arXiv research

Paper proves finite Morse index for certain self-shrinkers.

Study self shrinkers with medium entropy in 4D space.

Compact shrinkers with curvature pinching conditions proven.

The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.

Existence proof of noncompact self-shrinkers with arbitrary genus.

Proves unknottedness of certain 3D shapes with multiple ends.

Constructs self-shrinkers with unique asymptotic behavior.

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 paper proves uniqueness and convergence of asymptotically conical self-shrinkers and self-expanders.

We show that each end of a noncompact self-shrinker in $\mathbb{R}^3$ of finite topology is smoothly asymptotic to either a regular cone or a self-shrinking round cylinder.

New theorem shows noncompact self shrinkers are unknotted.

We construct Gaussian Harmonic forms of finite Gaussian weighted $L^2$-norm on non-compact surfaces that detect each asymptotically conical end. As an application we prove an extension of the index estimates of self-shrinkers in $[11]$ under the existence of such ends. We show that the Morse index of a self-shrinker is…

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.

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…

Study classifies 4D shrinkers with nonnegative Ricci curvature.

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…

Proves mean curvature flow from conical singularities to shrinkers.

We show that if a shrinking soliton is asymptotic to a cone along an end then the isometry group of the cross-section of the cone embeds in the isometry group of the end of the shrinker. We also provide sufficient conditions for the isometries of the end to extend to the entire shrinker.

The paper proves continuity of Morse index for Ricci shrinkers.

Let $C\subset\mathbb{R}^{n+1}$ be a regular cone with vertex at the origin. In this paper, we show the uniqueness for smooth properly embedded self-shrinking ends in $\mathbb{R}^{n+1}$ that are asymptotic to $C$. As an application, we prove that not every regular cone with vertex at the origin has a smooth complete pro…

Proves K-polystability for Kähler-Ricci shrinkers with decaying curvature.

Study of mean curvature flows with conical singularities using mathematical techniques.

Study shows Ricci flow's convergence and harmonic map heat flow's long-time existence.

Researchers found a family of Sp(2)-invariant solitons for Laplacian flow.

New expanders for mean curvature flow contradict genus-reduction conjecture.

The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.

In this paper we generalize the neck-stability theorem of Kleiner-Lott to a special class of four-dimensional nonnegatively curved Type I $κ$-solutions, namely, those whose asymptotic shrinkers are the standard cylinder $\mathbb{S}^3\times\mathbb{R}$. We use this stability result to prove a rigidity theorem: if a four-…

In [LW], we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak form…

We prove that a shrinking gradient Ricci soliton which is asymptotic to a Kähler cone along some end is itself Kähler on some neighborhood of infinity of that end. When the shrinker is complete, it is globally Kähler.

Researchers construct translators asymptotic to self-shrinkers, proving non-uniqueness and fattening.

Ricci flow modelled on specific singularities on closed manifolds.

Backward propagation rules for warped products under Ricci flow.

We show that the Bowl soliton in $\mathbb{R}^3$ is the unique translating solutions of the mean curvature flow which has the family of shrinking cylinders as an asymptotic shrinker at $-\infty$. As an application, we show that for a generic mean curvature flow, all (non-static) translating limit flows are the bowl soli…

We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of special Lagrangian…

We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness…

We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self…

Proves existence of shrinkers via mean curvature flow.

The study counts ends on shrinkers using geometric covering methods.

Study on stability of network flow shrinkers with findings on instability of specific shapes.

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

Strong Frankel theorem for shrinkers in all dimensions.

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

The paper proves bounded curvature for Kähler Ricci shrinker surfaces.

Given a smooth, symmetric, homogeneous of degree one function $f\left(λ_{1},\cdots,\,λ_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\,n$, and a rotationally symmetric cone $\mathcal{C}$ in $\mathbb{R}^{n+1}$, we show that there is a $f$ self-shrinker (i.e. a hypersurface $Σ$ in $\mathbb{R}^{n+1}$ which …

Local gap theorem for Ricci shrinkers ensures flatness if certain functionals are close to zero.

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

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