New result on Chern conjecture for minimal hypersurfaces and its application

We verify that if $M$ is a compact minimal hypersurface in $\mathbb{S}^{n+1}$ whose squared length of the second fundamental form satisfying $0\leq |A|^2-n\leq\frac{n}{22}$, then $|A|^2\equiv n$ and $M$ is a Clifford torus. Moreover, we prove that if $M$ is a complete self-shrinker with polynomial volume growth in $\mathbb{R}^{n+1}$ whose equation is given by (\ref{selfshr}), and if the squared length of the second fundamental form of $M$ satisfies $0\leq|A|^2-1\leq\frac{1}{21}$, then $|A|^2\equiv1$ and $M$ is a round sphere or a cylinder. Our results improve the rigidity theorems due to Q. Ding and Y. L. Xin \cite{DX1,DX2}.Comment: 21 page

Mean curvature flow of arbitrary codimension in spheres and sharp differentiable sphere theorem

In this paper, we investigate Liu-Xu-Ye-Zhao's conjecture [31] and prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves the convergence theorem of Baker [3] as well as the differentiable sphere theorems of Gu-Xu-Zhao [17, 51, 53].Comment: 20 page

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

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 $|A|^2\equiv1$ and $M$ is a round sphere or a cylinder. More generally, let $M$ be a complete $\lambda$-hypersurface with polynomial volume growth in $\mathbb{R}^{n+1}$ with $\lambda\neq0$. Then we prove that there exists an positive constant $\gamma$, such that if $|\lambda|\leq\gamma$ and the squared norm of the second fundamental form of $M$ satisfies $0\leq|A|^2-\beta_\lambda\leq\frac{1}{18}$, then $|A|^2\equiv \beta_\lambda$, $\lambda>0$ and $M$ is a cylinder. Here $\beta_\lambda=\frac{1}{2}(2+\lambda^2+|\lambda|\sqrt{\lambda^2+4})$.Comment: 14 page

A New Version of Huisken's Convergence Theorem for Mean Curvature Flow in Spheres

We prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous convergence theorem due to Huisken [9]. Moreover, we prove a convergence theorem under the weakly pinching condition. In particular, we obtain a classification theorem for weakly pinched hypersurfaces. It should be emphasized that our pinching condition implies that the Ricci curvature of the initial hypersurface is positive, but does not imply positivity of the sectional curvature.Comment: 20 page

On Chern's conjecture for minimal hypersurfaces in spheres

Using a new estimate for the Peng-Terng invariant and the multiple-parameter method, we verify a rigidity theorem on the stronger version of Chern Conjecture for minimal hypersurfaces in spheres. More precisely, we prove that if $M$ is a compact minimal hypersurface in $\mathbb{S}^{n+1}$ whose squared length of the second fundamental form satisfies $0\leq S-n\leq\frac{n}{18}$, then $S\equiv n$ and $M$ is a Clifford torus.Comment: 15 page

An Optimal Convergence Theorem for Mean Curvature Flow of Arbitrary Codimension in Hyperbolic Spaces

In this paper, we prove that if the initial submanifold $M_0$ of dimension $n(\ge6)$ satisfies an optimal pinching condition, then the mean curvature flow of arbitrary codimension in hyperbolic spaces converges to a round point in finite time. In particular, we obtain the optimal differentiable sphere theorem for submanifolds in hyperbolic spaces. It should be emphasized that our pinching condition implies that the Ricci curvature of the initial submanifold is positive, but does not imply positivity of the sectional curvature of $M_0$.Comment: 24 page

An Eigenvalue Pinching Theorem for Compact Hypersurfaces in A Sphere

In this article, we prove an eigenvalue pinching theorem for the first eigenvalue of the Laplacian on compact hypersurfaces in a sphere. Let $(M^n,g)$ be a closed, connected and oriented Riemannian manifold isometrically immersed by $\phi$ into $\S^{n+1}$. Let $q>n$ and $A>0$ be some real numbers satisfying $|M|^\frac{1}{n}(1+\|B\|_q)\leq A$. Suppose that $\phi(M)\subset B(p_0,R)$, where $p_0$ is a center of gravity of $M$ and radius $R<\frac{\pi}{2}$. We prove that there exists a positive constant \e depending on $q$, $n$, $R$ and $A$ such that if n(1+\|H\|_\infty^2)-\e\leq \l_1, then $M$ is diffeomorphic to $\S^n$. Furthermore, $\phi(M)$ is starshaped with respect to $p_0$, Hausdorff close and almost-isometric to the geodesic sphere S\(p_0,R_0\), where $R_0=\arcsin\frac{1}{\sqrt{1+\|H\|_\infty^2}}$.Comment: 13 page

Mean curvature flow of arbitrary codimension in complex projective spaces

In this paper, we investigate the mean curvature flow of submanifolds of arbitrary codimension in $\mathbb{C}\mathbb{P}^m$. We prove that if the initial submanifold satisfies a pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as $t \rightarrow \infty$. Consequently, we obtain a new differentiable sphere theorem for submanifolds in $\mathbb{C}\mathbb{P}^m$. Our work improves the convergence theorem for mean curvature flow due to Pipoli and Sinestrari {\cite{PiSi2015}}.Comment: 31 page

Galois Hulls of Linear Codes over Finite Fields

The $\ell$-Galois hull $h_{\ell}(C)$ of an $[n,k]$ linear code $C$ over a finite field $\mathbb{F}_q$ is the intersection of $C$ and $C^{{\bot}_{\ell}}$, where $C^{\bot_{\ell}}$ denotes the $\ell$-Galois dual of $C$ which introduced by Fan and Zhang (2017). The $\ell$- Galois LCD code is a linear code $C$ with $h_{\ell}(C) = 0$. In this paper, we show that the dimension of the $\ell$-Galois hull of a linear code is invariant under permutation equivalence and we provide a method to calculate the dimension of the $\ell$-Galois hull by the generator matrix of the code. Moreover, we obtain that the dimension of the $\ell$-Galois hulls of ternary codes are also invariant under monomial equivalence. %The dimension of $l$-Galois hull of a code is not invariant under monomial equivalence if $q>4$. We show that every $[n,k]$ linear code over $\mathbb F_{q}$ is monomial equivalent to an $\ell$-Galois LCD code for any $q>4$. We conclude that if there exists an $[n,k]$ linear code over $\mathbb F_{q}$ for any $q>4$, then there exists an $\ell$-Galois LCD code with the same parameters for any $0\le \ell\le e-1$, where $q=p^e$ for some prime $p$. As an application, we characterize the $\ell$-Galois hull of matrix product codes over finite fields

Surfaces pinched by normal curvature for mean curvature flow in space forms

In this paper, we investigate the mean curvature flow of compact surfaces in $4$-dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which generalise Baker-Nguyen's convergence theorem