For a subvariety of a smooth projective variety, consider the family of smooth hypersurfaces of sufficiently large degree containing it, and take the quotient of the middle cohomology of the hypersurfaces by the cohomology of the ambient variety and also by the cycle classes of the irreducible components of the subvariety. Using Deligne’s semisimplicity theorem together with Steenbrink’s theory...

We consider n-dimensional convex Euclidean hypersurfaces moving with normal velocity proportional to a positive power α of the Gauss curvature. We prove that hypersurfaces contract to points in finite time, and for α ∈ (1/(n + 2], 1/n] we also prove that in the limit the solutions evolve purely by homothetic contraction to the final point. We prove existence and uniqueness of solutions for non-...

In this paper we compute certain two-point integrals over a moduli space of stable maps into projective space. Computation of one-point analogues of these integrals constitutes a proof of mirror symmetry for genus-zero one-point Gromov-Witten invariants of projective hypersurfaces. The integrals computed in this paper constitute a significant portion in the proof of mirror symmetry for genus-on...

The 3-metric gij and extrinsic curvature (second fundamental form) Kij are the fundamental variables describing the geometry in any space+time decomposition of the Einstein equations. (g,K) describe the local geometry of a single space-like hypersurface M , and it is then natural to describe the evolution of the space-time geometry by a 1-parameter family (g(t), K(t)), describing the local geom...

We generalize some results of Coray on closed points cubic hypersurfaces. show that certain symmetric products hypersurfaces are stably birational.

We investigate the volume preserving mean curvature flow with Neumann boundary condition for hypersurfaces that are graphs over a cylinder. Through a center manifold analysis we find that initial hypersurfaces sufficiently close to a cylinder of large enough radius, have a flow that exists for all time and converges exponentially fast to a cylinder. In particular, we show that there exist globa...

Four constructions of constant mean curvature (CMC) hypersurfaces in Sn+1 are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. S...

The linear cosmological perturbation theory of an almost homogeneous and isotropic perfect fluid universe is reconsidered and formally simplified by introducing new covariant and gauge-invariant variables with physical interpretations on hypersurfaces of constant expansion, constant curvature or constant energy density. The existence of conserved perturbation quantities on scales larger than th...

We study connected orientable spacelike hypersurfaces $x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ where $L_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a fixed integer $0leq k

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.