Ancient solutions to the Curve Shortening Flow spanning the halfplane

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。

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。

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.

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.

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。

Nonconvex ancient solutions to Curve Shortening Flow

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.

Classifying convex compact ancient solutions to the affine curve shortening flow

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…

Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere

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…

圆盘上混合边界条件下的古代曲线缩短流

Ancient curve shortening flow in the disc with mixed boundary condition

本文在单位圆盘混合边界条件下构造并分类了固定非中心内点的凸古代曲线缩短流，证明仅有两个永恒解连接不稳定与稳定临界弧，并进一步分析了边界点退化情形下的有限时间坍缩奇点。

Classification of ancient flows by sub-affine-critical powers of curvature in $\mathbb{R}^2$

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.

Construction of High Codimension Ancient Mean Curvature Flows

We construct a class of compact ancient solutions to the mean curvature flow in Euclidean space with high codimension. In particular, we construct higher codimensional ancient curve shortening flows. Moreover, we characterize the asymptotic behavior of these solutions. Add on remark: the construction in this paper has been discovered by Altschuler D.-Altschuler S.-Angenent-Wu in [AAAW13].

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。

椭圆泛函的古代梯度流与Morse指数

Ancient gradient flows of elliptic functionals and Morse index

本文研究黎曼流形中椭圆泛函梯度流的闭古代解，对球面中平均曲率流及曲线缩短流在特定面积或长度界下进行了完全分类，揭示了其解仅为稳态或收缩的赤道球面、Clifford环面与圆。

Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$

Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, we prove codimension bounds for ancient mean curvature flows by their tangent flow at $-\infty$, generalizing a theorem for cylinders in [CM19b]. In the case of the $m$-covered circle, we apply this bound to prove a strong rigidity theorem. Furthermore, we extend this paradigm by showing that under the assumption of sufficiently rapid convergence, a compact ancient mean curvature flow is identical to its tangent flow at $-\infty$.

Convex ancient solutions to mean curvature flow

X.-J. Wang proved a series of remarkable results on the structure of convex ancient solutions to mean curvature flow. Some of his results do not appear to be widely known, however, possibly due to the technical nature of his arguments and his exploitation of methods which are not widely used in mean curvature flow. In this expository article, we present Wang's structure theory and some of its consequences. We shall simplify some of Wang's analysis by making use of the monotonicity formula and the differential Harnack inequality, and obtain an important additional structure result by exploiting…

$\boldsymbol{\text{R}^4}$中翼状平均曲率流的不存在性结果

A nonexistence result for wing-like mean curvature flows in $\mathbb{R}^4$

本文排除了$\text{R}^4$中3维超曲面平均曲率流的非塌缩翼状奇点模型，证明古老非塌缩流的爆破极限非楔体，提出精细气泡片分析，为$\text{R}^4$古老非塌缩流分类奠定基础。