Related papers: A note on the Chern Conjecture in dimension four

Let $M^4\to \mathbb{S}^5$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ in a $5$-dimensional sphere $\mathbb{S}^5$. In this paper, we prove that if $3$-mean curvature $H_3$ and the…

In this paper, we prove that any closed minimal hypersurface $M^4$ in the $5$-dimensional unit sphere $\mathbb{S}^5$ with constant scalar curvature and constant $3$-th mean curvature must be isoparametric. To be precise, $M^4$ is either an…

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…

Chern's conjecture states that a closed minimal hypersurface in the euclidean sphere is isoparametric if it has constant scalar curvature. When the number $g$ of distinct principal curvatures is greater than three, few satisfactory results…

For a closed hypersurface $M^n\subset S^{n+1}(1)$ with constant mean curvature and constant non-negative scalar curvature, the present paper shows that if $\mathrm{tr}(\mathcal{A}^k)$ are constants for $k=3,\ldots, n-1$ for shape operator…

In this paper, we prove that a closed minimally immersed hypersurface $M^4\subset\mathbb S^5$ with constant $S:=\sum\limits_{i=1}^4\lambda_i^2$ and $A_3:=\sum\limits_{i=1}^4\lambda_i^3$ whose scalar curvature $R_M$ is nonnegative must be…

In this paper, we study $n$-dimensional complete minimal hypersurfaces in a unit sphere. We prove that an $n$-dimensional complete minimal hypersurface with constant scalar curvature in a unit sphere with $f_3$ constant is isometric to the…

In this paper, we prove that for an $n$-dimensional closed minimal Willmore hypersurface $M^n$ with constant scalar curvature in the unit sphere $\mathbb{S}^{n+1}$, the squared norm $S$ of the second fundamental form of $M^n$ satisfies…

Let $M^n$ be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose $\mathfrak{a}$ is a symmetric $(0,2)$ tensor field whose dual $(1,1)$ tensor $\mathcal{A}$ has $n$ distinct eigenvalues, and…

If $M$ is an isoparametric hypersurface in a sphere $S^n$ with four distrinct principal curvatures, then the principal curvatures $\kappa_1,...,\kappa_4$ can be ordered so that their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and the…

We give a partial local description of minimal hypersurfaces $M^3$ with identically zero Gau\ss-Kronecker curvature function in the unit 4-sphere $\Bbb{S}^4(1)$, without assumption on the compactness of $M^3$.

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…

We consider the isometric deformation problem for oriented non simply connected immersed minimal surfaces $f:M \to S^{4}$. We prove that the space of all isometric minimal immersions of $M$ into $S^{4}$ with the same normal curvature…

We classify complete orientable hypersurfaces of constant isotropic curvature in space forms. We show that such a hypersurface has constant mean curvature only if it is an isoparametric hypersurface, and that it is minimal if and only if it…

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature in a unit sphere $S^{n+1}(1)$ and construct many compact nontrivial embedded hypersurfaces with constant $m^{\text{th}}$ mean curvature…

The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an…

In this paper, we study four dimensional hypersurface M^4_r with proper mean curvature vector field (i.e. \Delta\vec{H} is proportional to \vec{H}) in pseudo-Riemannian space form N^5_s(c), and show that it has constant mean curvature, and…

A hypersurface $M^{n-1}$ in a real space-form ${\bf R}^n$, $S^n$ or $H^n$ is isoparametric if it has constant principal curvatures. For ${\bf R}^n$ and $H^n$, the classification of isoparametric hypersurfaces is complete and relatively…

Let $M$ be an $n$-dimensional closed hypersurface with constant mean curvature and constant scalar curvature in an unit sphere. Denote by $H$ and $S$ the mean curvature and the squared length of the second fundamental form respectively. We…

Let $ M^n$ be a closed immersed minimal hypersurface in the unit sphere $\mathbb{S}^{n+1}$. We establish a special isoperimetric inequality of $M^n$. As an application, if the scalar curvature of $ M^n$ is constant, then we get a uniform…