arXiv research

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…

Paper proves finite Morse index for certain self-shrinkers.

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.

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…

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.

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

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.

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

In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with $|A|^2\le 1$ in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension.

We construct an immersed and non-embedded $S^2$ self-shrinker.

Study eigenvalues of drift Laplacian on symmetric self-shrinkers in R^3.

Study proves Łojasiewicz inequalities for self-shrinkers, aiding in their uniqueness.

For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the r…

The purpose of this paper is to study complete self-shrinkers of mean curvature flow in Euclidean spaces. In the paper, we give a complete classification for 2-dimensional complete Lagrangian self-shrinkers in Euclidean space $\mathbb R^4$ with constant squared norm of the second fundamental form.

In this note we first show a compactness theorem for rotationally symmetric self shrinkers of entropy less than 2, concluding that there are entropy minimizing self shrinkers diffeomorphic to $S^1 \times S^{n-1}$ for each $n \geq 2$ in the class of rotationally symmetric self shrinkers. Assuming extra symmetry, namely …

In this note we show that compact self shrinkers in $\mathbb{R}^3$ are "topologically standard" in that any genus $g$ compact self shrinker is ambiently isotopic to the standard genus $g$ embedded surface in $\mathbb{R}^3$. As a consequence self shrinking tori are unknotted.