In this paper, we study $n(n\ge 3)$-dimensional complete connected and oriented space-like hypersurfaces $M^n$ in an (n+1)-dimensional Lorentzian space form $M_1^{n+1}\left(c\right)$ with non-zero constant $k$-th $(k<n)$ mean curvature and two distinct principal curvatures $\lambda$ and $\mu$. We give some characterizations of Riemannian product $H^m\left(c_1\right)\times M^{n-m}\left(c_2\right)$ and show that the Riemannian product $H^m\left(c_1\right)\times M^{n-m}\left(c_2\right)$ is the only complete connected and oriented space-like hypersurface in $M_1^{n+1}\left(c\right)$ with constant $k$-th mean curvature and two distinct principal curvatures, if the multiplicities...

We give some optimal estimates of the height, curvature and volume of compact hypersurfaces in ${ℝ}^{n+1}$ with constant curvature bounding a planar closed (n-1)-submanifold.

We generalize a parametrization obtained by A. V. Corro in (2006) in the three-dimensional Euclidean space. Using this parametrization we study a class of oriented hypersurfaces $M^n$, $n\ge 2$, in Euclidean space satisfying a relation ${\sum }_{r=1}^n{(-1)}^{r+1}r{f}^{r-1}\left(\genfrac{}{}{0pt}{}{n}{r}\right)H_r=0,$ where $H_r$ is the $r$th mean curvature and $f\in C^{\infty }(M^n;ℝ)$, these hypersurfaces are called Weingarten hypersurfaces of the spherical type. This class of hypersurfaces includes the surfaces of the spherical type (Laguerré minimal surfaces). We characterize these hypersurfaces in terms...

In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space ${ℍ}^{n+1}$, that is, complete hypersurfaces of ${ℍ}^{n+1}$ whose mean curvature $H$ and normalized scalar curvature $R$ satisfy $R=aH+b$ for some $a$, $b\in ℝ$. In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of ${ℍ}^{n+1}$. Furthermore,...

Let Mⁿ be a hypersurface in $R^{n+1}$. We prove that two classical Jacobi curvature operators $J_x$ and $J_y$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $\left(K_{x,y}\circ K_{z,u}\right)\left(u\right)=\left(K_{z,u}\circ K_{x,y}\right)\left(u\right)$, where $K_{x,y}\left(u\right)=R(x,y,u)$, for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.

Let M be a closed connected surface in $R^3$ with positive Gaussian curvature K and let $K_{I}I$ be the curvature of its second fundamental form. It is shown that M is a sphere if $K_{I}I=c\surd H{K}^r$, for some constants c and r, where H is the mean curvature of M.

In the present paper we give some properties of $f$-biharmonic hypersurfaces in real space forms. By using the $f$-biharmonic equation for a hypersurface of a Riemannian manifold, we characterize the $f$-biharmonicity of constant mean curvature and totally umbilical hypersurfaces in a Riemannian manifold and, in particular, in a real space form. As an example, we consider $f$-biharmonic vertical cylinders in $S^2\times ℝ$.

We consider a sequence of multi-bubble solutions $u_k$ of the following fourth order equation $${\Delta }^2{u}_k={\rho }_k\frac{h\left(x\right)e^{u_k}}{{\int }_{\Omega }h{e}^{u_k}}\text{in}\Omega ,u_k=\Delta u_k=0\text{on}\partial \Omega ,(*)$$ where $h$ is a $C^{2,\beta }$ positive function, $\Omega$ is a bounded and smooth domain in ${ℝ}^4$, and ${\rho }_k$ is a constant such that ${\rho }_k\le C$. We show that (after extracting a subsequence), ${lim}_{k\to +\infty }{\rho }_k=32{\sigma }_{3}m$ for some positive integer $m\ge 1$, where ${\sigma }_3$ is the area of the unit sphere in ${ℝ}^4$. Furthermore, we obtain the following sharp estimates for ${\rho }_k$: $$\begin{array}{cc}\hfill {\rho }_k-32{\sigma }_{3}m& =c_0\sum _{j=1}^m{ϵ}_{k,j}^2\left(\sum _{l\ne j}\Delta G_4(p_j,p_l)+\Delta R_4(p_j,p_j)+\frac{1}{32{\sigma }_3}\Delta logh\left(p_j\right)\right)\hfill \\ & +o\left(\sum _{j=1}^m{ϵ}_{k,j}^2\right)\hfill \end{array}$$ where $c_0\&gt;0$, $log\frac{64}{{ϵ}_{k,j}^4}=\underset{x\in B_{\delta }\left(p_j\right)}{max}{u}_k\left(x\right)-log\left(\underset{\Omega }{\int }h{e}^{u_k}\right)$ and $u_k\to 32{\sigma }_3\sum _{j=1}^m{G}_4(·,p_j)$ in $C_{\mathrm{loc}}^4(\Omega \setminus \{p_1,...,p_m\})$. This yields a bound of solutions as...

We study lightlike hypersurfaces $M$ of an indefinite Kaehler manifold $\overline{M}$ of quasi-constant curvature subject to the condition that the characteristic vector field $\zeta$ of $\overline{M}$ is tangent to $M$. First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface $M$ of $\overline{M}$ such that (1) the screen distribution $S\left(TM\right)$ is totally umbilical or (2) $M$ is screen conformal.