Consider a compact embedded hypersurface ? t in R n+1 which moves with speed determined at each point by a function F (1 ; : : : ; n ; t) of its principal curvatures, for 0 t < T: We assume the problem is degenerate parabolic, that is, that F (; t) is nondecreasing in each of the principal curvatures 1 ; : : : ; n : We shall show that for t > 0 the hypersurface ? t sat-isses local a priori Lips...

For an (m+1)-dimensional space-time (X, g), define a mapped null hypersurface to be a smooth map ν : N → X (that is not necessarily an immersion) such that there exists a smooth field of null lines along ν that are both tangent and g-orthogonal to ν. We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle ST M of an immersed spacelike hypersurf...

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...

A cochain complex associated with the vector 1-form determined by the first and second fundamental tensors of a hypersurface M in E"+l is introduced. Its cohomology groups HP(M), called curvature groups, are isomorphic with the cohomology groups of M with coefficients in a subsheaf %R of the sheaf S of closed vector fields on M. If M is a minimal variety, the same conclusion is valid with S^ re...

Suppose that f defines a singular, complex affine hypersurface. If the critical locus of f is onedimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of f . This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.

In this article, we prove that either there exists at least one periodic orbit of Hamilton vector field on a given energy hypersurface in R 2n or there exist at least two periodic orbits on the near-by energy hypersurface in R. The more general results are also obtained.

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface X ⊂ P of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in P. We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.

We introduce a class extending the notion of Chern-Mather class to possibly nonreduced schemes, and use it to express the di erence between SchwartzMacPherson's Chern class and the class of the virtual tangent bundle of a singular hypersurface of a nonsingular variety. Applications include constraints on the possible singularities of a hypersurface and on contacts of nonsingular hypersurfaces, ...