arXiv research

Classifies ancient convex curves in convex domains.

New ancient solutions found for curvature flow in 2D.

Study ancient solutions to free boundary mean curvature flow in convex manifolds.

Ancient convex solutions to flow equations are limited to simple shapes.

Proves planarity and convexity for ancient solutions of mean curvature flow.

Compact, non-convex curve flows are created.

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 such conditions are: a uniform pinching condition on the curvatures, a suitable…

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 rotationally symmetric translating soliton.

Classifies ancient solutions to curvature flows, finding two main types.

Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.

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 pancakes solve mean curvature flow problem.

We prove that the only compact convex ancient solutions of the planar affine normal flow are contracting ellipses.

New convex ancient solutions found for flows by high powers of curvature.

In this paper we consider closed non-collapsed ancient solutions to the mean curvature flow ($n \ge 2$) which are uniformly two-convex. We prove that any two such ancient solutions are the same up to translations and scaling. In particular, they must coincide up to translations and scaling with the rotationally symmetr…

Ancient Lagrangian flows get limited convex solutions.

We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such a…

We construct a compact, convex ancient solution of mean curvature flow in $\mathbb R^{n+1}$ with $O(1)\times O(n)$ symmetry that lies in a slab of width $π$. We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, $O(n)$-invariant ancient solution that lies …

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

Unique ancient convex flow in a ball with free boundary found.

We study properly immersed ancient solutions of the codimension one mean curvature flow in $n$-dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or a…

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 …

Constructs ancient solutions to mean curvature flow with prescribed singular sets.

A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension $3$ which have positive sectional curvature and are $κ$-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $\mathbb{R}^3$, and prove that the rotationally symm…

We prove that the only compact, origin-symmetric, strictly convex ancient solutions of the planar $p$ centro-affine normal flows are contracting origin-centered ellipses.

Unique ancient solutions found for anisotropic curve shortening flow.

The study characterizes and rules out collapsing in convex ancient mean curvature 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 monotonicit…

Paper relaxes convexity assumptions in mean curvature flow results.

Two ancient solutions to Gauss curvature flow are identified for cylinders.

We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in $\mathbb{R}^{n+1}$ with $O(1)\times O(n)$ symmetry. We show they all have unique asymptotics as $t\to -\infty$ and we give precise asymptotic description of these solutions. In particular, solutions constructed by White, and Haslhofer …

We study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker have shown …

In this paper we study the classification of ancient convex solutions to the mean curvature flow in $\R^{n+1}$. An open problem related to the classification of type II singularities is whether a convex translating solution is $k$-rotationally symmetric for some integer $2\le k\le n$, namely whether its level set is a …

Ancient curve shortening flow in a disc with mixed boundary conditions is solved.

New translations defined; curve shortening flow solved in hyperbolic plane.

Derives Li & Yau estimates for heat equations on manifolds.

Proves local noncollapsing estimate for mean curvature flow.

We prove several sharp one-sided pinching estimates for immersed and embedded hypersurfaces evolving by various fully nonlinear, one-homogeneous curvature flows by the method of Stampacchia iteration. These include sharp estimates for the largest principal curvature and the inscribed curvature ('cylindrical estimates')…

Classifies ancient flows in a disc with boundary.

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 not solitons, meaning that they do not evolve by rigid motions or homotheties. …

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 o…

The paper confirms conjectures about ancient ovals and provides counterexamples.

We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed…

We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of Huisken--Sinestrari and the cylindrical estimate of Huisken--Sinestrari. Namely, we show …

Ancient grain boundaries resemble atoms in their formation and properties.

Ancient solutions to mean curvature flow have unique shapes.

Let $n\ge 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, $3$-parameters ancient solutions of the equation $v_t=(v^m)_{xx}+v-v^m$, $v>0$, in $\mathbb{R}\times (-\infty,T)$ for some $T\in\mathbb{R}$. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient so…

Classifies ancient ovals in higher dimensional mean curvature flow.