22 problems

Let $\mathcal{S}$ be a generic spacelike initial cosmological singularity, and let $J^+(p)$ denote the causal future of a point $p\in\mathcal{S}$. BKL proposal. Sufficiently close…

Let $(M,\mathcal{C})$ be a Riemann-Lorentz conformal space with polar end $D^{\infty}$. Suppose that $D^{\infty}$ is conformal-flat. A simultaneity distribution is a distribution w…

Consider the setting of Theorem … , the spacetime is future causally geodesically complete. The surrounding discussion says that the answer is expected to be affirmative for…

Let $k$ be the curvature of the symmetric surfaces, let $\Lambda$ be the cosmological constant, and consider generic initial data for the surface-symmetric Einstein-Vlasov system.…

A spacetime is maximally extended if it admits no further extension, and it is globally hyperbolic if it has a Cauchy surface. A constant mean curvature (CMC) hypersurface is a spa…

Let $(\widetilde{M}, \widehat{g})$ be a globally hyperbolic Lorentzian manifold with compact Cauchy hypersurfaces. Say that late-time observers are oblivious to topology if there e…

Let $(\widetilde{M}, \widehat{g})$ be a smooth, time-oriented, globally hyperbolic Lorentzian $(3+1)$-dimensional spacetime with $\widetilde{M} \cong M \times \mathbb{R}$, where…

Let $\mathcal{A}$ denote the attractor describing generic behaviour near the initial singularity, and let $\mathrm{K}^{\circ}$, $\mathcal{T}_{R_1}$, $\mathcal{T}_{R_3}$, and…

Let $\mathcal{A}$ denote the attractor describing the generic behaviour near the initial singularity, let $\mathcal{K}$ be the Bianchi type I Kasner subset, and let…

The dynamical system has plane-wave fixed points $\mathrm{PW}^{\pm}$, a fixed point source $\mathrm{RT}$, a fixed point $\mathrm{PW}^{0}$, a fixed point $\mathrm{T}_3$, and variabl…

Decay conjecture in the hyperbolic case.

Decay conjecture in the flat case.

Consider Bianchi type VIII and IX models in the subcritical and critical cases $v\in(0,1/2]$, and let ${\cal A}_-$ denote the attractor as $\tau_-\to\infty$. Define the invariant s…

Consider Bianchi type VIII and IX models in the supercritical case $v\in(1/2,1)$, with stable set $S\subset\mathrm{K}^{\ocircle}$, attractor ${\cal A}_-$, and $\tau_-\to\infty$. Su…

Consider Bianchi type VIII and IX models in the supercritical case $v\in(1/2,1)$, let $C$ be the Cantor set associated with infinite Bianchi type II heteroclinic chains, and let…

Consider Bianchi type VIII and IX models in the supercritical case $v\in(1/2,1)$, with time parameter satisfying $\tau_-\to\infty$. Period-two stable-set conjecture. Each heterocli…

Let $v\in(0,1/2)$, and let the Kasner circle eIFS be the iterated function system consisting of the maps ${\cal K}_{\mu\nu\zeta}:\mathrm{K}^{\ocircle}\to\mathrm{K}^{\ocircle}$ defi…

Let $(H(t),\mathbf x(t))$ be the solution of the quasi-standard system with arbitrary non-vacuum initial data satisfying $\Omega>0$, the Hamiltonian constraint, and $H>0$. Let…

Let $\mathcal A$ denote the class of initial data such that the corresponding maximal globally hyperbolic developments are future causally geodesically complete solutions to the Ei…

Cosmic no-hair conjecture. Generic elements of $\mathcal{A}$ yield maximal globally hyperbolic developments that are future asymptotically de Sitter like.

Let $n+1\leq 10$, and consider vacuum space-times. The conjecture concerns inextendible geodesics along which the curvature is unbounded and allows unspecified natural geometric va…

Let $(\mathcal{M},g)$ be the maximal Cauchy development of $T^2$-symmetric or $k=-1$ surface-symmetric initial data in the vacuum or with Vlasov matter and with $\Lambda \ge 0$. As…