Search for: All records

We prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized T 2 -symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant Λ . This stability result generalizes the results proven in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T 2 -symmetric vacuum spacetimes. Ann. Henri Poincaré . ( doi:10.1007/s00023-021-01142-0 )), which focus on the Λ = 0 case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for Λ = 0 , the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarized T 2 -symmetric vacuum solutions than those considered in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T 2 -symmetric vacuum spacetimes. Ann. Henri Poincaré . ( doi:10.1007/s00023-021-01142-0 )) and Fournodavlos G et al. (2020 Stable Big Bang formation for Einstein’s equations: the complete sub-critical regime . Preprint. ( http://arxiv.org/abs/2012.05888 )). Our results establish that the areal time coordinate takes all values in ( 0 , T 0 ] for some T 0 > 0 , for certain families of polarized T 2 -symmetric solutions with cosmological constant. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’. 

Abstract We present a numerical study of the local stability of mean curvature flow (MCF) of rotationally symmetric, complete noncompact hypersurfaces with type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs ‘numerically global’ (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviours under MCF. Firstly, there is a ‘near’ class of initial data which lead to the same singular behaviour as an unperturbed solution; in particular, the curvature at the tip of the hypersurface blows up at a type-II rate no slower than ( T − t ) −1 . Secondly, there is a ‘far’ class of initial data which lead to solutions developing a local type-I nondegenerate neckpinch under MCF. These numerical findings further suggest the existence of a ‘critical’ class of initial data which conjecturally lead to MCF of noncompact hypersurfaces forming local type-II degenerate neckpinches with the highest curvature blowup rate strictly slower than ( T − t ) −1 . 

Abstract We prove a nonpolarised analogue of the asymptotic characterisation of $$T^2$$ T 2 -symmetric Einstein flow solutions completed recently by LeFloch and Smulevici. In this work, we impose a condition weaker than polarisation and so our result applies to a larger class. We obtain similar rates of decay for the normalised energy and associated quantities for this class. We describe numerical simulations which indicate that there is a locally attractive set for $$T^2$$ T 2 -symmetric solutions not covered by our main theorem. This local attractor is distinct from the local attractor in our main theorem, thereby indicating that the polarised asymptotics are unstable.