Let Mⁿ (n ≥ 3) be an n-dimensional complete super stable minimal submanifold in ${ℝ}^{n+p}$ with flat normal bundle. We prove that if the second fundamental form A of M satisfies ${\int }_{M}i{\left|A\right|}^{\alpha }<\infty$, where α ∈ [2(1 - √(2/n)), 2(1 + √(2/n))], then M is an affine n-dimensional plane. In particular, if n ≤ 8 and ${\int }_M{\left|A\right|}^d<\infty$, d = 1,3, then M is an affine n-dimensional plane. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite $L^{\alpha }$-norm curvature in ℝ⁷ are considered.

In this paper, we characterize the $n$-dimensional $(n\ge 3)$ complete spacelike hypersurfaces $M^n$ in a de Sitter space $S_1^{n+1}$ with constant scalar curvature and with two distinct principal curvatures one of which is simple.We show that $M^n$ is a locus of moving $(n-1)$-dimensional submanifold $M_1^{n-1}\left(s\right)$, along $M_1^{n-1}\left(s\right)$ the principal curvature $\lambda$ of multiplicity $n-1$ is constant and $M_1^{n-1}\left(s\right)$ is umbilical in $S_1^{n+1}$ and is contained in an $(n-1)$-dimensional sphere $S^{n-1}\left(c\left(s\right)\right)=E^n\left(s\right)\cap S_1^{n+1}$ and is of constant curvature ${\left(\frac{d\{log|{\lambda }^2-(1-R){|}^{1/n}\}}{ds}\right)}^2-{\lambda }^2+1$,where $s$ is the arc length of an orthogonal trajectory of the family...

In this paper we investigate the geometry of conformal Killing graphs in a Riemannian manifold ${\overline{M}}_f^{n+1}$ endowed with a weight function $f$ and having a closed conformal Killing vector field $V$ with conformal factor ${\psi }_V$, that is, graphs constructed through the flow generated by $V$ and which are defined over an integral leaf of the foliation $V^{\perp }$ orthogonal to $V$. For such graphs, we establish some rigidity results under appropriate constraints on the $f$-mean curvature. Afterwards, we obtain some stability results...

It is still an open question whether a compact embedded hypersurface in the Euclidean space with constant mean curvature and spherical boundary is necessarily a hyperplanar ba1l or a spherical cap, even in the simplest case of a compact constant mean curvature surface in R3 bounded by a circle. In this paper we prove that this is true for the case of the scalar curvature. Specifica1ly we prove that the only compact embedded hypersurfaces in the Euclidean space with constant scalar curvature and...