Related papers: Classification of constant angle hypersurfaces in …
We classify hypersurfaces with rotational symmetry and positive constant $r$-th mean curvature in $\mathbb H^n \times \mathbb R$. Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also…
Given a Riemannian manifold $M,$ and an open interval $I\subset\mathbb{R},$ we characterize nontrivial totally umbilical hypersurfaces of the product $M\times I$ -- as well as of warped products $I\times_\omega M$ -- as those which are…
We show that for a generic $8$-dimensional Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the curvature condition, we show that for a dense set of 8-dimensional Riemannian metrics…
A submanifold is said to be tangentially biharmonic if the bitension field of the isometric immersion that defines the submanifold has vanishing tangential component. The purpose of this paper is to prove that a surface in Euclidean…
In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as "weighted'' mean curvatures, which extend the work…
We investigate ruled surfaces in 3d Riemannian manifolds, i.e., surfaces foliated by geodesics. In 3d space forms, we find the striction curve, distribution parameter, and the first and second fundamental forms, from which we obtain the…
In a seminal paper published in $1968$, J. Simons proved that, for $n\leq 5$, the Euclidean (minimal) cone $CM$, built on a closed, oriented, minimal and non totally geodesic hypersurface $M^n$ of $\mathbb S^{n+1}$ is unstable. In this…
In this paper, we introduce the notion of a prescribed angle hypersurface in a Riemannian manifold associated with a pair $(\mathcal{V},\theta)$, where $\mathcal{V}$ is a unit vector field along the hypersurface and $\theta$ denotes the…
We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant…
In the Minkowski space, we consider a compact, spacelike hypersurface with boundary, which can be written as a graph on a spacelike hyperplane. We prove that, if its $k$-th mean curvature is constant, and its boundary is on the hyperplane…
We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a (straight) ruled surface, i.e. it consists of straight lines tangent to a horizontal vector field along a…
We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied in \cite{Ji2}, \cite{CH}, \cite{CMO1},…
Superconformal surfaces in Euclidean space are the ones for which the ellipse of curvature at any point is a nondegenerate circle. They can be characterized as the surfaces for which a well-known pointwise inequality relating the intrinsic…
In 1968, Simons introduced the concept of index for hypersurfaces immersed into the Euclidean sphere S^{n+1}. Intuitively, the index measures the number of independent directions in which a given hypersurface fails to minimize area. The…
Consider the Euclidean space $\mathbb{R}^3$ endowed with a canonical semi-symmetric non-metric connection determined by a vector field $\mathsf{C}\in\mathfrak{X}(\mathbb{R}^3)$. We study surfaces when the sectional curvature with respect to…
In this paper, we study hypersurfaces in $\mathbb{H}^2\times\mathbb{H}^2$. We first classify the hypersurfaces with constant principal curvatures and constant product angle function. Then, we classify homogeneous hypersurfaces and…
In this paper, we study the elliptic Weingarten surfaces of minimal type immersed in the warped product space $\mathbb{R} \times_{h} \mathbb{R}$, when $h$ is a $C^{1}$-function in $\mathbb{R}^{2}$ with radial symmetry. That is, surfaces…
We derive the general formulas for a special configuration of the sequential warped product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a conformal metrics.…
We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $\Sigma$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The…
In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle…