Evolution of curves and surfaces by mean curvature
Brian White
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。
This article describes the mean curvature flow, some of the discoveries that have been made about it, and some unresolved questions.
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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.
Non-collapsing for nearly spherical closed curves under power curvature flow
利用两点方法证明了幂曲率流下\mathbb{R}^2中近似球形闭凸曲线的演化是不坍缩的。
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。
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。
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。
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.
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.
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。
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.
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。
We use ODEs and symmetry arguments to construct closed constant curvature space curves, first on cylinders, next on tori, at last with the Frenet-Serret equations.
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。
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$(k\geq 2)$ .
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.
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.
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。
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。
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.
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 $\mathbb{R}^2$.
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.
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。
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.
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。
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.
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。
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
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
Visualization of curvature on curve and surface by tangential angle parametrization
本文提出基于切向角参数的统一曲率可视化方法，通过在等切向角增量处放置标记或绘制曲率线，自然揭示平面曲线的拐点与顶点以及旋转曲面的脊线和抛物曲线，无需任意调参即可为分析与设计提供几何洞察。
Curvature Functionals for Curves in the Equi-Affine Plane
在给出了这类泛函的一般变分公式后，本文进一步研究了面积约束下等仿射曲率积分的临界点和全仿射弧长的临界点。
Finiteness of the total first curvature of a non-closed curve in $\mathbb{E}^{n}$
考虑$\mathbb{E}^n$中的正则平滑曲线，使得其坐标分量是$n$阶微分方程$ y^{(n)} (x) - y(x) = 0 ,$ $x \in \mathbb{R} $的基本解.我们证明了这条曲线的总第一曲率对于奇数$n$是无限的，对于偶数$n$是有限的。
Curves in R^n with finite total first curvature arising from the solutions of an ODE
我们利用一个n阶至少为2的实常微分方程的解集来构造R^n中的一条平滑曲线C。我们描述了当C是具有有限总第一曲率的无限长的真嵌入时。
Planar closed curves with prescribed curvature
本文利用变分方法并结合Struwe的单调性技巧，通过引入参数获得有界Palais-Smale序列，从而证明某些曲率函数类下具有指定曲率的平面闭曲线的存在性。
On the motion of a curve by its binormal curvature
我们提出了一个弱公式的副法向曲率流的曲线在$\R^3.$这个公式是足够广泛的积分流作为初始数据，并足够强大的弱-强唯一性举行，只要不发生自相交。我们还证明了一个整体存在定理在该框架。
Constant curvature curves in dual affine and dual Lorentz-Minkowski planes
本文研究了双仿射和双洛伦兹-闵可夫斯基平面中曲率恒为双常数的曲线性质，揭示了其几何特征和分类方法。
Criteria and Curvatures for Singularities of Finite Multiplicities of Curves in $\boldsymbol{R}^N$
本文提出构建有限乘数曲线奇点准则的系统方法，给出乘数二、三、四的显式准则，包括仅出现在三维及以上空间的特定尖点，并推广标准化曲率函数和尖点曲率到R^N中的奇异曲线，重新解释Fukui在R^N中有限乘数曲线的存在唯一性定理。
We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also, the angle of the parallel transport is a function with bounded variation, and its total variation is equal to an energy functional that depends on the "tangential" component of the derivative of the tantrix of the curve. We show that the total intrinsic curvature of irregular curves agrees with such an energy functional. By exploiting isometric embeddings, the…
In this paper we consider surfaces with one or two families of spherical curvature lines. We show that every surface with a family of spherical curvature lines can locally be generated by a pair of initial data: a suitable curve of Lie sphere transformations and a spherical Legendre curve. We then provide conditions on the initial data for which such a surface is Lie applicable, an integrable class of surfaces that includes cmc and pseudospherical surfaces. In particular we show that a Lie applicable surface with exactly one family of spherical curvature lines must be generated by the lift of …
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。
If a piece of the contour of a picture is missing to the eye vision, then the brain tends to complete it using some kind of sub-Riemannian geodesics of the unit tangent bundle of the plane, R2xS1. These geodesics can be obtained by lifting extremal curves of a total curvature type energy in the plane. We completely solve this variational problem, geometrically. Moreover, we also show a way of constructing rotational surfaces of constant negative curvature in R3 by evolving these extremal curves under their associated binormal flow with prescribed velocity. Finally, we prove that, locally, all …
Differential geometry of curves in dual space
该研究引入对偶空间$\boldsymbol{\text{d}}^3$中曲线的Frenet理论，定义曲率与挠率并分类常曲率对偶平面曲线，建立存在性基本定理且分类常曲率常挠率的对偶曲线。
Two nonlocal inverse curvature flows of convex closed plane curves
本文提出两类非局部逆曲率流，使闭凸曲线长度递减、面积递增，并在无奇点时光滑收敛至圆。
On an area-preserving inverse curvature flow for plane curves
本文研究凸闭平面曲线的1/κⁿ型保面积非局部逆曲率流，证明其全局存在、曲线长度非递增且最终在C^∞度量下收敛为圆，丰富了平面曲线保面积流的理论结果。
In this paper results from the differential geometry of curves are extended from normed planes to gauge planes which are obtained by neglecting the symmetry axiom. Based on the gauge analogue of the notion of Birkhoff orthogonality from Banach space theory, we study all curvature types of curves in gauge planes, thus generalizing their complete classification for normed planes. We show that (as in the subcase of normed planes) there are four such types, and we call them analogously Minkowski, normal, circular, and arc-length curvature. We study relations between them and extend, based on this,…
On a General Linear Nonlocal Curvature Flow of Convex Plane Curves
受Pan-Yang [PY]和Ma-Cheng [MC]的启发，我们研究了凸闭平面曲线的一般线性非局部曲率流，讨论了该流的短时存在性和渐近收敛性.由于流动的线性结构，该偏微分方程问题可以使用常微分方程方法来解决，同时借助于线性热方程的解的表示公式。
Spherical curves whose curvature depends on distance to a great circle
本文引入球面角动量，研究曲率由到大圆或点的距离决定的球面曲线，刻画多种已知曲线并得到可用初等或 Jacobi 椭圆函数表示的新曲线族及其弧长参数化。
On Vertices and Focal Curvatures of Space Curves
在欧氏空间$\R^{m+1}$中，浸入光滑曲线$γ:s\mapsto γ(s)$的{ \em焦点曲线}由其密切超球面的中心组成。聚焦曲线可以根据$γ$（${\bf t},{\bf n}_1, ...,{\bf n}_m$）的Frenet坐标系参数化为$C_γ(s)=(γ+c_1{\bf n}_1+c_2{\bf n}_2+...+c_m{\bf n}_m)(s)$，其中系数$c_1,...,c_{m-1}$是我们称为$γ$的{ \em聚焦曲率}的平滑函数。我们发现了一个将$γ$的欧几里德曲率$κ_i$、$i=1,...,m$与其焦曲率联系起来的公式。我们发现，焦点曲率满足一个系统的弗伦方程（不是矢量，但标量！）。我们使用的焦点曲率的属性，以便给，为$k=1,...,m$，密切$k$维球的半径是关键的必要和充分条件。给出了$γ$中的点是顶点的充分必要条件。最后，明确地给出了$γ$的Frenet标架和欧氏曲率与其焦线$C_γ$的Frenet标架和欧氏曲率的关系.
We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notation for polygonal curves, where the polarity property is exploited, and then make use of a density argument. Both our weak binormal and normal are rectifiable curves which naturally live in the projective plane. In particular, the length of the weak binormal agrees with the total absolute torsion of the given curve. Moreover, the weak normal is the vector …
Lc helices and harmonic curvatures in space forms (hypersurface)
在n维欧几里得空间E ^n中，给出了Özdamar和Hacisalihoellu [4]定义的非退化曲线的调和曲率.本文定义了一种新的曲线类型，即LC螺旋线，当这类曲线的切线与空间型中LC平行向量场的夹角为常数时，我们称之为LC螺旋线。进一步，利用调和曲率给出了这些曲线的几个特征。特别地，在三维空间形式中，我们得到了结果[5]。
It is known that the so-called rotation minimizing (RM) frames allow for a simple and elegant characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a certain linear equation involving the coefficients that dictate the RM frame motion (da Silva, da Silva in Mediterr J Math 15:70, 2018). Here, we shall prove the converse, i.e., we show that if all geodesic spherical curves on a Riemannian manifold are characterized by a certain linear equation, then all the geodesic spheres with a sufficiently small radius are totally umbilical and, consequently, th…
A note on expansion of convex plane curves via inverse curvature flow
最近，Andrews和Bryan [3]发现了一个比较函数，该函数允许他们将曲线缩短流将平面中嵌入的闭曲线收缩为圆形点这一众所周知的事实的经典证明缩短。使用这个比较函数，他们从下面根据端点之间的弧长和经过的时间来估计任何弦的长度。他们将这一估计应用于短线段，并直接推导出最大曲率以指数方式衰减到具有相同长度的圆的曲率。我们考虑凸曲线的逆（平均）曲率流下的扩展，并表明上述比较功能也适用于这种情况下，以获得一个新的证明的事实，即流动存在的所有时间，并成为圆形，即顺利收敛到单位圆后，适当的重新缩放。
In the article, we generalize some recent results of Colding and Minicozzi on generic singularities of mean curvature flow to curved ambient spaces. To do so, we make use of a weighted monotonicity formula to derive an "almost monotonicity" for the entropy upon embedding into $\R^\ell$. We are also lead to study the continuity of the entropy functional in certain cases.
Analytic Continuation of the Frenet Curvature Form of an Analytic Plane Curve
研究解析平面曲线能否由曲率确定特殊保形坐标网，通过弗伦内曲率1 - 形式的全纯延拓，利用重构公式等确定保形坐标网，给出对偶网映射导数表示，证明曲线可借此确定邻域内特殊保形坐标网。
The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed p…
Generalized Similar Frenet Curves
本文研究了在直接相似条件下确定Frenet曲线的微分几何不变量。这些不变量可用球指标的弧长表示的欧氏曲率来表示。然后，这些不变量表示为曲线的焦曲率。然后给出渐屈线曲率与形曲率的关系。然而，这些不变量给出了几何解释。
Immersed solutions of Plateau's problem for piecewise smooth boundary curves with small total curvature
在Gamma的全曲率小于6*Pi的条件下，给出了C^2类空间中任意分段可求长的闭Jordan曲线Gamma至少有一个盘型浸入极小曲面的新证明.在对比的方法，由于Osserman，格列佛和Alt，我们的证明依赖于折线逼近技术，利用存在的沉浸解决方案的高原问题的多边形边界曲线，提供了第一作者的成就卡尼尔的想法。
Given a pair of real functions $(k,f)$, we study the conditions they must satisfy for $k+λf$ to be the curvature in the arc-length of a closed planar curve for all real $λ$. Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied. Finally, the characterization obtained is used to show that a sufficient analogue of the 4-vertex theorem cannot be developed.
Locally nearly spherical surfaces are almost-positively $c$-curved
$c$ -在$C^2$规范中高斯曲率接近1的完整曲面的曲率几乎为正（在Kim-McCann的意义上）。我们的证明是通过仔细的案例分析，结合常曲率情况下的扰动参数，跟踪对接近曲率条件的估计。
Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in $\mathbb{L}^2$ whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular mom…
Contracting Convex Immersed Closed Plane Curves with Slow Speed of Curvature
研究了凸浸入平面曲线以速度（1/α）k^{α}的收缩，其中α在（0，1]中为常数，并证明了如果曲率的爆破率为1型，则它将收敛到位似自相似解.我们还讨论了第二型爆破的一种特殊对称情形，并证明了它收敛于平移自相似解。在曲线缩短流的情况下（即，当α=1时），这种平移自相似解是熟悉的“死神”（M. Grayson [GR]）。
Curvatures of embedded minimal disks blow up on subsets of C^1 curves
在欧氏空间的开子集U中适当嵌入的极小圆盘的任何序列都有一个子序列，使得曲率在U的相对闭子集K上爆炸，并且使得圆盘在K的补中收敛到U的极小层\K。利用Colding-Minicozzi的结果和Meeks的推广，我们证明了这样的爆破集K必包含在C^1嵌入曲线中.
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