The canonical foliation on null hypersurfaces in low regularity

Abstract

Let H denote the future outgoing null hypersurface emanating from a spacelike 2-sphere S in a vacuum spacetime (M, g). In this paper we study the so-called canonical foliation on H introduced in [13], [22] and show that the corresponding geometry is controlled locally only in terms of the initial geometry on S and the L 2 curvature flux through H. In particular, we show that the ingoing and outgoing null expansions trχ and trχ are both locally uniformly bounded. The proof of our estimates relies on a generalisation of the methods of [15] [16] [17] and [1] [2] [26] [32] where the geodesic foliation on null hypersurfaces H is studied. The results of this paper, while of independent interest, are essential for the proof of the spacelike-characteristic bounded L 2 curvature theorem [12]. STEFAN CZIMEK AND OLIVIER GRAF 2.11. Norms for the geodesic and canonical foliation geometry 2.12. Main results 2.13. Proof of Theorem 2.35. The bootstrap argument. 2.14. Organisation of the paper 3. Calculus Prerequisites 3.1. Uniform weak regularity of foliations 3.2. Littlewood-Paley theory and Besov spaces 3.3. Sobolev inequalities on 2-spheres 3.4. Sobolev inequalities on H 3.5. Null transport equations on H 3.6. L ∞ L 2 v Trace estimate 3.7. Uniform weak sphericality 3.8. Bochner identities on 2-spheres and consequences 3.9. Hodge systems on 2-spheres 4. Low regularity estimates 4.1. Bounds on S 1 for the connection coefficients. 4.2. Uniform weak regularity and sphericality of the canonical foliation 4.3. Bounds for the null curvature components of the canonical foliation on H 4.4. Schematic notation for null connection coefficients and null curvature components 4.5. Improvement of the estimates for Ω and A 4.6. Improvement of Υ 4.7. Improvement of L ∞ L 2 v estimates for A 4.8. Additional bounds for Υ 5. Higher regularity estimates 6. Local existence theorem 6.1. Geometric set up and theorem 6.2.

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

“…The existence and control of the bottom initial layer L bot follows from a small data local existence result for the spacelike-characteristic Cauchy problem. Such a result has been obtained in [CG19a,CG19b] under regularity assumptions weaker than in the present paper, and applies in particular in the present case. 10…”

“…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].…”

“…It is expected for the canonical foliation that we do not control all ∇ / 3 derivatives of the coefficient ω, since the foliation is only defined by an elliptic equation on 2-spheres. See also the estimates for the lapse Ω in [CG19a].…”

“…The need of this optimal regularity in the extension argument 4 of Section 8.1 motivates the choice of the canonical foliation. See also its use in [CG19a,CG19b,KN03] for similar tangential regularity reasons.…”