相关论文: Minimal hypersurfaces in $\mathbb{S}^5$ with vanis…

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…

We show that a closed minimal hypersurface in $\mathbb{S}^5$ with constant scalar curvature and zero Gauss curvature is totally geodesic.

We show that any complete minimal hypersurface in the five-dimensional hyperbolic space $\mathbb H^5$, endowed with constant scalar curvature and vanishing Gauss-Kronecker curvature, must be totally geodesic. Cheng-Peng [3] recently…

We investigate the structure of 3-dimensional complete minimal hypersurfaces in the unit sphere with Gauss-Kronecker curvature identically zero.

The aim of this paper is to complete the local classification of minimal hypersurfaces with vanishing Gauss-Kronecker curvature in a 4-dimensional space form. Moreover, we give a classification of complete minimal hypersurfaces with…

We investigate complete minimal hypersurfaces in the Euclidean space $% \ {R}^{4}$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\to {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature…

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 $M^3$.

In this paper, we completely classify the homothetical hypersurfaces having null Gauss-Kronocker curvature in a Euclidean (n+1)-space. Several applications to the production functions in economics are also given.

We show that a complete, two-sided, stable minimal hypersurface in $\mathbf{R}^5$ is flat.

In this paper we classify compact minimal surfaces in $S^5$ with non-negative Gaussian curvature using the notion of a contact angle.

We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface…

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 propose certain assumptions on the principal curvatures for a closed minimal hypersurface $M^5$ in $\mathbf{S}^6$ to be isoparametric, provided that the functions $S, f_3,f_4$ are constants. Our result removes the…

We study rotational hypersurfaces with constant Gauss-Kronecker curvature. We solve the ODE for the generating curves of such hypersurfaces and analyze several geometric properties of such hypersurfaces. In particular, we discover a class…

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…

We classify the homogeneous and isoparametric hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$. In the classification, besides the hypersurfaces $\mathbb{S}^1(r)\times\mathbb{S}^2,\,r\in (0,1]$, it appears a family of hypersurfaces with…

In this paper, we study hypersurfaces $M_{r}^{4}$ $(r=0, 1, 2, 3, 4)$ satisfying $\triangle \vec{H}=\lambda \vec{H}$ ($\lambda$ a constant) in the pseudo-Euclidean space $\mathbb{E}_{s}^{5}$ $(s=0, 1, 2, 3, 4, 5)$. We obtain that every such…

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…

Given a positive function $F$ on $\mathbb S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define for hypersurfaces in $\mathbb{R}^{n+1}$ the $r$-th anisotropic mean curvature function $H_{r; F}$, a generalization of the…

In this paper, we consider minimal hypersurfaces in the product space $\mathbb{H}^n \times \mathbb{R}$. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider…