A smooth, nondegenerate hypersurface in projective space contains no linear subvarieties of greater than half its dimension. It can contain linear subvarieties of half its dimension. This note proves that a smooth hypersurface of degree d ≥ 3 contains at most finitely many such subvarieties. Let k be a field. Let X ⊂ Pk be a hypersurface of degree d > 1. For each integer m > 0, denote by Fm(X) ...

We establish an integral formula on a smooth, precompact domain in a Kahler manifold. We apply this formula to study holomorphic extension of CR functions. Using this formula we prove an isoperimetric inequality in terms of a positive lower bound for the Hermitian curvature of the boundary. Combining with a Minkowski type formula on the complex hyperbolic space we prove that any closed, embedde...

Recommended by Christian Corda We obtain a height estimate concerning to a compact spacelike hypersurface Σ n immersed with constant mean curvature H in the anti-de Sitter space H n1 1 , when its boundary ∂Σ is contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic space H n. Our estimate depends only on the value of H and on the geometry of ∂Σ...

‎let $m^n$ be an $n(ngeq 3)$-dimensional complete connected and‎ ‎oriented spacelike hypersurface in a de sitter space or an anti-de‎ ‎sitter space‎, ‎$s$ and $k$ be the squared norm of the second‎ ‎fundamental form and gauss-kronecker curvature of $m^n$‎. ‎if $s$ or‎ ‎$k$ is constant‎, ‎nonzero and $m^n$ has two distinct principal‎ ‎curvatures one of which is simple‎, ‎we obtain some‎ ‎charact...

We deposit a prescribed amount of liquid on an umbilical hypersurface of the hyperbolic space Hn+1. Under the presence of a uniform gravity vector field directed towards , we seek the shape of such a liquid drop in a state of equilibrium of the mechanical system. The liquid-air interface is then modeled by a hypersurface under the condition that its mean curvature is a function of the distance ...

We study umbilic (space-like) submanifolds of pseudo-Riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo Euclidean space and relate this notion to umbilicity. Finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.A pseudo-Riemannian submanifold M in (a...

We prove that a hypersurface of a space form with almost constant mean curvature and almost constant scalar curvature is close to a geodesic sphere. In the case of Euclidean space, we deduce new characterizations of geodesic spheres.

I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open subset of such a space.

A hypersurface is said to be totally biharmonic if all its geodesics are curves in the ambient space. We prove that a into space form an isoparametric hypersurface, which allows us give full classification of hypersurfaces these spaces. Moreover, restricting ourselves 3-dimensional case, we show surfaces Bianchi–Cartan–Vranceanu spaces and their classification. In particular, that, leaving asid...

In this paper we develop the theory of finite-type invariants for homologically nontrivial 3-manifolds . We construct an infinite-dimensional affine space with a hypersurface in it corresponding to manifolds with Morse singularities. Connected components of the complement of this hypersurface correspond to homeomorphism type of spin 3-manifolds. This suggests the natural axiomatics of Vassiliev...