In this short communication, we announce results from our research on the structure of complete noncompact oriented weakly stable minimal hypersurfaces in a manifold of nonnegative sectional curvature. In particular, a complete oriented weakly stable minimal hypersurface in Rm, m > or = 4, must have only one end; any complete noncompact oriented weakly stable minimal hypersurface has only one e...

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We give necessary and sufficient conditions for such a hypersurface to be a pullback of a real-analytic curve in C via a meromorphic function. We define the rank of a real hypersurface and study the connections between rank, degree, and the type and size of the singularity for Levi-flat hypersurfaces. Finally, ...

We derive a general integral formula on an embedded hypersurface for general relativistic space-times. Suppose the hypersurface is foliated by two-dimensional compact “sections” Ss. Then the formula relates the rate of change of the divergence of outgoing light rays integrated over Ss under change of section to geometric (convexity and curvature) properties of Ss and the energy-momentum content...

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 ...

Let V be a degree d, reduced hypersurface in CP, n ≥ 1, and fix a generic hyperplane, H. Denote by U the (affine) hypersurface complement, CP−V ∪H, and let U be the infinite cyclic covering of U corresponding to the kernel of the linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules Hi(U ;Q) of the hypersurface complement and show t...

The Lewy extension theorem asserts the holomorphic extendability of CR functions defined in a neighborhood of a point on a hypersurface in C. The edge-of-the-wedge theorem asserts the extendability of holomorphic functions defined in wedges in C with edge a maximally real submanifold. In this article we prove under suitable hypotheses the holomorphic extendability to an open set in C of CR func...

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combina...

LetX be a compact, strictly convex C-hypersurface in the (n+1)-dimensional Euclidean space R. The Gauss map ofX maps the hypersurface one-to-one and onto the unit n-sphere S. One may parametrize X by the inverse of the Gauss map. Consequently, the Gauss curvature can be regarded as a function on S. The classical Minkowski problem asks conversely when a positive function K on S is the Gauss curv...

A new method for analytic inversion of Radon type integral transforms is proposed. Key words: Regular hypersurface family, Funk-Radon transform, Principal value integral, Reconstruction, Hyperbolic algebraic curve MSC 53C65 44A12 65R10 1 Introduction We present a uniform reconstruction method for a class of geometric integral transforms for submanifolds of codimension 1. The reconstruction does...