arXiv research

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

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

Paper proves finite Morse index for certain self-shrinkers.

Study shows generic surfaces avoid complex flow patterns.

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…

We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is comp…

Proves uniqueness of blowups for forced mean curvature flow.

Study classifies ruled surfaces from mean curvature flow solutions.

The paper constructs $λ$-hypersurfaces for $λ>0$ and $λ<0$.

Sharp curvature pinching for mean curvature flow in spheres proved.

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 …

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylin…

Researchers set entropy limits for specific types of self-shrinkers.

Study bounds on self-shrinkers with bounded HA for applications.

A rigidity theorem for smooth Legendrian self-shrinkers is proven.

The paper studies scalar curvature of self-shrinkers and proves curvature bounds.

Generalizes halfspace theorems to higher dimensions for self-shrinkers.

New theorem shows noncompact self shrinkers are unknotted.

We investigate Mean Curvature Flow self-shrinking hypersurfaces with polynomial growth. It is known that such self shrinkers are unstable. We focus mostly on self-shrinkers of the form $\mathbb S^k\times\R^{n-k}\subset \R^{n+1}$. We use a connection between the stability operator and the quantum harmonic oscillator Ham…

Study self shrinkers with medium entropy in 4D space.

Existence proof of noncompact self-shrinkers with arbitrary genus.

Using a maximum principle for self-shrinkers of the mean curvature flow, we give new proofs of a rigidity theorem for rotationally symmetric compact self-shrinkers and a result about the asymptotic behavior of self-shrinkers. This comparison argument also implies a linear bound for the second fundamental form of self-s…

In this paper, we study the Lagrangian F-stability of closed Lagrangian self-shrinkers immersed in complex Euclidean space. We show that any closed Lagrangian self-shrinker with first Betti number greater than one is Lagrangian F-unstable. In particular, any two-dimensional embedded closed Lagrangian self-shrinker is L…

New self-shrinkers found in higher dimensions.

Study classifies 3D self-shrinkers with constant second form norm.

We prove a local graphical theorem for two-dimensional self-shrinkers away from the origin. As applications, we study the asymptotic behavior of noncompact self-shrinkers with finite genus. Also, we show uniform boundedness on the second fundamental form of two-dimensional noncompact self-shrinkers with bounded mean cu…

The study proves properties of self-shrinkers with bounded curvature.

In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain sel…

The paper proves gap results for self-shrinkers in $r$-mean curvature flow.

Study proves spacelike self-shrinkers are hyperplanes under certain conditions.

In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotational symmetry. We also propose new existence and uniqueness problems for closed self-shrinkers with bi-rotational symmetry and provide…

Study bounds self-shrinker entropy using Li-Yau volume and Colding-Minicozzi entropy.

Self-shrinkers are important geometric objects in the study of mean curvature flows, while the Bernstein Theorem is one of the most profound results in minimal surface theory. We prove a Bernstein type result for graphical self-shrinker surfaces with codimension two in $\mathbb{R}^4$. Namely, under certain natural cond…

Researchers develop a numerical method to compute the index of self-shrinkers, finding it to be 5 for the Angenent torus.

Classifies self-shrinkers in arbitrary dimensions under specific curvature conditions.

Numerically estimates Colding-Minicozzi entropies of self-shrinkers.

Study entropy bounds and finiteness for symmetric self-shrinkers.

It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for $\mathcal{L}$-operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on \emph{polynomial volume growth}, which is…

The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.

Study proves rigidity of specific self-shrinkers under certain geometric conditions.

We obtain a Calabi-Yau type lower volume growth estimates for complete noncompact self-shrinkers of the mean curvature flow, more precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.

Proves unknottedness of certain 3D shapes with multiple ends.

Proves a pinching theorem for self-shrinkers of mean curvature flow.

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates, we can prove a compac…

Constructs self-shrinkers with unique asymptotic behavior.

The study classifies complete self-shrinkers in Euclidean space.

Logarithmic Sobolev inequality proven for non-compact self-shrinkers.

By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We prove rigidity results for squared norm of the second fundamental form of self-shrinkers, either under point-wise conditions or under integral conditions.