相关论文: Nonconvex ancient solutions to Curve Shortening Fl…
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 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.
In this note we construct new nonplanar ancient (in fact, eternal) solutions to the curve shortening flow in $\mathbb{R}^3$, built out of translating grim reapers laying in perpendicular planes.
In this note we construct an infinite family of ancient solutions to the Curve Shortening Flow which span the halfplane.
We construct a translating solution to anisotropic curve shortening flow and show that for a given anisotropic factor $g:S^1\to\mathbb{R}_+$, and a given direction and speed, this translator is unique. We then construct an ancient compact…
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…
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.…
We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in $\mathbb{R}^{n+1}$ for $n \geq 2$. These provide examples of mean convex yet nonconvex ancient solutions that are…
We consider an embedded convex ancient solution $\Gamma_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $\Gamma_t$ is either the family of contracting circles, which is a type I…
In this paper, we consider noncompact ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) which are strictly convex, uniformly two-convex, and noncollapsed. We prove that such an ancient solution is a…
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…
We address the classification of ancient solutions to fully nonlinear curvature flows for hypersurfaces. Under natural conditions on the speed of motion we classify ancient solutions which are convex, noncollapsing, uniformly two-convex and…
We address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross section. For each cylinder of convex bounded cross-section, we show that…
Given any non-central interior point $o$ of the unit disc $D$, the diameter $L$ through $o$ is the union of two linear arcs emanating from $o$ which meet $\partial D$ orthogonally, the shorter of them stable and the longer unstable (under…
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…
We define a new notion of translations in the hyperbolic plane and explicitly solve the equation of the curve shortening flow. Next, we consider the class of ancient convex solutions and solve the equation of the curve shortening flow when…
Motivated by Pan-Yang [PY] and Ma-Cheng [MC], we study a general linear nonlocal curvature flow for convex closed plane curves and discuss the short time existence and asymptotic convergence behavior of the flow. Due to the linear structure…
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…
We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of…
We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions.…