arXiv research

The study finds translators for higher order mean curvature flows in Euclidean and hyperbolic spaces.

New ancient curve shortening flows created from grim reapers.

This paper classifies grim reapers in a specific product space.

Constructing translating solitons from Lagrangian Grim Reapers.

Study shows only grim reaper cylinder for certain self-translating surfaces.

In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in $\mathbb{R}^{n+1}$($n=2,3)$ is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton $Σ$ with nonnegative scalar curva…

In this article we prove that a connected and properly embedded translating soliton in $\mathbb{R}^3$ with uniformly bounded genus on compact sets which is $C^1$-asymptotic to two planes outside a cylinder, either is flat or coincides with the grim reaper cylinder.

New families of translating solitons found in hyperbolic space.

The curve shortening flow transforms figure-eight curves into bowties.

Ancient solutions to curve shortening flow are constructed and analyzed.

Paper proves unique tangent flow at infinity for entropy-limited curve shortening.

Ancient curve flows classified into specific types.

The study proves stability of various graphical translators in mean curvature flow.

The paper extends convexity results for translating solitons in higher dimensions.

New curves defined by curvature powers studied for variational properties.

We construct new examples of self-translating surfaces for the mean curvature flow from a periodic configuration with finitely many grim reaper cylinders in each period. Because this work is an extension of the author's article on the desingularization of a finite family of grim reaper cylinders, we simply discuss the …

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial c…

The study classifies horo-shrinkers in hyperbolic space under different isometries.

We consider a curvature flow $V=H$ in the band domain $Ω:=[-1,1]\times \R$, where, for a graphic curve $Γ_t$, $V$ denotes its normal velocity and $H$ denotes its curvature. If $Γ_t$ contacts the two boundaries $\partial_\pm Ω$ of $Ω$ with constant slopes, in 1993, Altschular and Wu \cite{AW1} proved that $Γ_t$ converge…

The study characterizes and rules out collapsing in convex ancient mean curvature flow.

We describe a construction of complete embedded self-translating surfaces under mean curvature flow by desingularizing the intersection of a finite family of grim reapers in general position.

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.

Study translators in Generalised Robertson-Walker spacetimes, identifying warping functions and classifying examples.

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

Study curve shortening flow in high dimensions with boundary constraints.

The paper classifies solitons for mean curvature flow in hyperbolic space.

We desingularise the union of $3$ Grim paraboloids along Costa-Hoffman-Meeks surfaces in order to obtain complete embedded translating solitons of the mean curvature flow with $3$ ends and arbitrary finite genus.

Paper proves uniqueness of catenary cylinders based on their asymptotic shape.

It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.

Ancient grain boundaries resemble atoms in their formation and properties.

Existence of translating solutions shown for curve diffusion flow.

We prove the existence of classical solutions to the Dirichlet problem for the $α$-translating soliton equation defined in a strip of $\r^2$. We use the Perron method where a family of grim reapers are employed as barriers for solving the Dirichlet problem when the boundary data is formed by two copies of a convex func…

We construct a one-parameter family of singly periodic translating solutions to mean curvature flow that converge as the period tends to $0$ to the union of a grim reaper surface and a plane that bisects it lengthwise. The surfaces are semigraphical: they are properly embedded, and, after removing a discrete collection…

Study proves conditions for translating solitons to be planar.

We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give …

Curve shortening problem solved via Schwarz function.

Characterizes ruled translating solitons in Minkowski 3-space.

We study the contraction of a convex immersed plane curve with speed (1/α)k^{α}, where αin(0,1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converge…

The aim of this paper is to investigate the mean curvature flow soliton solutions on the Heisenberg group $\mathcal{H}$ when the initial data is a ruled surface by straight lines. We give a family of those solutions which are generated by $\mathfrak{Iso}_{0}(\mathcal{H})$ (the isometries of $\mathcal{H}$ for which the …

In this work we show that $2$-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of $\mathbb{R}^3$ with entropy strictly less than $3$ must be mean convex and thus, thanks to a result by J. Spruck and L. Xiao, are convex. Recently, such $2$-dimensional convex translating s…

The paper studies geometric flows of spacelike curves in Lorentz-Minkowski plane and proves their long-term behavior.

We classify all the translating solitons to the mean curvature flow in the three-dimensional Heisenberg group that are invariant under the action of some one-parameter group of isometries of the ambient manifold. The problem is solved considering any canonical deformation of the standard Riemannian metric of the Heisen…

In this paper, we introduce a concept of B-minimal sub-manifolds and discuss the stability of such a sub-manifold in a Riemannian manifold $(M,g)$. Assume $B(x)$ is a smooth function on $M$. By definition, we call a sub-manifold $Σ$ {\em B-minimal} in $(M,g)$ if the product sub-manifold $Σ\times S^1$ is a {\em minimal}…

A translating soliton is a hypersurface $M$ in $\mathbb{R}^{n+1}$ such that the family $M_t= M- t \,\mathbf{e}_{n+1}$ is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point $\mathbf{H}=\mathbf{e}_{n+1}^{\perp}.$ In this paper we obtain a cha…

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show …

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. …

Infinite-time blow-up in high-dimensional mean curvature flow.

Pseudo-rehearsal allows neural networks to learn a sequence of tasks without forgetting how to perform in earlier tasks. Preventing forgetting is achieved by introducing a generative network which can produce data from previously seen tasks so that it can be rehearsed along side learning the new task. This has been fou…