arXiv research

AI helps complete ancient tablets with missing parts.

Handwriting disorder (termed dysgraphia) is a far from a singular problem as nearly 8.6% of the population in France is considered dysgraphic. Moreover, research highlights the fundamental importance to detect and remediate these handwriting difficulties as soon as possible as they may affect a child's entire life, und…

Ancient curve flows classified into specific types.

Ancient solutions to mean curvature flow have unique shapes.

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

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.

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 Ricci flow found from Taub-Bolt metric.

Ancient curves span halfplanes via flow.

Classifies ancient flows in a disc with boundary.

Classifies ancient convex curves in convex domains.

Ancient solutions to Kähler Ricci flow classified completely.

Ancient symplectic solutions to mean curvature flow are flat.

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

Ancient Ricci flows are identified without curvature sign condition.

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

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

New ancient solutions found for curvature flow in 2D.

Ancient flows by curvature powers in 2D have finite entropy.

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

New ancient curve shortening flows created from grim reapers.

Ancient pancake solutions found for curvature flows.

Classifies ancient ovals in higher dimensional mean curvature 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 …

Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.

Proves rigidity of ancient solutions in mean curvature flow.

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

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…

The paper confirms conjectures about ancient ovals and provides counterexamples.

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…

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…

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…

The study disproves rotating ancient flows in 4D space.

Enhanced estimates for ancient ovals and translators in 3D and 4D.

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.

Ancient curve shortening flows have entropy and curvature bounds equivalent.

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinkin…

Ancient solutions to curve shortening flow are constructed and analyzed.

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

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

Ancient solutions and translators identified for Lagrangian flow.

Compact, non-convex curve flows are created.

New proof classifies ancient flows in 3D space.

Study higher-dimensional Ricci flow solutions, proving uniqueness.

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

Classifies ancient and expanding Ricci flows with specific groups.