Complete Self-shrinkers Research Articles

Classification of complete 3-dimensional self-shrinkers in the Euclidean space ℝ4

In this paper, we completely classify $3$-dimensional complete self-shrinkers with constant norm $S$ of the second fundamental form and constant $f_{3}$ in Euclidean space $\mathbb R^{4}$, where $h_{ij}$ are components of the second fundamental form, $S=\sum_{i,j}h^{2}_{ij}$ and $f_{3}=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$.

Rigidity of complete self-shrinkers whose tangent planes omit a nonempty set

Rigidity of complete self-shrinkers whose tangent planes omit a nonempty set

Complete Self-similar Hypersurfaces to the Mean Curvature Flow with Nonnegative Constant Scalar Curvature

As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. In this paper we give complete classifications of codimension one complete self-shrinkers with nonnegative constant scalar curvature. We also give alternative proofs of the classifications of codimension one translating solitons and self-expanders with nonnegative constant scalar curvature.

The second gap on complete self-shrinkers

In this paper, we study complete self-shrinkers in Euclidean space and prove that an n n -dimensional complete self-shrinker in Euclidean space R n + 1 \mathbb {R}^{n+1} is isometric to either R n \mathbb {R}^{n} , S n ( n ) S^{n}(\sqrt {n}) , or S k ( k ) × R n − k S^k (\sqrt {k})\times \mathbb {R}^{n-k} , 1 ≤ k ≤ n − 1 1\leq k\leq n-1 , if the squared norm S S of the second fundamental form, f 3 f_3 are constant and S S satisfies S > 1.83379 S>1.83379 . We should remark that the condition of polynomial volume growth is not assumed.

Complete Lagrangian self-shrinkers in $${\mathbf {R}}^4$$

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.

Complete self-shrinkers with constant norm of the second fundamental form

In this paper, we classify 3-dimensional complete self-shrinkers in Euclidean space \({\mathbb {R}}^{4}\) with constant squared norm of the second fundamental form S and constant \(f_{4}\).

欧氏空间中完备自收缩子的刚性定理

欧氏空间中完备自收缩子的刚性定理

A new pinching theorem for complete self-shrinkers and its generalization

In this paper, we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in ℝn+1, and if the squared norm of the second fundamental form of M satisfies \(0\leqslant S - 1\leqslant {1 \over {18}}\), then S ≡ 1 and M is a round sphere or a cylinder. More generally, let M be a complete λ-hypersurface of codimension one with polynomial volume growth in ℝn+1 with λ ≠ 0. Then we prove that there exists a positive constant γ, such that if |λ| ⩽ γ and the squared norm of the second fundamental form of M satisfies \(0\leqslant S - \beta_\lambda\leqslant {1 \over {18}}\), then S ≡ βλ, λ > 0 and M is a cylinder. Here \({\beta _\lambda} = {1 \over 2}(2 + {\lambda ^2} + \left| \lambda \right|\sqrt {{\lambda ^2} + 4})\).

Rigidity and curvature estimates for graphical self-shrinkers

Self-shrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It it presently known that an entire self-shrinking graph must be a hyperplane. In this paper we show that the hyperplane is rigid in an even stronger sense, namely: For $2 \leq n \leq 6$, any smooth, complete self-shrinker $\Sigma^n\subset\mathbf{R}^{n+1}$ that is graphical inside a large, but compact, set must be a hyperplane. In fact, this rigidity holds within a larger class of almost stable self-shrinkers. A key component of this paper is the procurement of linear curvature estimates for almost stable shrinkers, and it is this step that is responsible for the restriction on $n$. Our methods also yield uniform curvature bounds for translating solitons of the mean curvature flow.

On Chern's conjecture for minimal hypersurfaces and rigidity of self-shrinkers

Using the generalized Yau parameter method and the Sylvester theory, we verify that if M is a compact minimal hypersurface in Sn+1 whose squared length of the second fundamental form satisfies 0≤|A|2−n≤n22, then |A|2≡n and M is a Clifford torus. Moreover, we prove that if M is a complete self-shrinker with polynomial volume growth in Rn+1, and if the squared length of the second fundamental form of M satisfies 0≤|A|2−1≤121, then |A|2≡1 and M is a round sphere or a cylinder. Our results improve the rigidity theorems due to Ding and Xin [21,22].

SELF-SHRINKERS WITH SECOND FUNDAMENTAL FORM OF CONSTANT LENGTH

We give a new and simple proof of a result of Ding and Xin, which states that any smooth complete self-shrinker in $\mathbb{R}^{3}$ with the second fundamental form of constant length must be a generalised cylinder $\mathbb{S}^{k}\times \mathbb{R}^{2-k}$ for some $k\leq 2$. Moreover, we prove a gap theorem for smooth self-shrinkers in all dimensions.

2-Dimensional complete self-shrinkers in $$\mathbf {R}^3$$ R 3

It is our purpose to study complete self-shrinkers in Euclidean space. By making use of the generalized maximum principle for \(\mathcal {L}\)-operator, we give a complete classification for 2-dimensional complete self-shrinkers with constant squared norm of the second fundamental form in \(\mathbb R^3\). Ding and Xin (Trans Am Math Soc 366:5067–5085, 2014) have proved this result under the assumption of polynomial volume growth, which is removed in our theorem.

Stability and compactness for complete 𝑓-minimal surfaces

Let ( M , g ¯ , e − f d μ ) (M,\overline {g}, e^{-f}d\mu ) be a complete metric measure space with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove that in M M there is no complete two-sided L f L_f -stable immersed f f -minimal hypersurface with finite weighted volume. Further, if M M is a 3 3 -manifold, we prove a smooth compactness theorem for the space of complete embedded f f -minimal surfaces in M M with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in R 3 \mathbb {R}^3 by Colding-Minicozzi.

Classification and rigidity of self-shrinkers in the mean curvature flow

In this paper, we first use the method of Colding and Minicozzi [5] to show that K. Smoczyk's classification theorem [16] for complete self-shrinkers in higher codimension also holds under a weaker condition. Then as an application, we give some rigidity results for self-shrinkers in arbitrary codimension.

Complete self-shrinkers of the mean curvature flow

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 polynomial volume growth, which is assumed in Cao and Li [5]. Thus, we can obtain the rigidity theorems on complete self-shrinkers without assumption on polynomial volume growth. For complete proper self-shrinkers of dimension 2 and 3, we give a classification of them under assumption of constant squared norm of the second fundamental form.

Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space

Letf(x)be a smooth strictly convex solution ofdet(∂2f/∂xi∂xj)=exp(1/2)∑i=1nxi(∂f/∂xi)-fdefined on a domainΩ⊂Rn; then the graphM∇fof∇fis a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean spaceRn2nwith the indefinite metric∑dxidyi. In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graphM∇fis complete inRn2nand passes through the origin then it is flat.

ESTIMATES FOR EIGENVALUES OF $\mathfrak L$ OPERATOR ON SELF-SHRINKERS

In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator [Formula: see text], which is introduced by Colding and Minicozzi in [Generic mean curvature flow I; generic singularities, Ann. Math.175 (2012) 755–833], on an n-dimensional compact self-shrinker in R n+p. Estimates for eigenvalues of the differential operator [Formula: see text] are obtained. Our estimates for eigenvalues of the differential operator [Formula: see text] are sharp. Furthermore, we also study the Dirichlet eigenvalue problem of the differential operator [Formula: see text] on a bounded domain with a piecewise smooth boundary in an n-dimensional complete self-shrinker in R n+p. For Euclidean space R n, the differential operator [Formula: see text] becomes the Ornstein–Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein–Uhlenbeck operator.

Complete self-shrinkers confined into some regions of the space

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 compact, we deduce spectral information on the natural drifted Laplacian associated to the self-shrinker. These results go in the direction of verifying the validity of a conjecture by H. D. Cao concerning the polynomial volume growth of complete self-shrinkers. A finite strong maximum principle in case the self-shrinker is confined into a cylindrical product is also presented.