This article is the second of two in which we develop a geometric framework for analysing silent and anisotropic big bang singularities. In the present article, we record geometric conclusions obtained by combining the geometric framework with Einstein's equations. The main features of the results are the following: The assumptions do not involve any symmetry requirements and are weak enough to be consistent with most big bang singularities for which the asymptotic geometry is understood. The framework gives a clear picture of the asymptotic geometry. It also reproduces the Kasner map, conjectured in the physics literature to constitute the essence of the asymptotic dynamics for vacuum solutions to Einstein's equations. When combined with Einstein's equations, the framework yields partial improvements of the assumptions concerning, e.g., the expansion normalised Weingarten map $\mathcal{K}$ (one of the central objects of the framework, defined as the Weingarten map of the leaves of the foliation divided by the mean curvature). For example, the expansion normalised normal derivative of $\mathcal{K}$ can, under suitable assumptions concerning the eigenvalues of $\mathcal{K}$, be demonstrated to decay exponentially and $\mathcal{K}$ can be demonstrated to converge exponentially, even though we initially only impose weighted bounds on these quantities. Finally, the framework gives a unified perspective on the existing results. Moreover, in $3+1$-dimensions, the only parameters necessary to interpret the results are the eigenvalues of $\mathcal{K}$ and an additional scalar function determined by the geometry induced on the leaves of the foliation. In the companion article, we obtain conclusions concerning the asymptotic behaviour of solutions to linear systems of wave equations on the backgrounds consistent with the framework.

The subject of this article is the structure of big bang singularities in spatially homogeneous solutions to the Einstein non-linear scalar field equations. In particular, we focus on Bianchi class A; i.e., developments arising from left invariant initial data on unimodular $3$-dimensional Lie groups. We prove that solutions are either vacuum or matter dominated, depending on whether the limit of an expansion normalised normal derivative of the scalar field is zero or not, respectively. The main result concerning the asymptotics in the direction of the singularity is, essentially, that solutions induce data on the singularity, with two exceptions: vacuum dominated Bianchi type VIII and IX without additional symmetries (they are neither isotropic nor locally rotationally symmetric) exhibit BKL-type oscillations. Disregarding the exceptions, there is in fact a bijection between initial data on the singularity and developments. Initial data on the singularity thus play a central role in the analysis; they both parameterise developments and give optimal asymptotic information. However, the main point of the article is to prove that the set of isometry classes of initial data on the singularity (of a fixed Bianchi type and symmetry (such as isotropy, local rotational symmetry etc.)) has a smooth structure; that the set of isometry classes of developments (similarly restricted) has a smooth structure which fits together with the natural smooth structure of isometry classes of regular initial data with fixed mean curvature; and that the Einstein flow generates a diffeomorphism between the two sets. However, the article contains substantial additional information, such as, e.g., the construction of a large class of spatially locally homogeneous solutions that can be demonstrated to be globally non-linearly stable (in the absence of symmetries) both to the future and to the past.

The purpose of this article is to develop a local existence theory for a class of CMC gauges for the Einstein-non-linear scalar field equations. We do so in the context of closed and parallelisable initial manifolds. The assumption that the initial manifold is parallelisable is not a topological restriction in the case of closed oriented 3-manifolds, but it is a restriction in higher dimensions. The results include local existence, uniqueness, a continuation criterion and Cauchy stability. Previous results concerning related gauges have been restricted to the case of n-torus spatial topology. Moreover, they have not covered the Einstein-non-linear scalar field setting. The main motivation for developing this theory is that it forms the basis for a family of past global non-linear stability results we derive in a separate article.

We introduce a geometric notion of initial data on the singularity for the Einstein-scalar field equations and demonstrate that previous notions of data on the singularity constitute special cases. The definition thus gives a unified geometric perspective on existing results. The formulation also leads to a natural geometric initial value formulation of Einstein's equations with initial data on the singularity. However, in order for any notion of initial data on the singularity to parametrise convergent solution, the crucial question is: do convergent solutions necessarily induce such data? There are several notions of data on the singularity for the Einstein-scalar field equations and corresponding existence results. The existence results demonstrate that the corresponding notions are sufficient to ensure convergent behaviour. However, none of the existing notions of data on the singularity are necessary. A central part of the article is therefore to demonstrate the necessity of our requirements.

General relativity is an area at the interface of partial differential equations, differential geometry, global analysis, mathematical physics and dynamical systems. It interacts with astrophysics, cosmology, high energy physics, and numerical analysis. The field is rapidly expanding and has witnessed remarkable developments and interconnections with other fields in recent years.The workshop Mathematical Aspects of General Relativity was organised by Carla Cederbaum (Tübingen), Mihalis Dafermos (Cambridge/Princeton), Jim Isenberg (Eugene) and Hans Ringström (KTH Stockholm). There were 48 on-site and 4 online participants. There were 16 one hour talks, nine 30 minute talks and four 10 minute talks.

The goal of this article is to parametrise solutions to Einstein's equations with big bang singularities and quiescent asymptotics. To this end, we introduce a notion of initial data on big bang singularities and conjecture that it can be used to parametrise quiescent solutions. A mathematical statement of the conjecture presupposes a precise definition of the class of quiescent solutions as well as a proof of existence and uniqueness of developments corresponding to initial data on a big bang singularity. We provide one definition of quiescence here. We also appeal to existing results in order to illustrate that, in certain cases, there are unique developments corresponding to initial data on the singularity. However, our perspective leads to a large class of open problems corresponding to the general conjecture. An additional benefit of the notion of initial data developed here is that it can be used to give a unified perspective on the existing results concerning quiescent singularities. In fact, we provide several examples of how existing results can be considered to be special cases of the framework developed here. A second, potential, application is to oscillatory and spatially inhomogeneous big bang singularities. Considering the existing arguments in the spatially homogeneous setting, a crucial first step in the study of oscillatory behaviour is to understand how solutions approach the Kasner circle along a stable manifold, and then depart via an unstable manifold. In order to carry out a similar analysis in the spatially inhomogeneous setting, it is of central importance to first identify the stable manifold. Building on the work of Fournodavlos and Luk, we here propose such an identification.

Mathematical general relativity, the subject of this workshop, is a remarkable confluence of different areas of mathematics. Einstein’s equation, the focus of mathematical relativity, is one of the most fruitful nonlinear hyperbolic PDE systems under study. As well, some of the most challenging geometric analysis problems in Reimannian geometry and elliptic PDE theory arise from the study of the initial data for Einstein’s equations. In addition, these studies play a crucial role in modeling the physics of astrophysical and cosmological systems. This workshop reflected the rapid progress seen in the field in recent years, and highlighted some of the most interesting questions under study in mathematical relativity.

General Relativity is one of the triumphs of twentieth century physics. Its spectacular predictions include gravitational waves, black holes, and spacetime singularities. The mathematical study of this theory leads to deep problems connecting the areas of partial differential equations, geometry and topology with physics. The talks of the workshop illustrated the rapid progress in this subject over the last few years.

The subject of the book is the topology and future stability of models of the universe. In standard cosmology, the universe is assumed to be spatially homogeneous and isotropic. However, it is of interest to know whether perturbations of the corresponding initial data lead to similar solutions or not. This is the question of stability. It is also of interest to know what the limitations on the global topology imposed by observational constraints are. These are the topics addressed in the book. The theory underlying the discussion is the general theory of relativity. Moreover, in the book, matter is modelled using kinetic theory. As background material, the general theory of the Cauchy problem for the Einstein–Vlasov equations is therefore developed.

In 1952, Yvonne Choquet-Bruhat demonstrated that it makes sense to consider Einstein's vacuum equations from an initial value point of view; given initial data, there is a globally hyperbolic development. Since there are many developments, one does, however, not obtain uniqueness. This was remedied in 1969 when Choquet-Bruhat and Robert Geroch demonstrated that there is a unique maximal globally hyperbolic development (MGHD). Unfortunately, there are examples of initial data for which the MGHD is extendible, and, what is worse, extendible in inequivalent ways. Thus it is not possible to predict what spacetime one is in simply by looking at initial data and, in this sense, Einstein's equations are not deterministic. Since the examples exhibiting this behaviour are rather special, it is natural to conjecture that for generic initial data, the MGHD is inextendible. This conjecture is referred to as the strong cosmic censorship conjecture and is of central importance in mathematical relativity. In this paper, we shall describe this conjecture in detail, as well as its resolution in the special case of T3-Gowdy spacetimes.

In a recent paper by Thomas Jurke, it was proved that the asymptotic behaviour of a solution to the polarized Gowdy equation in the expanding direction is of the form αln t + β + t−1/2ν + O(t−3/2), where α and β are constants and ν is a solution to the standard wave equation with zero mean value. Furthermore, it was proved that α, β and ν uniquely determine the solution. Here we wish to point out that given α, β and ν with the above properties, one can construct a solution to the polarized Gowdy equation with the above asymptotics. In other words, we show that α, β and ν constitute data at the moment of infinite expansion. We then use this fact to make the observation that there are polarized Gowdy spacetimes such that in the areal time coordinate, the quotient of the maximum and the minimum of the mean curvature on a constant time hypersurface is unbounded as time tends to infinity.

We consider Gowdy spacetimes under the assumption that the spatial hypersurfaces are diffeomorphic to the torus. The relevant equations are then wave map equations with the hyperbolic space as a target. In a paper by Grubiš;ić and Moncrief, a formal expansion of solutions in the direction towards the singularity was proposed. Later, Kichenassamy and Rendall constructed a family of real analytic solutions with the maximum number of free functions and the desired asymptotics at the singularity. The condition of real analyticity was subsequently removed by Rendall. In a previous paper, we proved that one can put a condition on initial data that leads to asymptotic expansions. However, control of up to and including three derivatives in L2 was necessary, and the condition was rather technical. The main point of the present paper is to demonstrate the existence of certain monotone quantities and to illustrate how these can be used to weaken the assumptions to one derivative in the sup norm. Furthermore, we demonstrate that the false spikes do not appear in the disc model. Finally, we show that knowledge concerning the behaviour of the solution (as time tends to the singularity) for one fixed spatial point in some situations can be used to conclude that there are smooth expansions in the neighbourhood of that spatial point.

Bianchi VIII vacuum solutions to Einstein's equations are causally geodesically complete to the future, given an appropriate time orientation, and the objective of this paper is to analyse the asymptotic behaviour of solutions in this time direction. For the Bianchi class A spacetimes, there is a formulation of the field equations that was presented in an article by Wainwright and Hsu, and we will analyse the asymptotic behaviour of solutions in these variables. We also try to give the analytic results a geometric interpretation by analysing how a normalized version of the Riemannian metric on the spatial hypersurfaces of homogeneity evolves.