Marcus Kriele

Outer trapped surfaces and their apparent horizon

We give a new definition of “closed outer trapped surface’’ with respect to a hypersurface and show that the boundary of the trapped region (the apparent horizon) is a marginally trapped surface, i.e., has vanishing outer null expansion. While this is an important and well known result, there does not seem to exist a proof in the literature.

Lorentzian affine hyperspheres with constant affine sectional curvature

We study ane hyperspheres M with constant sectional curvature (with respect to the ane metric h). A conjecture by M. Magid and P. Ryan states that every such ane hypersphere with nonzero Pick invariant is anely equivalent to either where the dimension n satises n =2 m 1o rn =2 m .U p to now, this conjecture was proved if M is positive denite or if M is a 3-dimensional Lorentz space. In this paper, we give an armative answer to this conjecture for arbitrary dimensional Lorentzian ane hyperspheres.

Black holes, cosmological singularities and change of signature

There exists a widespread belief that signature-type change could be used to avoid spacetime singularities. We show that signature change cannot be utilized to this end unless the Einstein equation is abandoned at the suface of signature-type change. We also discuss how to solve the initial-value problem and show to what extent smooth and discontinuous signature changing solutions are equivalent.

Generalized spherical functions on projectively flat manifolds

Local and global properties of the first order spherical functions are generalized to projectively flat manifolds.

On Square Integrability of Mean Curvature for Surfaces with Positive Gaussian Curvature

We show that {ie319-1} H2dµ = ∞ for any complete surface M ⊂ R3 which has positive curvature outside a compact subset of R3. This proves a conjecture of Friedrich.

Physical properties of geometric singularities

We study a class of singularities of prescribed behaviour and investigate physical properties such as their strength, and the validity of energy conditions in a neighbourhood of these singularities. As applications of our theory, we show that shell (crossing) singularities are at variance with a new (but plausible) energy condition. We also show that for perfect fluid shell singularities, the energy density always exists as a spacetime average. In another application we prove that smooth, signature-type changing spacetimes are non-singular at the surface of signature change if the dominant energy condition holds.

Shell singularities of three-dimensional dust spacetimes

We study irrotational, three-dimensional dust spacetimes and show that their only dynamical singularities are shell singularities in the sense of Kriele, Lim and Scott (1994). These singularities are weak and generically naked.

The fundamental theorem forC∞ immersed affine hypersurfaces with type changing Blaschke metric

We investigate the conormal geometry of relative affine hypersurfaces whose relative metric (Blaschke metric) degenerates on a codimension 1 submanifold. Such hypersurfaces arise in the investigation of compact hypersurfaces which are not diffeomorphic to the sphere. We give a fundamental theorem in terms of the conormal structure. Finally, we present a new, affinely invariant tensor which is defined at the set where the relative metric is degenerate.

A stable class of spacetimes with naked singularities

We present a stable class of spacetimes which satisfy the conditions of the singularity theorem of Hawking & Penrose (1970), and which contain naked singularities. This offers counterexamples to a geometric version of the strong cosmic censorship hypothesis.

A geometric theory of shell singularities

We introduce and give the motivation for the concept of shell singularities. Shell singularities are a generalization of shell crossing singularities to spacetimes which are not necessarily spherically symmetric. We give a definition of strong cosmic censorship for shell singularities and show under which conditions shell singularities are censored. We study pregeodesics and geodesics which cross a shell singularity and introduce coordinates which are adapted to the singularity. We then specialize to spherical symmetry and discuss the family of perfect fluid spacetimes with shell crossing singularities given by Muller zum Hagen et al. We prove that spherically symmetric perfect fluid spacetimes with shell singularity have vanishing velocity of sound at the singularity. This lends further support to cosmic censorship.