和谱曲线

Darboux transforms and spectral curves of constant mean curvature surfaces revisited

研究了常中曲率曲面的达布变换的几何性质，并利用这些达布变换得到了常中曲率环面的代数几何表示.我们发现，所有的达布变换的CMC环面的空间有一个自然的子集，这是一个代数曲线（称为频谱曲线）和所有的达布变换表示的频谱曲线上的点是CMC环面。使用达布变换得到的谱曲线不是双有理的，但具有与使用更传统的可积系统方法得到的谱曲线相同的归一化。

闭合流形的欧拉特征与近非负曲率算子

Euler Characteristic of Closed Manifolds with Almost Nonnegative Curvature Operator

研究具有近非负曲率算子的闭合流形，解决Herrmann-Sebastian-Tuschmann关于其欧拉特征符号的问题，证明在特定条件下欧拉特征非负，并在无限基本群条件下证明欧拉特征、签名及$\widehat{A}$-特征 vanish。

广义旋转型闭中曲率自收缩曲面

Closed Mean Curvature Self-Shrinking Surfaces of Generalized Rotational Type

对于每个$n\geq 2$，我们构造了一个新的闭嵌入中曲率自收缩超曲面在$\mathbb{R}^{2n}$。这些自收缩体与$S^{n-1}\times S^{n-1}\times S^1$同形，并且是$SO(n)\times SO(n)$不变的。该方法的灵感来自Hsiang的构造，这些曲面推广了自收缩环面，它与Angenent构造的$S^{n-1}\times S^1$同构。

Uniqueness of two-convex closed ancient solutions to the mean curvature flow

In this paper we consider closed non-collapsed ancient solutions to the mean curvature flow ($n \ge 2$) which are uniformly two-convex. We prove that any two such ancient solutions are the same up to translations and scaling. In particular, they must coincide up to translations and scaling with the rotationally symmetric closed ancient non-collapsed solution constructed by Brian White in (2000), and by Robert Haslhofer and Or Hershkovits in (2016).

对数线性密度平面上常加权曲率曲线的分类

The classification of constant weighted curvature curves in the plane with a log-linear density

本文对平面上具有对数线性密度的常权曲率曲线类进行了分类，换句话说，对$\Bbb R^2.$中具有常强迫项的所有行进曲线锋进行了分类。分类给出了一些有趣的现象和结果，包括：当曲线的加权曲率（或等价地，行进曲锋的强迫项）趋于无穷大，这是文[13]中一个主要结果的简单证明，也是关于有密度平面上等周问题$e^y.$的一些已知事实

具有正标量曲率的闭流形的直径估计

Diameter estimate for closed manifolds with positive scalar curvature

本文研究正标量曲率单连通闭黎曼流形的直径上界，利用标量曲率积分、Yamabe常数和维数建立估计，并证明共形浸入球面时可去除Yamabe常数依赖且积分幂次尖锐。

渐近非负曲率流形中闭超曲面的一个锐利几何不等式

A sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature

本文利用黎曼几何标准比较方法，在渐近非负曲率的完备非紧流形中为闭超曲面建立了锐利几何不等式，并简化了非负里奇曲率情形下Willmore型不等式的证明。

Closed almost-Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler

We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature $k\geq 0$ with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for $k<0$ if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern-Weil theory implies useful integral formulas, which are then combined with results from Seiberg--Witten theory.

Examples of manifolds of positive Ricci curvature with quadratically nonnegatively curved infinity and infinite topological type

In this paper, we construct a complete n-dim Riemannian manifold with positive Ricci curvature, quadratically nonnegatively curved infinity and infinite topological type. This gives a negative answer to a conjecture by Jiping Sha and Zhongmin Shen in the case of n greater than or equal to 6.

$\mathbb{S}^{m+1}$中具有两个不同主曲率的闭Moebius型超曲面的分类

Classification of Hypersurfaces with Two Distinct Principal Curvatures and Closed Moebius Form in $\mathbb{S}^{m+1}$

设$x$是$(m+1)$维单位球面$\mathbb{S}^{m+1}(m\geq3)$中的$m$维无脐超曲面.其中一个重要的问题是对具有两个不同主曲率的超曲面进行分类。本文对具有两个不同主曲率和闭Moebius型的超曲面进行了分类和显式表示，进而对维数大于3的共形平坦超曲面进行了刻画和分类。