The existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers. 0. Introduction Consider the problem of finding a closed hypersurface of prescribed curvature F in a globally hyperbolic (n+1)-dimensional Lorentzian manifold N having a compact Cauchy hypersurface S0. To be more precise, let Ω be a connected op...

We construct solutions of u = e which blow-up precisely on a given space-like hypersurface of class H. For this purpose, we prove a general existence theorem for Fuchsian PDE in Sobolev spaces. The precise relation between the regularity of the data and that of the solution is shown to involve logarithmic symbols, in a model situation. A few further results on power nonlinearities are also incl...

Numerical relativity describes a discrete initial value problem for general relativity. A choice of gauge involves slicing spacetime into space-like hypersurfaces. This introduces past and future gauge relative to the hypersurface of present time. Here, we propose solving the discretized Einstein equations with a choice of gauge in the future and a dynamical gauge in the past. The method is ill...

1480 NOTICES OF THE AMS VOLUME 42, NUMBER 12 T he theory of functions (what we now call the theory of functions of a complex variable) was one of the great achievements of nineteenth century mathematics. Its beauty and range of applications were immense and immediate. The desire to generalize to higher dimensions must have been correspondingly irresistible. In this desire to generalize, there w...

We apply Cartan’s method of equivalence to construct invariants of a given null hypersurface in a Lorentzian space-time. This enables us to fully classify the internal geometry of such surfaces and hence solve the local equivalence problem for null hypersurface structures in 4-dimensional Lorentzian space-times. ∗Research supported by Komitet Badań Naukowych (Grant nr 2 P03B 060 17), the Erwin ...

We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in Pn of degree 2d ≤ min(n+4, 2n−2) and dimension at least three is irreducible and of the expected dimension.

Let SUX(3) be the moduli space of semi-stable vector bundles of rank 3 and trivial determinant on a curve X of genus 2. It maps onto P and the map is a double cover branched over a sextic hypersurface called the Coble sextic. In the dual P there is a unique cubic hypersurface, the Coble cubic, singular exactly along the abelian surface of degree 1 line bundles on X. We give a new proof that the...

The concept of time is discussed in the context of the canonical formulation of the gravitational field. Using a hypersurface orthogonal foliation, the arbitrariness of the lapse function is eliminated and the shift vector vanishes, allowing a consistent definition of time. I. ON LAPSES AND SHIFTS As it well know, the space-time of Newtonian mechanics is foliated by globally defined 3-dimension...

— A point P on a smooth hypersurface X of degree d in PN is called a star point if and only if the intersection of X with the embedded tangent space TP (X) is a cone with vertex P . This notion is a generalization of total inflection points on plane curves and Eckardt points on smooth cubic surfaces in P3. We generalize results on the configuration space of total inflection points on plane curv...