Related papers: Rigidity of $4$-dimensional complete self-shrinker…

In this paper, we firstly verify that if $M$ is a complete self-shrinker with polynomial volume growth in $\mathbb{R}^{n+1}$, and if the squared norm of the second fundamental form of $M$ satisfies $0\leq|A|^2-1\leq\frac{1}{18}$, then…

Let $(M^n, g, f)$ be an $n$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \frac{1}{2}g$. 1. If its scalar curvature is $\frac{k}{2}$, Ricci curvature is nonnegative and sectional…

We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S^4 or a quotient of S^3 cross R. This gives a clean classification result removing the earlier additional…

In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three dimensional Euclidean space is a round sphere, provided its mean curvature and the norm of its position vector have an upper…

In dimension $4$, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform…

We classify four-dimensional shrinking Ricci solitons satisfying $Sec \geq \frac{1}{24} R$, where $Sec$ and $R$ denote the sectional and the scalar curvature, respectively. They are isometric to either $\mathbb{R}^{4}$ (and quotients),…

In this paper, we obtain a classification theorem of $2$-dimensional complete Lagrangian self-expanders with constant squared norm of the second fundamental form in $\mathbb C^{2}$.

In this paper, we prove that any Ricci shrinker that is sufficiently close to $(\mathbb C P^N,g_{FS})$ in the Gromov-Hausdorff sense must itself be isometric to $(\mathbb C P^N,g_{FS})$.

In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder as the only self-shrinkers satisfying a classic geometric assumption, namely the union of all tangent affine submanifolds of a complete…

Given a smooth, symmetric, homogeneous of degree one function $f=f\left(\lambda_{1},\cdots,\,\lambda_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\, n$, and an oriented, properly embedded smooth cone $\mathcal{C}^n$ in…

Let $(M, g, f)$ be a $5$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \lambda g$, where $\text{Ric}$ is the Ricci tensor and $\nabla^2f$ is the Hessian of the potential function $f$. We…

We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a…

Let $M^4$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ and constant 3-mean curvature $H_3$ in $\mathbb{S}^{5}$. If $H_3^2\leq \frac{1}{2}.$ and Gauss-Kronecker curvature $K_M$…

We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^{4}$ with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with…

Let M be a closed minimal hypersurface in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature. We prove that, if the sum of the cubes of all principal curvatures and the number of distinct principal curvatures are…

In this paper, we prove that the round cylinders are rigid in the space of Ricci shrinkers. Namely, any Ricci shrinker that is sufficiently close to $S^{n-1}\times \mathbb R$ in the pointed-Gromov-Hausdorff topology must itself be isometric…

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.

Suppose $(M^n, g, f)$ is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in \cite{Petersen-Wylie,Guan-Lu-Xu}. Using maximum principle, we prove that…

In this work, we establish several rigidity results for spacelike self-shrinkers immersed in the pseudo-Euclidean space $\mathbb{R}^{n+p}_p$. Under suitable boundedness conditions on either the mean curvature vector or the second…

In this article, we first classify Legendrian self-shrinkers in $\mathbb{R}% ^{3}$ and $\mathbb{R}^{5}$. We then proved a Legendrian rigidity theorem, which can be regarded as an analogue of the result of Li-Wang \cite{lw}. More precisely,…