Rongli Huang, Juanjuan Chen
The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in $\mathbb{R}^{2}$.
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It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.
High Codimension Curve Shortening Flow with Free Boundary
该研究探讨了高余维下自由边界曲线缩短流的性质，通过熵界分析证明了流在低熵情况下收敛或产生半圆奇点。
We formulate a uniqueness conjecture for curve shortening flow of proper curves on certain symmetric surfaces and give an example of a non-flat metric on the plane with respect to which curve shortening flow is not unique. That is, with respect to a suitably chosen metric, we construct a non-static solution to curve shortening flow starting from a properly embedded geodesic.
Lectures on curve shortening flow
本文通过曲线缩短流的几何热方程，展示非线性偏微分方程中的最大原理估计、单调性公式、Harnack不等式和奇点分析等核心方法，阐述Huisken证明Grayson定理的过程，即任何平面闭合嵌入曲线最终收缩为一个圆点。
Singularities of low entropy high codimension curve shortening flow
本文研究欧氏背景下任意余维的曲线缩短流，证明其在奇点附近渐近平面并分析爆破极限，进而得出具有熵界的初始曲线在有限时间内收敛至圆点。
An example of a compact, non-convex, embedded ancient solution for the curve shortening flow, which is asymptotic to Yin-Yang curve, is constructed.
We consider an embedded convex ancient solution $Γ_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $Γ_t$ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II ancient solution to the curve shortening flow. We also give a necessary and sufficient curvature condition for an embedded, closed ancient solution to the curve shortening flow to be convex.
We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open curves with fixed endpoints evolving in certain convex domains on surfaces of constant curvature. Specifically, we show that such curves do not develop singularities, and evolve to geodesics.
In this paper we find solutions $u_ε$ to a certain class of vector-valued parabolic Allen-Cahn equation that as $ε\to 0$ develops as interface a given triod evolving under curve shortening flow.
In the study of the curve shortening flow on general closed curves, Abresch and Langer posed a conjecture that the homothetic curves can be regarded as saddle points between multi-folded circles and some singular curves. In other words, these homothetic curves are the watershed between curves with a nonsingular future and those with singular future along the flow. In this article, we provide an affirmitive proof to this conjecture.
Monotonicity of the modulus under curve shortening flow
研究了平面中两个嵌入的闭曲线在曲线缩短流下的模长单调性，证明其随时间递增，并推广到满足下曲率界的 ambient 表面。
We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form 120 degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.
A new energy method for shortening and straightening complete curves
本文提出了一种新型能量方法，通过将曲线缩短重新诠释为切线对齐，研究了曲线缩短流和弹性流的长期行为及收敛性。
In this paper we study the (asymptotic and exponential) stability of the $m$-fold circle as a solution of the $p$-curve shortening flow ($p\geq 1$ an integer).
We construct an ancient solution to planar curve shortening. The solution is at all times compact and embedded. For $t\ll0$ it is approximated by the rotating Yin-Yang soliton, truncated at a finite angle $α(t) = -t$, and closed off by a small copy of the Grim Reaper translating soliton.
In this paper, we study a family of curves on $S^2$ that defines a two-dimensional smooth projective plane. We use curve shortening flow to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of smooth projective planes into one which is isomorphic to the real projective plane. In addition, as a consequence of our main result, we show that any two smooth embedded curves on $RP^2$ which intersect transversally at exactly one point converge to two different geodesics under the flow.
Curve Shortening Flow of Space Curves with Convex Projections
本文研究空间曲线缩短流，证明投影到二维平面保持凸性的闭合浸入曲线在收缩成点前始终光滑且投影凸，并据此说明任意闭合空间曲线可经升维扰动获得收缩至一点的演化。
In this paper we classify convex compact ancient solutions to the affine curve shortening flow: namely, any convex compact ancient solution to the affine curve shortening flow must be a shrinking ellipse. The method combines a rescaling argument inspired by \cite{Wang}, affine invariance of the equation and monotonicity of the affine isoperimetric ratio. We will give two proofs. The essential ideas are related, but the first one uses level set represenation of the evolution. The second proof employs Schauder's estimates, and it also provides a new simple proof for the corresponding classificat…
We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial curves are convex, then an appropriate rescaling at the finite maximal time of existence yields a grim reaper or half a grim reaper as limit flow. We construct examples of initial curves satisfying the mentioned criterion.
Space curve motion describes dynamics of material defects or interfaces, can be found in image processing or vortex dynamics. This article analyses some properties of space curves evolved by the curve shortening flow. In contrast to the classical case of shrinking planar curves, space curves do not obey the Avoidance principle in general. They can lose their convexity or develop non-circular singularities even if they are simple. In the first part of the text, we show that even though the convexity of space curves is not preserved during the motion, their orthogonal projections remain convex. …
In this paper we introduce a geometric quantity, the $r$-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow. The length estimates we obtain are used to prove results about the level set flow in the plane. If $K$ is locally-connected, connected and compact, then the level set flow of $K$ either vanishes instantly, fattens instantly or instantly becomes a smooth closed curve. If the compact set in question is a Jordan curve $J$, then the proof proceeds by using the $r$-multiplicity to show that if $γ_n$ is a sequence of smooth curves converging unifor…
Delayed parabolic regularity for curve shortening flow
本文针对曲线缩短流提出延迟抛物正则性原则，证明从时间$A/π$起一条曲线的正则性可由另一条控制，并以此构造了初始数据仅为$L^1$函数的解。
We prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve shortening flow: If the isoperimetric profile of the region enclosed by the initial curve is greater than that of some `model' convex region with exactly four vertices and with reflection symmetry in both axes, then the inequality remains true for the isoperimetric profiles of the evolved regions. We apply this using the Angenent solution as the model region to deduce sharp time-dependent upper bounds on curvature for arbitrary embedded closed curves evolving by the normalized …
We prove that the only closed, embedded ancient solutions to the curve shortening flow on $\mathbb{S}^2$ are equators or shrinking circles, starting at an equator at time $t=-\infty$ and collapsing to the north pole at time $t=0$. To obtain the result, we first prove a Harnack inequality for the curve shortening flow on the sphere. Then an application of the Gauss-Bonnet, easily allows us to obtain curvature bounds for ancient solutions leading to backwards smooth convergence to an equator. To complete the proof, we use an Aleksandrov reflection argument to show that maximal symmetry is preser…
A new isoperimetric estimate is proved for embedded closed curves evolving by curve shortening flow, normalized to have total length $2π$. The estimate bounds the length of any chord from below in terms of the arc length between its endpoints and elapsed time. Applying the estimate to short segments we deduce directly that the maximum curvature decays exponentially to 1. This gives a self-contained proof of Grayson's theorem which does not require the monotonicity formula or the classification of singularities.
We consider convex symmetric lens-shaped networks in R^2 that evolve under curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks converge to a self-similarly shrinking network, which we prove to be unique in an appropriate class. We also include a classification result for some self-similarly shrinking networks.
We apply the stabilization technique, developed by T. Zelenyak in 1960s for parabolic equations, on curve shortening flow in $\R^3$, and derive several new monotonicity formulas. All of them share one main feature: the dependence of the "energy" term on the angle between the position vector and the plane orthogonal to the tangent vector. The first formula deals with the projection of the curve on the unit sphere, and computes the derivative of its length. The second formula is the generalization of the classical formula of G. Huisken, while the third one is the generalization of the monotonici…
Stability analysis for the anisotropic curve shortening flow of planar networks
该文研究三曲线平面网络的各向异性曲线缩短流，通过证明各向异性曲率能量满足Łojasiewicz-Simon梯度不等式，得出接近局部能量极小值的初始数据对应流全局存在且收敛于能量极小值的稳定性结论。
A delayed interior area-to-height estimate for the Curve Shortening Flow
本文在图形设定下推广了Sobnack与Topping(2024)的结果，证明了曲线缩短流的内部图形估计，并应用该估计证明了从Radon测度弱起始的图形曲线缩短流的存在性。
Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow
该研究针对三维扭转曲线的曲线缩短流，定义挠率-曲率熵泛函，证明流的奇点要么对应拐点要么是满足τ/κ²趋于无穷的II型奇点，与经典平面性定理的τ/κ趋于零的结论形成鲜明对比。
The Harnack inequality without convexity for curve shortening flow
本文提出了一种无需凸性假设的曲线缩短流哈纳克不等式，用于分析曲线演化过程中的正则性及图形化时间。
Curve shortening flow with an ambient force field
本文研究了环境力场下平面各向异性曲线缩短流，证明了高曲率闭合嵌入曲线收缩至圆点，并给出环境力场使初始凸曲线最终变非凸的充分条件。
In this paper we give rates of convergence for the $p$-curve shortening flow for $p\geq 1$ an integer, which improves on the known estimates and which are probably sharp.
The Affine Shape of a Figure-Eight under the Curve Shortening Flow
本文研究一类具二面体对称、凸瓣和曲率单调性的8字形曲线的曲线缩短流，证明重整化后极限收敛于领结四边形，且适当弧段缩放后收敛到Grim Reaper Soliton，给出了Angenent局部凸情形定理的类似结果。
Curve shortening flow on Riemann surfaces with conical singularities
研究具有有限个共形锥形奇点的黎曼曲面上的曲线缩短流，针对初始曲线穿过奇点的情况，建立了流的短时存在性、唯一性和正则性，并证明演化曲线在曲面奇点处保持固定，得到了一些坍塌和收敛结果。
We consider a motion of non-closed planar curves with infinite length. The motion is governed by a steepest descent flow for the geometric functional which consists of the sum of the length functional and the total squared curvature. We call the flow shortening-straightening flow. In this paper, first we prove a long time existence result for the shortening-straightening flow for non-closed planar curves with infinite length. Then we show that the solution converges to a stationary solution as time goes to infinity. Moreover we give a classification of the stationary solution.
We give a rather elementary proof of a Huisken-type monotonicity formula for curve shortening flow in 3D.
The method proposed by T. I. Zelenjak is applied to the mean curvature flow in the plane. A new type of monotonicity formula for star-shaped curves is obtained.
We study the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain in the Euclidean plane. Under certain conditions on the initial curve the flow does not develop any singularity, and it subconverges smoothly to an arc of a circle sitting outside of the given fixed domain and enclosing the same area as the initial curve.
We study an $L^{2}$-type gradient flow of an immersed elastic curve in $\mathbb{R}^{2}$ whose endpoints repel each other via a Coulomb potential. By De Giorgi's minimizing movements scheme we prove long-time existence of the flow. The work is complemented by several numerical experiments.
A curve shortening equation related to the evolution of grain boundaries is presented. This equation is derived from the grain boundary energy by applying the maximum dissipation principle. Gradient estimates and large time asymptotic behavior of solutions are considered. In the proof of these results, one key ingredient is a new weighted monotonicity formula that incorporates a time-dependent mobility.
We provide several examples of dissipative systems that can be obtained from conservative ones through a simple, quadratic,change of time. A typical example is the curve-shortening flow in R^d, which is a particular case ofmean-curvature flow with co-dimension higher than one (except in the case d=2).Through such a change of time, this flow can be formally derived from the conservative model of vibrating strings obtainedfrom the Nambu-Goto action. Using the concept of "relative entropy" (or "modulated energy"), borrowed from the theoryof hyperbolic systems of conservation laws, we introduce a …
In this paper we prove that if $γ$ is a Jordan curve on $\mathbb{S}^2$ then there is a smooth curve shortening flow defined on $(0,T)$ which converges to $γ$ in $\mathcal{C}^0$ as $t\to 0^+ $. Another perspective is that the level-set flow of $γ$ is smooth. This is a generalization of the author's previous work where the planar case was studied. If a Jordan curve on $\mathbb{S}^2$ has Lebesgue measure zero then we show that the level-set flow instantly becomes a smooth closed curve. If the Lebesgue measure is positive then for small time the level-set flow is an annulus with smooth boundary. T…
Which shapes can appear in a Curve Shortening Flow Singularity?
本文研究平面曲线缩短流奇点中可能出现的缠结，提出 n-环曲线并猜想其渐近收敛于挤压蝴蝶结，通过形式分析和数值模拟提供支持。
In this paper we explore the nature of self-similar solutions of the Curve Shortening Flow and the Vortex Filament Equation, also known as the Binormal Flow. We explore some of their fundamental conservation properties and describe the behavior of their self-similar solutions. For Curve Shortening Flow we mainly expose the results of Huisken, Grayson, and Halldorsson concerning the equation's basic properties and self-similar solutions in the plane. For the Vortex Filament Equation we present the results by Banica and Vega, Arms and Hama, and Hasimoto. We also derive the evolution equations of…
We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time $T$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter $γ>1/2$, the…
Huisken's Distance Comparison Principle in Higher Codimension
本文将 Huisken 距离比较原理推广到高余维反射对称曲线缩短流，并利用该原理证明具有一一凸投影的对称流会形成 I 型奇点且渐近趋于圆形。
With few exceptions, known explicit solutions of the curve shortening flow (CSE) of a plane curve, can be constructed by classical Lie point symmetry reductions or by functional separation of variables. One of the functionally separated solutions is the exact curve shortening flow of a closed, convex "oval"-shaped curve and another is the smoothing of an initial periodic curve that is close to a square wave. The types of anisotropic evaporation coefficient are found for which the evaporation-condensation evolution does or does not have solutions that are analogous to the basic solutions of the…
Non-uniqueness of geodesic limits and a question of Grayson and Gage
针对Grayson与Gage提出的闭曲面上不朽曲线缩短流的极限测地线是否唯一的问题，在球面$\rm S^2$上构造反例，证明该极限测地线不唯一，否定回答了该问题。
In this paper we define the continuous Level Lines Shortening evolution of a two-dimensional image as the Curve Shortening operator acting simultaneously and independently on all the level lines of the initial data, and show that it computes a viscosity solution for the mean curvature motion. This provides an exact analytical framework for its numerical implementation, which runs online on any image at http://www.ipol.im/. Analogous results hold for its affine variant version, the Level Lines Affine Shortening.
On the Gage-Hamilton-Grayson Theorem for free boundary curves
本文为凸平面域中嵌入自由边界曲线缩短流的Gage-Hamilton-Grayson定理提供了基于White爆破论证的独立证明。
Grim Raindrop: A Translating Solution to Curve Diffusion Flow
本文证明了平面中曲线扩散流存在一个正则浸入的平移解，并指出该流是曲线缩短流的高阶版本。
The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature
本文通过时间离散化构造了曲线缩短流的全局近似解，该近似解对奇点不敏感且更易分析，并据此恢复经典结果、证明新结论，同时提出了一种高效的数值计算方法。
We classify closed convex $α$-curve shortening flows for sub-affine-critical powers $α\leq \frac{1}{3}$. In addition, we show that closed convex smooth finite entropy $α$-curve shortening flows with $\frac{1}{3}<α$ is a shrinking circle. After normalization, the ancient flows satisfying the above conditions converge exponentially fast to smooth closed convex shrinkers at the backward infinity. In particular, when $α=\frac{1}{k^2-1}$ with $3\leq k \in \mathbb{N}$, the round circle shrinker has non-trivial Jacobi fields, but the ancient flows do not evolve along the Jacobi fields.
We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve has fixed contact angle $α\in (0, π)$ with that line and satisfies a no-flux condition. The proof is led by contradiction: A compactness argument combined with the short time existence result enables us to extend the flow, which contradicts the maximality of the solution.
Using area-preserving curve shortening flow, and a new inequality relating the potential generated by a set to its curvature, we study a non-local isoperimetric problem which arises in the study of di-block copolymer melts, also referred to as the Ohta-Kawasaki energy. We are able to show that the only connected critical point is the ball under mild assumptions on the boundary, in the small energy/mass regime. In particular this class includes all rectifiable, connected 1-manifolds in $\mathbb{R}^2$. We also classify the simply connected critical points on the torus in this regime, showing the…
We extend the slow blow up solutions of Krieger, Schlag, and Tataru to semilinear wave equations on a curved background. In particular, for a class of manifolds $(M,g)$ we show the existence of a family of blow-up solutions with finite energy norm to the equation {equation} \partial_t^2 u - Δ_g u = |u|^4 u, \notag {equation} with a continuous rate of blow up. In contrast to the case where $g$ is the Minkowski metric, the argument used to produce these solutions can only obtain blow up rates that are bounded above.
We consider the focusing quintic nonlinear Schrödinger equation posed on a rotationally symmetric surface, typically the sphere $S^2$ or the two dimensional hyperbolic space $H^2$. We prove the existence and the stability of solutions blowing up on a suitable curve with the log log speed. The Euclidean case is handled in Raphaël (2006) and our result shows that the log log rate persists in other geometries with the assumption of a radial symmetry of the manifold.
The continuity of curvature flow in Fréchet distance
本文研究平面Jordan曲线在曲率流及水平集流下的演化，证明其演化在Fréchet距离下连续依赖于初始曲线。
We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $d\geq 3$ and find two critical curves intersecting at one same point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time below these curves (i.e. critical and supercritical cases). In addition, th…
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