arXiv research

Proves uniqueness of cylindrical tangent flows in mean curvature flow.

This paper studies mean curvature flows near cylindrical singularities.

Study shows generic surfaces avoid complex flow patterns.

Study resolves flow through cylindrical singularities, proving nonfattening.

Resolves conjecture on cylindrical mean curvature flows in all dimensions.

The paper studies cylindrical singularities in mean curvature flow and proves their local regularity.

Constructs a mean curvature flow with surgery for compact mean convex hypersurfaces.

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

Study classifies ruled surfaces from mean curvature flow solutions.

Localizes curvature estimates for evolving hypersurfaces under various flows.

New bifurcation found in perturbations of non-generic closed self-shrinkers.

Study proves uniqueness and rigidity of cylindrical self-shrinkers using Łojasiewicz inequalities.

We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindric…

We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in higher codimension is a nonlinear parabolic system where many of the methods used f…

Proves convergence of mean curvature flow on cylinders with unique continuation.

We study high codimension mean curvature flow of a submanifold $\mathcal{M}^n$ of dimension $n$ in Euclidean space $\mathbb{R}^{n+k}$ subject to the quadratic curvature condition $|A|^{2}\leq c_n |H|^{2}, c _n = \min\{ \frac{4}{3n} , \frac{1}{n-2}\}$. This condition extends the notion of two-convexity for hypersurface…

The paper analyzes self-similar solutions for mean curvature flow in 3D.

Sharp curvature pinching for mean curvature flow in spheres proved.

We prove a complete family of `cylindrical estimates' for solutions of a class of fully non-linear curvature flows, generalising the cylindrical estimate of Huisken-Sinestrari for the mean curvature flow. More precisely, we show that, for the class of flows considered, an $(m+1)$-convex ($0\leq m\leq n-2$) solution bec…

Study on stability of cylindrical singularities in MCF of finite codimensions.

Self-shrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a strong sense - that is, any self-shrinker that is mean convex with uniformly bounded curva…

The paper shows that ancient noncollapsed mean curvature flows have a blowdown of at most n-2 dimensions.

We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^{n-1}$-estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis…

Proves uniqueness of blowups for forced mean curvature flow.

We show that any complete, immersed self-expander to the inverse mean curvature flow, which has one end asymptotic to a cylinder, or has two ends asymptotic to two coaxial cylinders, must be rotationally symmetric.

New approach analyzes ancient solutions and singularities of mean curvature flow.

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In …

Sharp curvature estimates for mean curvature flow in spheres.

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

In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $\varepsilon=\varepsilon(X)>0$ such that the flow is mean convex in a …

Classifies ancient ovals in higher dimensional mean curvature flow.

New 1-parameter family of ovals identified in 4d Ricci flow classification.

In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a "neck" which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.

For any $n$-dimensional smooth manifold $Σ$, we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in $Σ$ are cylindrical (of convex type) if the flow converges to a smooth hypersurface $M_{\infty}$ (maybe empty) at infinity. Previously this was shown (i) for $n\leq 7$,…

Low-entropy surfaces can be flowed into spheres and cylinders.

Study nondegenerate singularities in mean curvature flow.

The paper classifies ruled surfaces in a specific space that move in a special way.

The study quantizes ancient flows in cylinders, revealing their asymptotic behavior.

We show that any strictly mean convex translator of dimension $n\geq 3$ which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature…

The paper studies hypersurfaces in spheres using mean curvature flow with surgery.

In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In the author's thesis the stability of cylinders under the flows was considered; it was found that they are stable provi…

Proves Arnold-Thom conjecture for surfaces' arrival times.

Researchers classify translators and rotators in hyperbolic 3-space for mean curvature flow.

Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-…

Study high codimension mean curvature flow in Riemannian manifolds, proving limiting flow in Euclidean space.

This paper establishes geometric obstructions to the existence of complete, properly embedded, mean curvature flow self-translating solitons $Σ^n\subseteq \mathbb{R}^{n+1}$, generalizing previously known non-existence conditions such as cylindrical boundedness.

We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additio…