Solving the constraint equation for general free data
Xuantao Chen, Sergiu Klainerman
TL;DR
The paper develops a novel method to solve the vacuum Einstein constraint equations by prescribing four dynamical scalars (modulo ℓ≤1 modes) and fixing two gauge scalars. It rewrites the constraints as a well-posed system of coupled transport and elliptic equations on 2-spheres (the Horizontal Constraint System, HCS) and solves it via a carefully crafted linear iteration that exploits a triangular block structure. The authors prove the existence of a large class of exterior asymptotically flat initial data sets that can be glued to prescribed interiors, and they demonstrate how the resulting data evolve toward black holes with controlled ADM charges, extending prior gluing and trapped-surface formation results. The approach is compatible with arbitrary fast decay and adaptable to slower decay regimes, highlighting potential sharpness results and broad applicability to initial data construction within the gluing framework.
Abstract
We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that once appropriate gauge conditions have been chosen and four scalars freely specified (modulo $\ell\leq 1$ modes), we can rewrite the constraint equations as a well-posed system of coupled transport and elliptic equations on $2$-spheres, which we solve by an iteration procedure. Our method provides a large class of exterior solutions of the constraint equations that can be matched to given interior solutions, according to the existing gluing techniques. As such, it can be applied to provide a large class of initial Cauchy data sets evolving to black holes, generalizing the well-known result of the formation of trapped surfaces due to Li and Yu. Though in our main theorem, we only specify conditions consistent with $g-g_{Schw}=O(r^{-1-δ})$, $k=O(r^{-2-δ})$, the method is flexible enough to be applied in many other situations. It can, in particular, be easily adapted to construct arbitrarily fast decaying data. We expect, moreover, that our method can also be applied to construct data with slower decay, such as that used by Shen. In fact, an important motivation for developing our method is to show that the result of Shen is sharp, i.e., construct small, smooth initial data sets which violate Shen's decay conditions, and for which the stability of the Minkowski space result is wrong.