In this paper we modify the maximum principal of (Galloway, 2000) for totally geodesic null hypersurfaces by proving a geometric maximum principle which obeys mean curvature inequalities of a family of totally umbilical null hypersurfaces of a spacetime manifold (Theorem 6). As a physical interpretation we show that, in particular, for a prescribed class of spacetimes the geometric inequality o...

We consider spacetimes with compact Cauchy hypersurfaces and with Ricci tensor bounded from below on the set of timelike unit vectors, and prove that the results known for spacetimes satisfying the timelike convergence condition, namely, foliation by CMC hypersurfaces, are also valid in the present situation, if corresponding further assumptions are satisfied. In addition we show that the volum...

The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of Z generated by the class of a point. Thus, the span of the graph hypersurfaces in the Grothendieck ring is nearly killed by setting the Lefschetz motive L to...

In this note we show that there are algebraic families of hyperbolic, FermatWaring type hypersurfaces in P of degree 4(n − 1), for all dimensions n ≥ 2. Moreover, there are hyperbolic Fermat-Waring hypersurfaces in P of degree 4n − 2n + 1 possessing complete hyperbolic, hyperbolically embedded complements. Many examples have been given of hyperbolic hypersurfaces in P (e.g., see [ShZa] and the ...

We describe convex quadric hypersurfaces in Rn and characterize them as convex hypersurfaces with quadric sections by a continuous family of hyperplanes.

We construct normal hypersurfaces whose local cohomology modules have infinitely many associated primes. These include hypersurfaces of characteristic zero with rational singularities, as well as F-regular hypersurfaces of positive characteristic. As a consequence, we answer a question on the associated primes of certain families of ideals which arose from the localization problem in tight clos...

We discuss how one can extend to hypersurfaces with isolated singularities the methods of Gerkmann, Abbott-Kedlaya-Roe and Lauder for counting points on smooth hypersurfaces.

Motivated by the physical concept of special geometry two mathematical constructions are studied, which relate real hypersurfaces to tube domains and complex Lagrangean cones respectively. Methods are developed for the classification of homogeneous Riemannian hypersurfaces and for the classification of linear transitive reductive algebraic group actions on pseudo Riemannian hypersurfaces. The t...

In this note we show that there are algebraic families of hyperbolic, Fermat-Waring type hypersurfaces in P of degree 4(n− 1)2, for all dimensions n 2. Moreover, there are hyperbolic Fermat-Waring hypersurfaces in P of degree 4n2−2n +1 possessing complete hyperbolic, hyperbolically embedded complements. Many examples have been given of hyperbolic hypersurfaces in P3 (e.g., see [ShZa] and the li...