The spacelike-characteristic Cauchy problem of general relativity in low regularity

Abstract

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface Σ B(0, 1) ⊂ R 3 and the outgoing null hypersurface H emanating from ∂Σ, the time of existence of a solution to the Einstein vacuum equations is controlled by low regularity bounds on the initial data at the level of curvature in L 2 . The proof uses the bounded L 2 curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces.

Global nonlinear stability of Minkowski space for spacelike-characteristic initial data

“…In the case of Theorem 1.8, no such global time function is available, and we rely instead on a construction of maximal hypersurfaces in the interior region, by prescribing their boundaries on the transition hypersurface, which is the timelike boundary between the interior and the exterior region. This is close in spirit to the proof of the spacelike-characteristic bounded L 2 curvature result from [CG19b].…”

“…The control at the interface T of the difference of vectorfields T ext − T int is obtained by a control of the slope between the maximal hypersurfaces and the boundary T (see a similar result in [CG19b]), X ext − X int8 (and subsequently S ext − S int and K ext − K int ) is obtained by a control on S * using the harmonic coordinates and the control of the slope, and by integration along T , using that D 2 X 0,…”

“…The canonical foliations on the last slices in [CK93,KN03] are build to satisfy the same transverse regularity features. See also the discussions in [CG19a,CG19b].…”

“…Spacetime regions as M int top with conical degeneracies and outside of spherical symmetry are not treated as such in other works. Our procedure is in the spirit of [CG19a,CG19b].…”

“…In the literature, the characteristic Cauchy problem outside of spherical symmetry is rather studied under a local (e.g. [Ren90,CCM11,CG19b]) or semi-global perspective -i.e. in a size 1 region from the initial null hypersurface -as in [Luk12,LZ18] or [CN05,CN10].…”