Piotr T. Chruściel

On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations

The regularity of the solutions to the Yamabe Problem is considered in the case of conformally compact manifolds and negative scalar curvature. The existence of smooth hyperboloidal initial data for Einsteins field equations is demonstrated.

Semi-global existence and convergence of solutions of the Robinson-Trautman (

It is shown that for smooth initial data solutions of the Robinson-Trautman equation (also known as the two-dimensional Calabi equation) exist for all positive “times,” and asymptotically converge to a constant curvature metric.

2

Abstract General space-times evolving from U (1) × U (1) symmetric Cauchy data prescribed on compact Cauchy surfaces are studied. Existence and properties of solutions of the constraint equations are analyzed. Some “canonical” forms of the metric are derived. When the spatial topology is S 3 or S 2 × S 1 or L ( p , q ) we show that no singularities form before “the spacelike boundary of Gowdys square” is reached.

-dimensional Calabi) equation

We provide an introduction to selected recent advances in the mathematical understanding of Einsteins theory of gravitation.

On space-times with U(1)×U(1) symmetric compact Cauchy surfaces

We study Killing vector fields in asymptotically flat space–times. We prove the following result, implicitly assumed in the uniqueness theory of stationary black holes. If the conditions of the rigidity part of the positive energy theorem are met, then in such space–times there are no asymptotically null Killing vector fields, except if the initial data set can be embedded in Minkowski space–time. We also give a proof of the nonexistence of nonsingular (in an appropriate sense) asymptotically flat space–times that satisfy an energy condition and that have a null ADM four‐momentum, under conditions weaker than previously considered.

MATHEMATICAL GENERAL RELATIVITY: A SAMPLER

Abstract.We prove that the space-time developments of generic solutions of the vacuum constraint Einstein equations do not possess any global or local Killing vectors, when Cauchy data are prescribed on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that nonexistence of global symmetries implies, generically, non-existence of local ones. As part of the argument, we prove that generic metrics do not possess any local or global conformal Killing vectors.

Killing vectors in asymptotically flat space–times. I. Asymptotically translational Killing vectors and the rigid positive energy theorem

We present a systematic study of static solutions of the vacuum Einstein equations with negative cosmological constant which asymptotically approach the generalized Kottler (“Schwarzschild–anti-de Sitter”) solution, within (mainly) a conformal framework. We show connectedness of conformal infinity for appropriately regular such spacetimes. We give an explicit expression for the Hamiltonian mass of the (not necessarily static) metrics within the class considered; in the static case we show that they have a finite and well-defined Hawking mass. We prove inequalities relating the mass and the horizon area of the (static) metrics considered to those of appropriate reference generalized Kottler metrics. Those inequalities yield an inequality which is opposite to the conjectured generalized Penrose inequality. They can thus be used to prove a uniqueness theorem for the generalized Kottler black holes if the generalized Penrose inequality can be established.

KIDs are Non-Generic

We extend the validity of Dain’s angular-momentum inequality to maximal, asymptotically flat, initial data sets on a simply connected manifold with several asymptotically flat ends which are invariant under a U(1) action and which admit a twist potential.

Towards the classification of static vacuum spacetimes with negative cosmological constant

The relationship between the geometric properties of “hyperboloidal” Cauchy data for vacuum Einstein equations at the conformal boundary of the initial data surface and between the space-time geometry is analyzed in detail. We prove that a necessary condition for existence of a smooth or a polyhomogeneous Scri (i.e., a Scri around which the metric is expandable in terms ofr−j logir rather than in terms ofr−j) is the vanishing of the shear of the conformal boundary of the initial data surface. We derive the “boundary constraints” which have to be satisfied by an initial data set for compatibility with Friedrichs conformal framework. We show that a sufficient condition for existence of a smooth Scri (not necessarily complete) is the vanishing of the shear of the conformal boundary of the initial data surface and smoothness up to boundary of the conformally rescaled initial data. We also show that the occurrence of some log terms in an asymptotic expansion at the conformal boundary of solutions of the constraint equations is related to the non-vanishing of the Weyl tensor at the conformal boundary.

Mass and angular-momentum inequalities for axi-symmetric initial data sets. II. Angular momentum

Abstract We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditions. Our results include sharp solvability criteria, over both compact and non-compact manifolds; weighted Poincaré and Schrödinger-Lichnerowicz inequalities provide asymptotic control in the noncompact case.