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In this paper, we establish new characterization results concerning totally umbilical hypersurfaces of the hyperbolic space
, under suitable constraints on the behavior of the Lorentzian Gauss map of complete hypersurfaces having some constant higher order mean curvature. Furthermore, working with different warped product models for
and supposing that certain natural inequalities involving two consecutive higher order mean curvature functions are satisfied, we study the rigidity and the nonexistence of complete hypersurfaces immersed in
We extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino–Schoen/Chrusciel–Delay. We combine conformal techniques and vector field methods: a naive adaptation of the ‘Morawetz vector field’ to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino–Schoen/Chrusciel–Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows larger classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino–Schoen/Chrusciel–Delay space-times.
The structure of space–time is examined by extending the standard Lorentz connection group to its complex covering group, operating on a 16-dimensional “spinor” frame. A Hamiltonian variation principle is used to derive the field equations for the spinor connection. The result is a complete set of field equations which allow the sources of the gravitational and electromagnetic fields, and the intrinsic spin of a particle, to appear as a manifestation of the space–time structure. A cosmological solution and a simple particle solution are examined. Further extensions to the connection group are proposed.
In this work we study the behaviour of compact, smooth, orientable, spacelike hypersurfaces without boundary, which are immersed in cosmological spacetimes and move under the inverse mean curvature flow. We prove longtime existence and regularity of a solution to the corresponding nonlinear parabolic system of partial differential equations.
We prove a priori estimates for the gradient and curvature of spacelike hypersurfaces moving by mean curvature in a Lorentzian manifold. These estimates are obtained under much weaker conditions than have been previously assumed. We also use mean curvature flow in the construction of maximal slices in asymptotically flat spacetimes. An essential tool is a maximum principle for sub-solutions of a parabolic operator on complete Riemannian manifolds with time-dependent metric.
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