The Canonical Foliation On Null Hypersurfaces in Low Regularity

Abstract

Let ${{\mathcal {H}}}$ denote the future outgoing null hypersurface emanating from a spacelike 2-sphere S in a vacuum spacetime $({{\mathcal {M}}},\textbf{g})$. In this paper we study the so-called canonical foliation on ${{\mathcal {H}}}$ 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 ${{\mathcal {H}}}$. In particular, we show that the ingoing and outgoing null expansions ${\textrm{tr}}\chi$ and ${\textrm{tr}}{{{\underline{\chi }}}}$ 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 ${{\mathcal {H}}}$ 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].

Appendices

Appendix A. Proof of Propositions 2.32 and 2.33

In this section we prove the formulas from Proposition 2.32 and 2.33. The following computations are standard and can be found in various forms in [1, 22, 25] for instance. In what follows, we use the formulas from [8] pp. 149-150. We have

$$\begin{aligned} \alpha _{AB}&= \textbf{R}(L,e_A,L,e_B) = \textbf{R}(L,e'_A+\Upsilon _AL,L,e'_B+\Upsilon _BL) \\&= \textbf{R}(L,e'_A,L,e'_B) = (\alpha ')^{\dagger }_{AB}, \end{aligned}$$

and

$$\begin{aligned} \beta _A =&\frac{1}{2}\textbf{R}(e_A,L,{\,\underline{L}},L) \\ =&\frac{1}{2}\textbf{R}(e'_A+ \Upsilon _AL,L,{\,\underline{L}}'+2\Upsilon _Be'_B+|\Upsilon |^2L,L) \\ =&\frac{1}{2}\textbf{R}(e'_A,L,{\,\underline{L}}',L) + \Upsilon _B \textbf{R}(e'_A,L,e'_B,L) \\ =&\beta '_A + \Upsilon _B\alpha '_{AB}\\ =&(\beta ')^{\dagger }_A+\Upsilon _B(\alpha ')^{\dagger }_{AB}, \end{aligned}$$

and

$$\begin{aligned} \rho =&\frac{1}{4}\textbf{R}({\,\underline{L}},L,{\,\underline{L}},L) \\ =&\frac{1}{4}\textbf{R}({\,\underline{L}}'+2\Upsilon _Ae_A'+|\Upsilon |L,L,{\,\underline{L}}'+2\Upsilon _Be_B' + |\Upsilon |^2L,L) \\ =&\frac{1}{4}\textbf{R}({\,\underline{L}}',L,{\,\underline{L}}',L) + \frac{1}{2}\Upsilon _A\textbf{R}(e'_A,L,{\,\underline{L}}',L) \\&+ \frac{1}{2}\Upsilon _B\textbf{R}({\,\underline{L}}',L,e'_B,L) + \Upsilon _A\Upsilon _B\textbf{R}(e_A',L,e'_B,L) \\ =&\rho ' + (\beta ')^{\dagger }\cdot \Upsilon + (\alpha ')^{\dagger }\cdot \Upsilon \cdot \Upsilon . \end{aligned}$$

Following the previous computation for $\rho$ we also obtain

$$\begin{aligned} \sigma= & {} \frac{1}{4} {{}^*}\textbf{R}({\,\underline{L}},L,{\,\underline{L}},L) = \sigma ' + \Upsilon _A {{}^*}\textbf{R}(e_A',L,{\,\underline{L}}',L) + \Upsilon _A\Upsilon _B {{}^*}\textbf{R}(e_A',L,e_B',L) \\= & {} \sigma ' - \left( {{}^*}\beta '\right) ^{\dagger }\cdot (\Upsilon ) - ({{}^*}\alpha ')^{\dagger }\cdot \Upsilon \cdot \Upsilon . \end{aligned}$$

We have

$$\begin{aligned} {{\underline{\beta }}}_A =&\frac{1}{2}\textbf{R}(e_A,{\,\underline{L}},{\,\underline{L}},L) {=} \frac{1}{2}\textbf{R}(e'_A {+} \Upsilon _AL,{\,\underline{L}}' {+} 2\Upsilon _Be'_B {+} |\Upsilon |^2L,{\,\underline{L}}'+2\Upsilon _Ce'_C + |\Upsilon |^2L, L) \\ =&\frac{1}{2}\textbf{R}(e'_A,{\,\underline{L}}',{\,\underline{L}}',L) + \frac{1}{2}\Upsilon _A\textbf{R}(L,{\,\underline{L}}',{\,\underline{L}}',L) \\&+ \Upsilon _B\textbf{R}(e'_A,e'_B,{\,\underline{L}}',L) + \Upsilon _C\textbf{R}(e'_A,{\,\underline{L}}',e'_C,L) \\&+ \Upsilon _A \Upsilon _B\textbf{R}(L,e'_B,{\,\underline{L}}',L) + \Upsilon _A\Upsilon _C\textbf{R}(L,{\,\underline{L}}',e'_C,L) + 2\Upsilon _B\Upsilon _C\textbf{R}(e'_A,e'_B,e'_C,L) \\&+ \frac{1}{2}|\Upsilon |^2\textbf{R}(e'_A,L,{\,\underline{L}}',L) + 2\Upsilon _A\Upsilon _B\Upsilon _C \textbf{R}(L,e'_B,e'_C,L) + |\Upsilon |^2\Upsilon _C\textbf{R}(e'_A,L,e'_C,L) \\ =&{{\underline{\beta }}}'_A - 2\Upsilon _A\rho ' + 2 {{}^*}\Upsilon _A\sigma ' + (-\Upsilon _A\rho '+{{}^*}\Upsilon _A\sigma ') - 2\Upsilon _A \Upsilon \cdot (\beta ')^{\dagger }-2\Upsilon _A\Upsilon \cdot (\beta ')^{\dagger }\\&- 2{{}^*}\Upsilon _A\Upsilon \cdot ({{}^*}\beta ')^{\dagger }+ |\Upsilon |^2\beta '_A- 2\Upsilon _A\Upsilon \cdot \Upsilon \cdot (\alpha ')^{\dagger }+ |\Upsilon |^2\Upsilon \cdot (\alpha ')^{\dagger }_{A} \\ =&({{\underline{\beta }}}')^{\dagger }_A -3\Upsilon _A\rho ' + 3{{}^*}\Upsilon _A\sigma ' - 4\Upsilon _A \Upsilon \cdot (\beta ')^{\dagger }-2{{}^*}\Upsilon _A \Upsilon \cdot ({{}^*}\beta ')^{\dagger }\\&+ |\Upsilon |^2(\beta ')^{\dagger }_A -2\Upsilon _A\Upsilon \cdot \Upsilon \cdot (\alpha ')^{\dagger }+ |\Upsilon |^2\Upsilon \cdot (\alpha ')^{\dagger }_A. \end{aligned}$$

This finishes the proof of Proposition 2.32. We turn to the connection coefficients. We have immediately $\chi _{AB} = \chi '_{AB}$. We also have

$$\begin{aligned} \zeta _A =&\frac{1}{2}\textbf{g}(\textbf{D}_AL,{\,\underline{L}}) \\ =&\frac{1}{2}\textbf{g}(\textbf{D}_{e'_A + \Upsilon _A L}L,{\,\underline{L}}' +2\Upsilon _Be'_B +|\Upsilon |^2L) \\ =&\frac{1}{2}\textbf{g}(\textbf{D}_{e'_A}L,{\,\underline{L}}') + \Upsilon _B \textbf{g}(\textbf{D}_{e'_A}L,e'_B) \\ =&(\zeta ')^{\dagger }_A + \Upsilon \cdot \chi _A, \end{aligned}$$

and

Finally, we have

This finishes the proof of Proposition 2.33.

Appendix B. Proof of Lemmas 3.5 and 3.20

1.1 B.1. Proof of Lemma 3.5

This section is dedicated to the proof of Lemma 3.5.

In fact, we prove the following more general estimate

(B.1)

for $-1< s < 0$.

Remark B.1

As it will be clear from what follows, the proof below does not work for other ranges of exponents s, and would require additional regularity assumptions on the 2-sphere S.

From Proposition 2.3 in [26], we have the following characterisation of $H^s(S)$ using the Littlewood-Paley projectors defined in Section 3.2

$$\begin{aligned} \left\| F \right\| ^2_{H^s(S)} \simeq \sum _{k\ge 0} 2^{2sk}\left\| P_kF \right\| ^2_{L^2(S)} + \left\| P_{<0}F \right\| _{L^2(S)}^2. \end{aligned}$$

(B.2)

From Section 2.2 and Proposition 2.1 in [26], we recall the following properties of the Littlewood-Paley projection operators defined in Section 3.2. For all $k\in {{\mathbb {Z}}}$, we have

$$\begin{aligned} P_k = P_kP_{k-1} + P_kP_k + P_kP_{k+1}, \end{aligned}$$

(B.3)

and for F an S-tangent tensor and for all $k\in {{\mathbb {Z}}}$, we have

$$\begin{aligned} \begin{aligned} \left\| P_kF \right\| _{L^2(S)} \lesssim \left\| F \right\| _{L^2(S)}, \,\,\,\,\,\, \left\| P_{<0}F \right\| _{L^2(S)} \lesssim \left\| F \right\| _{L^2(S)}, \end{aligned} \end{aligned}$$

(B.4)

and,

(B.5)

We turn to the proof of estimate (B.1). Using (3.4), (B.2) and (B.5), we have

(B.6)

The first term in the right-hand side of (B.6) can be estimated using (B.3), (B.5) and that $-1<s<0$

For the second term in the right-hand side of (B.6), using (B.4), we first write the following decomposition

The first term can be estimated, using that preserves the support of the projectors $P_k$ (see also Section 2.2 in [26]) and (3.4),

and similarly, we deduce for the last two terms, using (B.4) and (B.5)

Using this, (B.4) and that $-1<s<0$, we therefore deduce that for the second term of (B.6) we have

Finally, plugging the above estimates into (B.6) and using (B.2), we obtain

This finishes the proof of Lemma 3.5.

1.2 B.2. Proof of Lemma 3.20

This section is dedicated to the proof of Lemma 3.20. We assume that f is a scalar function satisfying the elliptic equation (3.11)

Multiplying equation (3.11) by $f-\overline{f}$ and integrating by part, we have

Using Lemma 3.19, we have the following Poincaré inequality

Therefore, using Sobolev Lemma 3.8, we have

and the bound holds by a standard absorption argument. The bound on ${{\mathcal {H}}}$ follows by integration in v. This finishes the proof of Lemma 3.20.