arXiv research

Study on null hypersurfaces in complex contact manifolds.

The paper finds conditions for certain hypersurfaces to be totally umbilical.

Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.

Characterizes totally umbilical hypersurfaces in product spaces.

Study of hypersurfaces in Sol4_0 geometry, classifying parallel and totally umbilical types.

Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.

For $n\geq 2$ we define a notion of umbilicity for hypersurfaces in the Heisenberg group $H_{n}$. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant $p$(or horizontal)-mean curvature in $H_{n}$ up to Heisenberg translations.

In this article, we prove new stability results for almost-Einstein hypersurfaces of the Euclidean space, based on previous eigenvalue pinching results. Then, we deduce some comparable results for almost umbilical hypersurfaces.

Study null energy condition impacts on special hypersurfaces in static spacetimes.

The paper studies biharmonic conformal hypersurfaces in Riemannian manifolds.

Study of $f$-biharmonic hypersurfaces in conformally flat spaces.

We show that there is a correspondence between totally umbilic null hypersurfaces in generalized Robertson-Walker spaces and twisted decompositions of the fibre. This allows us to prove that nullcones are the unique totally umbilic null hypersurfaces in the closed Friedmann Cosmological model. We also apply this kind o…

In this paper we have studied the properties of covariant almost analytic vector field on Q - quasi umbilical hypersurface $M$ of a Sasakian manifold $\tilde M$ with $(φ, g, u, v, λ)-$structure and obtained the scalars $α$ and $β$ using $1 - form u, v$ covariant almost analytic for the hypersurface $M$ to be totally um…

We find explicitly all bi-umbilical foliated semi-symmetric hypersurfaces in the four-dimensional Euclidean space.

Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.

We show that any compact quaternionic contact (qc) hypersurfaces in a hyper-Kähler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. We also show that any nowhere umbilical qc hypersurface in a hyper-Kähler manifold is endowed with an involutiv…

Paper solves a mixed boundary value problem in space forms with umbilical boundaries.

Normal forms and invariants for nondegenerate hypersurfaces in C^2.

Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.

We study the principal configurations around an isolated $η$-umbilical point on a generic spacelike surface $S$ immersed in a null hypersurface $M$ of Minkowski space $\mathbb{R}_1^4$ relative to a well-defined null vector field $η$ orthogonal to the surface $S$. In the particular case of $M$ being a null rotation hype…

We introduce two classes of null hypersurfaces of an indefinite Sasakian manifold, $(\overline{M}, \overlineφ,ζ, η)$, tangent to the characteristic vector field $ζ$, called; {\it contact screen conformal} and {\it contact screen umbilic} null hypersurfaces. These hypersurfaces come in to fill the existing gap in screen…

We prove that a Riemannian product of type M x R (where R denotes the Euclidean line) admits totally umbilical hypersurfaces if and only if M has locally the structure of a warped product and we give a complete description of the totally umbilical hypersurfaces in this case. Moreover, we give a necessary and sufficient…

Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.

The paper examines stable capillary hypersurfaces in hyperbolic space.

In [2] we have classified the Blaschke quasi-umbilical submanifolds in the conformal space ${\mathbb Q}^n_s$. In this paper we shall classify the Blaschke para-umbilical hypersurfaces in the conformal space ${\mathbb Q}^n_s$. That may be also considered as the extension of the classification of the conformal isotropic …

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin$^c$ manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin$^c$ case the result of O. Kowalski stating that, every totally umbilical hypersurface of an Einstein manif…

We study immersed, connected, umbilic hypersurfaces in the Heisenberg group $H_{n}$ with $n$ $\geq$ $2.$ We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigm…

In this paper we have studied the properties of Quasi umbilical hypersurface $M$ of a Sasakian manifold $\tilde M$ with $(φ, g, u, v, λ)-$structure and established the relation for $M$ to be cylindrical.

This paper establishes the geometric structure of the lines of principal curvature of a hypersurface immersed in ${\mathbb R}^4$ in a neighborhood of the set $\mathcal{S}$ of its principal curvature singularities, consisting of the points at which atF least two principal curvatures are equal. Under generic conditions d…

Study of spacelike submanifolds with umbilical lightlike normals in Lorentzian spacetimes.

We consider closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using a new integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These…

It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1}, different from an Euclidean sphere, must have stability index greater than or equa…

The study examines spacelike foliations on Lorentz manifolds under specific conditions.

Let $Σ$ be a smooth closed hypersurface with non-negative Ricci curvature, isometrically immersed in a space form. It has been proved in \cite{P}, \cite{CZ}, and \cite{C2} that there are some $L^2$ inequalities on $Σ$ which measure the stability of closed umbilical hypersurfaces or more generally, closed hypersurfaces …

The paper classifies hypersurfaces in Heisenberg groups with rotational symmetry.

In this note, we give a geometric characterization of the compact and totally umbilical hypersurfaces that carry a non trivial locally static Killing Initial Data (KID). More precisely, such compact hypersurfaces have constant mean curvature and are isometric to one of the following manifolds: (i) Sn the standard spher…

In the present paper, we show that the geometry of a screen integrable null hypersurface can be generated from an isometric immersion of a leaf of its screen distribution into the ambient space. We prove, under certain geometric conditions, that such immersions are contained in semi-Euclidean spheres or hyperbolic spac…

In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on the compact hypersurfaces of ambient spaces with bounded sectional curvature. As application we deduce rigidity results for stable constant mean curvature hypersurfaces $M$ of these spaces $N$. Indeed, we prove that if $M$ is i…

The paper proves isoparametric functions on Finsler space forms under specific conditions.

We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \geq 2$, $p\in (1, \, +\infty)$ and $Σ$ an $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$, boundary of a convex, open set. We show that …

We define a complex connection on a real hypersurface of $\C^{n+1}$ which is naturally inherited from the ambient space. Using a system of Codazzi-type equations, we classify connected real hypersurfaces in $\C^{n+1}$, $n\ge 2$, which are Levi umbilical and have non zero constant Levi curvature. It turns out that such …

In this paper we study curvature properties of semi-symmetric type of totally umbilical radical transversal lightlike hypersurfaces $(M,g)$ and $(M,\widetilde g)$ of a Kähler-Norden manifold $(\overline M,\overline J,\overline g,\overline {\widetilde g})$ of constant totally real sectional curvatures $\overline ν$ and …

We show that every spherical 2-Dupin submanifold that is not a hypersurface is conformally congruent to the standard embedding of the real, complex, quaternionic or octonionic projective plane. We also classify 2-CPC, 2-umbilical and weakly 2-umbilical submanifolds in space forms.

Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.

New method classifies hypersurfaces that can bend infinitesimally.

Study on stability of mean curvature flow in hyperbolic space.

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds $N^n$ contained in hypersurfaces $M^{n+1}$ of the $(n+2)$-space. We give condition under which this affine focal set is a regular hypersurface and, for curves in $3$-space, we describe its stable singulariti…

The paper explores geometric properties of free boundary hypersurfaces in balls.