arXiv research

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…

Ancient pancakes solve mean curvature flow problem.

Ancient solutions to Ricci flow in higher dimensions are mostly cylinders or solitons.

Ancient solutions on a strip are constant if polynomial, and have finite-dimensional space for slower growth.

Ancient solutions to Kähler Ricci flow classified completely.

New ancient solutions found for curvature flow in 2D.

Ancient solutions of Ricci flow with Type I growth are classified.

Ancient symplectic solutions to mean curvature flow are flat.

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

Study on ancient Ricci flows with positive curvature, proving noncollapsedness.

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.

Ancient pancake solutions found for curvature flows.

Ancient Ricci flow found from Taub-Bolt metric.

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 …

Study higher-dimensional Ricci flow solutions, proving uniqueness.

We consider an ancient solution $g(\cdot,t)$ of the Ricci flow on a compact surface that exists for $t\in (-\infty,T)$ and becomes spherical at time $t=T$. We prove that the metric $g(\cdot,t)$ is either a family of contracting spheres, which is a type I ancient solution, or a Rosenau solution, which is a type II ancie…

Proves rigidity of ancient solutions in mean curvature flow.

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

Classifies ancient convex curves in convex domains.

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. They are type I ancient solutions.

Ancient solutions to curve shortening flow are constructed and analyzed.

In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere…

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. They are type I ancient solutions.

In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota's work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative c…

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

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

Ancient solutions to Ricci flow with isotropic curvature conditions are classified.

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…

We study the time analyticity of ancient solutions to heat equations on graphs. Analogous to Dong and Zhang [DZ19], we prove the time analyticity of ancient solutions on graphs under some sharp growth condition.

Study on Ricci flow of invariant metrics on spheres, finding new ancient solutions.

Unique ancient solutions found for anisotropic curve shortening flow.

Ancient solutions and translators identified for Lagrangian flow.

In this paper, we study the classification of $κ$-noncollapsed ancient solutions to three-dimensional Ricci flow on $S^3$. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.

Type II (ancient) solutions to the Ricci flow on surfaces are not yet classified. It is conjectured that the Rosenau solution and the cigar are the only solutions, modulo scaling. In this paper, we mainly study the backward limit and the circumference at spatial infinity of Type II ancient solutions on noncompact surfa…

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

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…

Liouville theorems extended to graphs with bounded geometry.

New ancient curve shortening flows created from grim reapers.

Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.

An explicit representation formula for all positive ancient solutions of the heat equation in the Euclidean case is found. In the Riemannian case with nonnegative Ricci curvature, a similar but less explicit formula is also found. Here it is proven that any positive ancient solution is the standard Laplace transform of…

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.

We consider compact ancient solutions to the three-dimensional Ricci flow which are noncollapsed. We prove that such a solutions is either a family of shrinking round spheres, or it has a unique asymptotic behavior as $t \to -\infty$ which we describe. This analysis applies in particular to the ancient solution constru…

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

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension greater than one, and for some nonlinea…

Compact, non-convex curve flows are created.

In this paper, we study $κ$-noncollapsed ancient solutions to the Ricci flow with nonnegative curvature operator in higher dimensions. We impose one further assumption: one of the asymptotic shrinking gradient Ricci solitons is the standard cylinder $\mathbb{S}^{n-1}\times\mathbb{R}$. By making use of the properties of…