arXiv subjects
Publications and source records attributed to F. Manfio.
In this paper, we classify umbilic-free hypersurfaces $f: M^{n}\to\mathbb{R}^{n+1}$, $n \geq 4,$ with semi-parallel Moebius second fundamental form and three distinct principal curvatures.
We will present an estimate for the first eigenvalue of the Dirichlet and Neumann problems in terms of the Bakry-\'Emery Ricci curvature for a compact weighted manifold. As an application we will establish a stability condition for a h-minimal hypersurface.
In this paper we establish conditions on the length of the traceless part of the second fundamental form of a complete constant mean curvature hypersurface immersed in a space of constant sectional curvature in order to show that it is totally umbilical.
We consider hypersurfaces of products $M\times\mathbb R$ with constant $r$-th mean curvature $H_r\ge 0$ (to be called $H_r$-hypersurfaces), where $M$ is an arbitrary Riemannian $n$-manifold. We develop a general method for constructing them, and employ it to produce many examples for a variety of manifolds $M,$ including all simply connected space forms and the hyperbolic spaces $\mathbb{H}_{\mathbb F}^m$ (rank $1$ symmetric spaces of noncompact type). We construct and classify complete rotational $H_r(\ge 0)$-hypersurfaces in $\mathbb{H}_{\mathbb F}^m\times\mathbb R$ and in $\mathbb S^n\times\mathbb R$ as well. They include spheres, Delaunay-type annuli and, in the case of $\mathbb{H}_{\mathbb F}^m\times\mathbb R,$ entire graphs. We also construct and classify complete $H_r(\ge 0)$-hypersurfaces of $\mathbb{H}_{\mathbb F}^m\times\mathbb R$ which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett-Liebmann-type theorem by showing that a compact, connected and strictly convex $H_r$-hypersurface of $\mathbb H^n\times\mathbb R$ or $\mathbb S^n\times\mathbb R$ $(n\ge 3)$ is a rotational embedded sphere. Other uniqueness results for complete $H_r$-hypersurfaces of these ambient spaces are obtained.