Self-similar solutions to the curve shortening flow

关于具有凸投影的渐近锥形曲线缩短流

On asymptotically conical curve shortening flows with convex projections

研究三维空间中具有凸投影的渐近锥形曲线在曲线缩短流下的长期演化行为。

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.

双曲平面中曲线缩短流的几何特性

Geometric aspects of the curve shortening flow in the hyperbolic plane

本文研究了双曲平面中曲线缩短流的翻译概念，解决了曲线缩短流方程，并探讨了古早凸解的曲率函数分离变量情况下的解法，以及闭古早解的面积估计。

解析曲面上的不朽曲线缩短流

Immortal Curve Shortening Flow on Analytic Surfaces

本文证明在闭可定向实解析黎曼曲面上，任何光滑浸入的闭不朽曲线缩短流均收敛至唯一的非常数闭测地线。

Convex ancient solutions to curve shortening flow

We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Sesum and X.-J. Wang

The curve shortening flow in the metric-affine plane

We investigate for the first time the curve shortening flow in the metric-affine plane and prove that under simple geometric condition it shrinks a closed convex curve to a "round point" in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in Euclidean plane.

收缩半圆在自由边界曲线缩短流中的稳定性

Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

研究了自由边界曲线缩短流在二维凸域中收敛到圆半点的稳定性及收敛速率。

Curve shortening flows in warped product manifolds

We study curve shortening flows in two types of warped product manifolds. These manifolds are $S^1\times N$ with two types of warped metrics where $S^1$ is the unit circle in $R^2$ and $N$ is a closed Riemannian manifold. If the initial curve is a graph over $S^1$, then its curve shortening flow exists for all times and finally converges to a geodesic closed curve.

无穷远处非凸曲面上的曲线缩短流

Curve shortening flows on surfaces that are not convex at infinity

本文研究了无穷远处非凸曲面上的曲线缩短流，针对单位圆与开区间的扭曲乘积，证明了曲线在有限时间后会变为图，推广了平面曲线缩短流的经典行为。

Affine Self-similar solutions of the affine curve shortening flow I: The degenerate case

In this paper, we consider affine self-similar solutions for the affine curve shortening flow in the Euclidean plane. We obtain the equations of all affine self-similar solutions up to affine transformations and solve the equations or give descriptions of the solutions for the degenerate case. Some new special solutions for the affine curve shortening flow are found.

各向异性曲线缩短流的凸古老解

Convex Ancient Solutions to Anisotropic Curve Shortening Flow

本文构造了各向异性曲线缩短流的平移解和紧致古老解，并证明在给定各向异性因子、方向、速度及带状宽度条件下，这些解是唯一的。

Harnack inequalities for the curve shortening flow

We offer an algorithmic approach for determining Harnack quantities for the curve shortening flow and we show how, following this procedure, one can obtain Hamilton's Harnack inequality for this flow $κ_t + \frac{1}{2t}κ\geq \frac{κ_s^2}κ$, where $κ$ is the curvature of the curve being deformed by the flow.

A Short Note on Contracting Self-Similar Solutions of the Curve Shortening Flow

By the curve shortening flow, the only closed embedded contracting self-similar solutions are circles: we give a very short and intuitive geometric proof of this basic and classical result using an idea of Gage.

面积保持曲线缩短流的收缩器：存在性和鞍点性质

Shrinkers of the area-preserving curve-shortening flow: Existence and saddle-point property

本文研究面积保持曲线缩短流的收缩器存在性及鞍点性质，通过λ曲线理论证明非圆形收缩器的存在，并提出类似Abresch-Langer分类的分类方案。

旋转曲面上曲线缩短流的旋转孤子

Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces

该研究刻画了三维欧氏空间中旋转曲面上曲线缩短流的旋转孤子，并证明这类开曲线的两端渐近于平行测地线，完善了相关几何流的孤子理论。

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 $\mathbb{R}^2$.

Convergence of Curve Shortening Flow to Translating Soliton

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\mathbb{R}^2$ under the $α$-curve shortening flow for exponents $α>\frac12$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $α$-curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case $α=1$, and we prove for all exponents up to the critical case $α>\frac12$.

Curve shortening flow on Riemann surfaces with possible ambient conic singularities

In this paper, we study the curve shortening flow (CSF) on Riemann surfaces. We generalize Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. We reprove the Gage-Hamilton-Grayson theorem on surfaces. We also prove that for embedded simple closed curves, CSF can not touch conic singularities with cone angles $\leq π$.

Distance comparison principle and Grayson type theorem in the three dimensional curve shortening flow

In this paper, we use the distance comparison principle, first been developed by G. Huisken, to study the spatial curve shortening flow. We have got the result that if the initial curve is the helix, then the local minimum of the ratio of the extrinsic and intrinsic distance is non-decreasing. And we have proved a Grayson type theorem for a variety of spatial curves.

Curve shortening and the topology of closed geodesics on surfaces

We study "flat knot types" of geodesics on compact surfaces M^2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M^2. We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial.

曲线缩短流中凸投影的奇点

Singularities of Curve Shortening Flow with Convex Projections

研究证明了在R^n中具有单射凸投影到某个2平面的光滑闭合浸入曲线在曲线缩短流下会产生I型奇点并渐近趋于圆形，同时证明了在R^n中曲线缩短流下的Huisken猜想的类比形式，即任何光滑闭合浸入曲线可通过光滑扰动转化为在R^{n+2}中收缩至圆点的闭合浸入曲线。

$\mathbb{R}^3$ 中源于 grim reapers 的非平面古老曲线缩短流

Nonplanar ancient curve shortening flows in $\mathbb{R}^3$ from grim reapers

本文通过组合位于垂直平面上的平移 grim reapers，构造了 $\mathbb{R}^3$ 中曲线缩短流的新的非平面古老（永恒）解。

Curve shortening flows on rotational surfaces generated by monotone convex functions

In this paper, we study curve shortening flows on rotational surfaces in $\mathbb{R}^3$. We assume that the surfaces have negative Gauss curvatures and that some condition related to the Gauss curvature and the curvature of embedded curve holds on them. Under these assumptions, we prove that the curve remains a graph over the parallels of the rotational surface along the flow. Also, we prove the comparison principle and the long-time existence of the flow.

A note on self-similar solutions of the curve shortening flow

This article gives an alternative approach to the self-shrinking and self-expanding solutions of the curve shortening flow, which are related to singularity formation of the mean curvature flow. The motivation for the self-similar solutions arises from natural area preserving rescaling. Further we describe the self-similar solutions in terms of a simple ODE and give an alternative proof that they lie in planes.

Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

We prove that a closed immersed plane curve with total curvature $2πm$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2πm$ whose entropy is less than $m$ times the entropy of the embedded circle. As an application, we extend Colding-Minicozzi's notion of a generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF whose only singularities are embedded circles and type II singularities.

Soliton Solutions to the Curve Shortening Flow on the 2-dimensional hyperbolic plane

We show that a curve is a soliton solution to the curve shortening flow if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. We use this characterization to provide a qualitative study of the solitons. We show that for each fixed vector there is a 2-parameter family of soliton solutions to the curve shortening flow on the 2-dimensional hyperbolic space. Moreover, we prove that each soliton is defined on the entire real line, it is embedded and its geodesic curvature converges to a con…

The curve shortening flow with parallel 1-form

Let $M$ be a closed Riemannian manifold with a parallel 1-form $Ω$. We prove two theorems about the curve shortening flow in $M$. One is that the {\csf} $\ct$ in $M$ exists for all $t$ in $[0, \infty)$, if it satisfies $Ω(T)\geq 0$ on the initial curve $\co$. Here $T$ is the unit tangent vector on $\co$. The other one is about the convergence. It says that in a closed {\Rm} $\tilde{M}$, assume the curve shortening flow $\ct$ exists for all $t\in[0,\infty)$ and its length converges to a positive limit, then $ \lim\limits_{t\rightarrow\infty}max_{\ct}|\nabla^{m}A|^{2}=0$ for all $m=0,1,...$. Her…

球面上曲线缩短流的精确距离比较

Sharp distance comparison for curve shortening flow on the round sphere

该研究证明了球面上曲线缩短流的精确弦弧改进性质，由此得到曲率控制，进而直接证明简单球面曲线的两种演化结局，完善了球面曲线缩短流的相关结论。

具有自由边界的曲线缩短流的距离比较原理

A distance comparison principle for curve shortening flow with free boundary

本文通过引入反射弦弧剖面并建立弦弧估计，证明了凸平面区域中嵌入自由边界曲线缩短流要么无限时间收敛到唯一临界弦，要么有限时间收缩为边界上的圆形半点。

A Condition Ensuring Spatial Curves Develop Type-II Singularities Under Curve Shortening Flow

We provide a condition for spatial curves which rules out the development of a type I singularity. The condition is that after the last time for which an inflection point develops, if the torsion is ever everywhere non-negative, the curve cannot develop a type I singularity.

Curve Shortening Flow in a Riemannian Manifold

In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the global behavior of the flow. In particular, we show the following results. 1). Let $\mathbf{M}$ be a compact locally symmetric space. If the curve shortening flow exists for infinite time, and $$ \lim_{t\to\infty}L(γ_{t})>0, $$ then for every $n>0$, $$ \lim_{t\to \infty}\sup(|\frac{D^{n}T}{\partial s^{n}}|)=0. $$ In particular, the limiting curve exists and is a …

双曲面上的逆曲线缩短流

The inverse curve shortening flow on the hyperbolic plane

本文研究双曲面上的逆曲线缩短流，分类了所有与双曲面的抛物线性和共形向量场相关的孤立子，并证明抛物线性孤立子在y轴上为图，共形孤立子在x轴上为图，分析了这些孤立子的凹性和接近坐标轴的行为。

Solitons of discrete curve shortening

For a polygon in Euclidean space we consider a transformation T which is obtained by applying the midpoints polygon construction twice and using an index shift. For a closed polygon this is a curve shortening process. A polygon is called (affine) soliton of the transformation T if its image under T is an affine image of the polygon. We describe a large class of solitons by considering smooth curves which are solutions of a linear system of differential equations of second order with constant coefficients. As examples we obtain solitons lying on spiral curves which under the transformation T ro…

The curve shortening problem associated to a density

In $\mathbb{R}^n$ with a density $e^ψ$, we study the mean curvature flow associated to the density ($ψ$-mean curvature flow or $ψ$MCF) of a hypersurface. The main results concern with the description of the evolution under $ψ$MCF of a closed embedded curve in the plane with a radial density, and with a statement of subconvergence to a $ψ$-minimal closed curve in a surface under some general circumstances.

The curve shortening flow with density of a spherical curve in codimension two

In the present paper we carry out a systematic study about the flow of a spherical curve by the mean curvature flow with density in a 3-dimensional rotationally symmetric space with density $(M^3_w,\:g_w,\:ξ)$ where the density $ξ$ decomposes as sum of a radial part $φ$ and an angular part $ψ$. We analyse how either the parabolicity or the hyperbolicity of $(M^3_w,\:g_w)$ condition the behaviour of the flow when the solution goes to infinity.

Non-uniqueness of curve shortening flow

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.

有限熵缩短曲线在无穷远处切流的唯一性

Uniqueness of tangent flows at infinity for finite-entropy shortening curves

本文证明了有限熵古代曲线缩短流在无穷远处切流的唯一性，并通过重标度流的指数收敛及对尖端、顶点、拐点和图形半径增长率的精确刻画，排除了收缩圆、静态直线、回形针和平移冷酷收割者等例外情形。

A Compact Non-convex Ancient Curve Shortening Flow

An example of a compact, non-convex, embedded ancient solution for the curve shortening flow, which is asymptotic to Yin-Yang curve, is constructed.

The blow up analysis of the general curve shortening flow

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.

Type I singularities in the curve shortening flow associated to a density

We define Type I singularities for the mean curvature flow associated to a density $ψ$ ($ψ$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $ψ$-curvature due to the density. We describe a family of curves whose evolution under $ψ$MCF (in a Riemannian surface of non-negative curvature with a density which is singular at a geodesic of the surface) produces only type I singularities and study the limits of their blow-ups.

The fundamental solutions of the curve shortening problem via the Schwarz function

Curve shortening in the $z$-plane in which, at a given point on the curve, the normal velocity of the curve is equal to the curvature, is shown to satisfy $S_tS_z=S_{zz}$, where $S(z,t)$ is the Schwarz function of the curve. This equation is shown to have a parametric solution from which the known explicit solutions for curve shortening flow; the circle, grim reaper, paperclip and hairclip, can be recovered.

Classification of compact ancient solutions to the curve shortening flow

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.

曲线缩短流讲义

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。

低熵高余维曲线缩短流的奇点

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。

The Dirichlet problem for curve shortening flow

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.

Blow-up rates for the general curve shortening flow

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 $\mathbb{R}^{2}$.

Relaxation of the flow of triods by Curve Shortening Flow via the vector-valued parabolic Allen-Cahn equation

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.

On the Saddle Point Property of Abresch-Langer curves under the 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.

曲线缩短流下模长的单调性

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。

Homotopic curve shortening and the affine curve-shortening flow

We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call "homotopic curve shortening" (HCS), starts with a closed curve (which might self-intersect) in the presence of a set $P\subset \mathbb R^2$ of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent. We find experimentally that, if the initial curve is held fixed and $P$ is chosen to be either a very fine regular grid or a uniformly r…

Self-similarly expanding networks to curve shortening flow

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.

一种缩短和使曲线直线化的新型能量方法

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。

On the stability of $m$-fold circles and the dynamics of generalized curve shortening flows

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).

古代有限熵曲线缩短流的分类

Classification of ancient finite-entropy curve shortening flows

本文分类了古代有限熵曲线缩短流，证明其只能是静态直线、收缩圆、纸夹、平移型 grim reaper 或图形古代 trombone，并指出紧致流为凸，非紧致流为静态直线或完整图。