arXiv research

Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.

Study on hypersurfaces in pseudo-Euclidean space with constant curvature or rotational properties.

Study on unique minimal hypersurfaces in rotational domains.

Study classifies rotational hypersurfaces with prescribed mean curvature.

The paper classifies CMC free boundary hypersurfaces in rotational domains.

The paper defines and calculates fourth fundamental form and i-th curvatures for hypersurfaces in 4D Euclidean space.

In this paper, we are concerned with hypersurfaces in $H^n\times R$ with constant r-mean curvature, to be called $H_r$-hypersurfaces. We construct examples of complete $H_r$-hypersurfaces which are invariant by parabolic screw motion or by rotation. We prove that there is a unique rotational strictly convex entire $H_r…

The paper confirms Yau's conjecture for minimal rotational hypersurfaces.

Paper shows rigidity of rotationally symmetric minimal hypersurfaces in 5D spaces.

We prove that any piece of a rotational hypersurface with prescribed mean curvature function in a Euclidean space can be uniquely extended infinitely, which generalizes the results by Euler and Delaunay for surfaces of revolution with constant mean curvautre. Next, we prove the same kind of theorem for generalized rota…

Paper proves existence of a special 3D shape with rotational symmetry.

The paper characterizes hypersurfaces in curved spaces using their geometry.

The paper classifies and studies conformally flat hypersurfaces in 4D space.

The paper classifies rotational hypersurfaces in n-space using a modified Laplacian operator.

We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kaehler manifolds with the above mentioned metrics. New examples o…

The paper classifies hypersurfaces in product spaces with specific curvature properties and finds that only rotational ones admit almost Ricci solitons.

Classifies hypersurfaces with positive constant mean curvature in hyperbolic space.

The study explores special hypersurfaces in Riemannian products, focusing on elliptic Weingarten conditions.

Explains minimal surfaces and their properties.

The paper computes spectra of Laplacian and Jacobi operators on rotational cmc hypersurfaces of spheres.

In the previous paper, it has been proved that the generalized rotational hypersurfaces of O(n-1)-type and O (l+1) x O(m+1)-type, for which the mean curvature is any prescribed continuous function. This paper is a sequel, and a similar existence result is shown for any type.

The study constructs and classifies hypersurfaces with constant curvature in product spaces.

The study classifies special Lorentz surfaces in a 4-space with neutral metric.

We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+1}=0$ and we classify them. Then we prove some uniqueness theorems for $r$-minimal hypersurfaces with a given (finite or asymptotic) boundary. In particular, we obtain a Schoen-type Theorem for two ended complete hypersurfa…

The paper studies eigenvalues and stability of hypersurfaces in spheres.

We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also con…

Derives a sharp inequality for trace-free matrices with applications to hypersurfaces.

We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hyper…

We describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at…

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces …

In this article we obtain a classification of strictly locally convex affine hypersurfaces in A^{n+1} for which the geometrical structure is pointwise invariant under the group SO(n-1) represented by rotations around a fixed axis in the tangent space.

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 minimal hypersurfaces with finite total curvature. This assumption implies that the …

Classifies weakly Einstein hypersurfaces in spaces of constant curvature.

In this work, we prove a version of the fundamental theorem of submanifolds to target manifolds with warped structure.

The paper classifies hypersurfaces in Heisenberg groups with rotational symmetry.

The study explores Einstein hypersurfaces in warped product spaces and their properties.

Study finds non-CMC biconservative hypersurfaces in spheres, proving their existence but not embeddability.

Characterizes curves on geodesic spheres and totally geodesic hypersurfaces in hyperbolic and spherical spaces.

Study on biconservative hypersurfaces with constant scalar curvature in space forms.

Given an isometric immersion $f\colon M^n\to \R^{n+1}$ of a compact Riemannian manifold of dimension $n\geq 3$ into Euclidean space of dimension $n+1$, we prove that the identity component $Iso^0(M^n)$ of the isometry group $Iso(M^n)$ of $M^n$ admits an orthogonal representation $Φ\colon Iso^0(M^n)\to SO(n+1)$ such tha…

In this note we characterize compact hypersurfaces of dimension $n\geq 2$ with constant mean curvature $H$ immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when $n\geq 3$ and $H\neq 0$, they are locally contained in a rotational h…

We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in a canonic…

We flow a hypersurface in Euclidean space by mean curvature flow with a Neumann boundary condition, where the boundary manifold is any torus of revolution. If we impose the conditions that the initial manifold is compatible and does not contain the rotational vector field in its tangent space, then mean curvature flow …

A hypersurface without umbilics in the n+1 dimensional Euclidean space is known to be determined by the Moebius metric and the Moebius second fundamental form up to a Moebius transformation when n>2. In this paper we consider Moebius rigidity for hypersurfaces and deformations of a hypersurface preserving the Moebius m…

Study principal configurations near special points on spacelike surfaces in null hypersurfaces.

In this paper we describe all rotation $H$-hypersurfaces in $H^n \times R$ and use them as barriers to prove existence and characterization of certain vertical $H$-graphs and to give symmetry and uniqueness results for compact $H$-hypersurfaces whose boundary is one or two parallel submanifolds in slices. We also descr…

Study finds new minimal surfaces in Schwarzschild space.

Minimal hypersurfaces in spheres generated by isoparametric foliations are found.