19 problems

In solutions of Einstein's equations near a Big Bang singularity, asymptotically velocity term dominated (AVTD) behavior means that, with respect to a given coordinate system, the…

Near a Big Bang singularity in solutions of Einstein's equations, observers approaching the singularity may encounter successive epochs in which the solution behaves spatially homo…

$L_{\infty}$-algebra conjecture. There exists an $L_{\infty}$ structure on the complex $({\bf F}^{\bullet}_{{\bf b}^-},\mathcal{Q})$, such that its bilinear operation is the symmet…

Let $V\subset M$ be a precompact open set. For sufficiently small $\epsilon>0$, let $g_\epsilon$ be the formal solution on $M_\epsilon$ obtained by gluing small Kerr black holes al…

The Einstein constraint equations admit asymptotically flat initial data $(g,k)$, where $g$ is a Riemannian metric and $k$ is the second fundamental form, localized in a region sma…

The setting is a maximal globally hyperbolic development of initial data for Einstein's equations, with the initial manifold either asymptotically flat or spatially compact. The de…

Consider initial data for the Einstein vacuum equations on intersecting null hypersurfaces, with outgoing and ingoing shears denoted by $\hat{\chi}$ and $\hat{\underline{\chi}}$, r…

Let $(\mathcal{M},g)$ be an asymptotically flat solution of the Einstein vacuum equations … An event horizon is the boundary separating the region from which observers can reach in…

A cosmological solution is a globally hyperbolic Lorentzian manifold with closed Cauchy hypersurfaces. Oscillatory behavior conjecture. General solutions should have oscillatory be…

Let $\Psi=(\mathbb{M},\eta,\phi,\bar{\phi})$ be a solution of the generalized Maurer-Cartan equation for the $L_{\infty}$-algebra on ${\bf F}^{\cdot}_-$, with … Assume that…

Let $(\Sigma,q,K)$ be asymptotically flat Cauchy data for the $3+1$-dimensional vacuum Einstein equations admitting a spacelike translational Killing field, and let its maximal Cau…

Let ${\bf F}^{\cdot}_-$ be the invariant subcomplex from Conjecture 4.1a, and let its degree-two Maurer–Cartan elements be written as $\Psi=(\mathbb{M},\eta,\phi,\bar{\phi})$. Set…

A Beltrami-Courant differential determines a corresponding $L_{\infty}$ subalgebra and its generalized Maurer–Cartan equation. Generalized Maurer–Cartan conjecture. The generalized…

Let $(\Sigma_0,g,k)$ be an initial data set for the Einstein-vacuum equations, where $g$ is the Riemannian metric on the initial hypersurface and $k$ is its second fundamental form…

$L^2$-conjecture. The optimal regularity for the initial data set is

Let $(\mathcal{M},g)$ be a spacetime foliated by spacelike hypersurfaces $\mathcal H_t$, with lapse $n$ and second fundamental form encoded by the tensor $\boldsymbol{\pi}$. Let…

L²-conjecture. The optimal regularity for the initial data set is $Ric(g)\in L^2$, corresponding to an initial metric $g\in H^2$.

CMC conjecture. There is a foliation in $\textbf{M}$ by CMC Cauchy surfaces whose mean curvatures take all allowable values: all values in $(-\infty,0)$ if $\Sigma_0$ is of Yamabe…

Let $\Sigma$ be a three-manifold, let $Y(\Sigma)$ denote its Yamabe invariant, and let $k$ be the mean-curvature parameter of a CMC flow. The maximal range of $k$ is a connected op…