A far-from-CMC existence result for the constraint equations on manifolds with ends of cylindrical type

Initial data sets with ends of cylindrical type: II. The vector constraint equation

We construct solutions of the vacuum vector constraint equations on manifolds with cylindrical ends.

On the CMC foliation of future ends of a spacetime

We consider spacetimes with compact Cauchy hypersurfaces and with Ricci tensor bounded from below on the set of timelike unit vectors, and prove that the results known for spacetimes satisfying the timelike convergence condition, namely, foliation by CMC hypersurfaces, are also valid in the present situation, if corresponding further assumptions are satisfied. In addition we show that the volume of any sequence of spacelike hypersurfaces, which run into the future singularity, decays to zero provided there exists a time function covering a future end, such that the level hypersurfaces have non…

Initial data sets with ends of cylindrical type: I. The Lichnerowicz equation

We construct large classes of vacuum general relativistic initial data sets, possibly with a cosmological constant Lambda, containing ends of cylindrical type.

具有渐近柱状端的流形的带电Penrose不等式与带电正质量定理

The Penrose inequality and positive mass theorem with charge for manifolds with asymptotically cylindrical ends

本文基于加倍论证及常规带电Penrose不等式，为具有最外极小曲面边界和有限个渐近柱状端的时间对称初始数据集建立了带电Penrose不等式及刚性结论，并证明了此类流形的带电正质量定理。

Non-constant mean curvature trumpet solutions for the Einstein constraint equations

We prove the existence of a large class of initial data for the vacuum Einstein equations which possess a finite number of asymptotically Euclidean and asymptotically conformally cylindrical or periodic ends. Aside from being asymptotically constant, only mild conditions on the mean curvature of these initial data sets are imposed.

A limit equation criterion for applying the conformal method to asymptotically cylindrical initial data sets

We prove that in a certain class of conformal data on an asymptotically cylindrical manifold, if the conformally decomposed Einstein constraint equations do not admit a solution, then one can always find a nontrivial solution to the limit equation first explored by Dahl, Gicquaud, and Humbert in [DGH11]. We also give an example of a Ricci curvature condition on the manifold which precludes the existence of a solution to this limit equation, showing that such a limit criterion can be a useful tool for studying the Einstein constraint equations on manifolds with asymptotically cylindrical ends.