The bounded $L^2$ curvature conjecture

Abstract

This is the main paper in a sequence in which we give a complete proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^2$-norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the scaling of the Einstein equations, it is nevertheless critical with respect to its causal geometry. Indeed, $L^2$ bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than $1+1$ (based on Strichartz estimates) were obtained in Bahouri and Chemin (Am J Math 121:1337–1777, 1999; IMRN 21:1141–1178, 1999), Tataru (Am J Math 122:349–376, 2000; JAMS 15(2):419–442, 2002), Klainerman and Rodnianski (Duke Math J 117(1):1–124, 2003) and optimized in Klainerman and Rodnianski (Ann Math 161:1143–1193, 2005), Smith and Tataru (Ann Math 162:291–366, 2005), the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. To achieve our goals we recast the Einstein vacuum equations as a quasilinear $so(3,1)$-valued Yang–Mills theory and introduce a Coulomb type gauge condition in which the equations exhibit a specific new type of null structure compatible with the quasilinear, covariant nature of the equations. To prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure. These require a careful construction and control of parametrices including $L^2$ error bounds which is carried out in Szeftel (Parametrix for wave equations on a rough background I: regularity of the phase at initial time, arXiv:1204.1768, 2012; Parametrix for wave equations on a rough background II: construction of the parametrix and control at initial time, arXiv:1204.1769, 2012; Parametrix for wave equations on a rough background III: space-time regularity of the phase, arXiv:1204.1770, 2012; Parametrix for wave equations on a rough background IV: control of the error term, arXiv:1204.1771, 2012), as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in Szeftel (Sharp Strichartz estimates for the wave equation on a rough background, arXiv:1301.0112, 2013). It is at this level that our problem is critical. Indeed, any known notion of a parametrix relies in an essential way on the eikonal equation, and our space-time possesses, barely, the minimal regularity needed to make sense of its solutions.

Appendices

Appendix A: Proof of Lemma 4.2

Let $X$ a vectorfield on $\Sigma _0$. We use the harmonic coordinate system on $\Sigma _t$ of Lemma 6.2 with $t=0$. We have in a coordinate patch $U$:

$$\begin{aligned} {\text{ div }}X&= \frac{1}{\sqrt{|g|}}\dot{{\partial }}_i(\sqrt{|g|}g^{ij}X_j)\\&= \frac{1}{\sqrt{|g|}}\dot{{\partial }}_i(\sqrt{|g|}(g^{ij}-\delta ^{ij})X_j+\delta ^{ij}(\sqrt{|g|}-1)X_j)+\frac{1}{\sqrt{|g|}}\dot{{\text{ div }}}X \end{aligned}$$

where $\dot{\partial }$ and $\dot{{\text{ div }}}$ denote the derivatives and the flat divergence relative to the coordinate system defined above, as opposed to the frame derivatives $\partial$ and the divergence ${\text{ div }}$. This yields

$$\begin{aligned} \dot{{\text{ div }}}X = \sqrt{|g|}{\text{ div }}X-\dot{{\partial }}_i\Big ((\sqrt{|g|}(g^{ij}-\delta ^{ij})+\delta ^{ij}(\sqrt{|g|}-1))X_j\Big ). \end{aligned}$$

Also, we have

$$\begin{aligned}{\hbox {curl}}X=\dot{{\hbox {curl}}}X.\end{aligned}$$

Let $\varrho$ a smooth cut-off function localized in the coordinate patch $U$. Then, we have

$$\begin{aligned}&(\dot{{\text{ div }}}(\varrho X), \dot{{\hbox {curl}}}(\varrho X))\\&\quad = \varrho \Big (\sqrt{|g|}{\text{ div }}X-\dot{{\partial }}_i\Big ((\sqrt{|g|}(g^{ij}-\delta ^{ij})+\delta ^{ij}(\sqrt{|g|}-1))X_j\Big ), {\hbox {curl}}(X)\Big )\\&\qquad +(\dot{\nabla }\varrho X, \dot{\nabla }\varrho \dot{\wedge } X)\\&\quad \!=\!\Big (\varrho \sqrt{|g|}{\text{ div }}X-\dot{{\partial }}_i\Big ((\sqrt{|g|}(g^{ij}\!-\!\delta ^{ij})X_j\!+\!\delta ^{ij}(\sqrt{|g|}\!-\!1))\varrho X_j\Big ), {\hbox {curl}}(X)\Big )\\&\qquad +((\sqrt{|g|}(g^{ij}-\delta ^{ij})X_j+\delta ^{ij}(\sqrt{|g|}-1))\dot{{\partial }}_i\varrho \varrho X_j+\dot{\nabla }\varrho X, \dot{\nabla }\varrho \dot{\wedge } X). \end{aligned}$$

Let us denote by $\dot{\mathcal {D}}$ the div-curl system in coordinates, i.e.

$$\begin{aligned}\dot{\mathcal {D}}=(\dot{{\text{ div }}}(\varrho X), \dot{{\hbox {curl}}}(\varrho X)).\end{aligned}$$

We obtain

$$\begin{aligned} \varrho X&= \dot{\mathcal {D}}^{-1} \Big (\varrho \sqrt{|g|}{\text{ div }}X-\dot{{\partial }}_i\Big ((\sqrt{|g|}(g^{ij}-\delta ^{ij})X_j+\delta ^{ij}(\sqrt{|g|}-1))\varrho X_j\Big ), \\&\qquad {\hbox {curl}}(X)\Big )\\&\quad +\dot{\mathcal {D}}^{-1} ((\sqrt{|g|}(g^{ij}-\delta ^{ij})X_j+\delta ^{ij}(\sqrt{|g|}-1))\dot{{\partial }}_i\varrho \varrho X_j+\dot{\nabla }\varrho X, \dot{\nabla }\varrho \dot{\wedge } X). \end{aligned}$$

Now, we use the following standard elliptic estimates on ${\mathbb R}^3$:

$$\begin{aligned} \dot{\mathcal {D}}^{-1}\in {\mathcal L}(L^{\frac{6}{5}}({\mathbb R}^3), L^2({\mathbb R}^3)),\,\, \dot{\mathcal {D}}^{-1}\dot{\partial }\in {\mathcal L}(L^2({\mathbb R}^3)), \end{aligned}$$

where the notation $\mathcal {L}(X,Y)$ stands for the set of bounded linear operators from the space $X$ to the space $Y$. Together with our assumption on the harmonic coordinates (6.2), this yields in a coordinate Patch $U$

$$\begin{aligned}&||\varrho X||_{L^2(U)} \\&\quad \lesssim ||{\text{ div }}X||_{L^{\frac{6}{5}}(U)}+||{\hbox {curl}}X||_{L^{\frac{6}{5}}(U)}+||(\sqrt{|g|}(g^{ij}-\delta ^{ij})\\&\qquad +\delta ^{ij}(\sqrt{|g|}-1))\varrho X||_{L^2(U)}+||\dot{{\partial }}\varrho X||_{L^{\frac{6}{5}}(U)}\\&\quad \lesssim ||{\text{ div }}X||_{L^{\frac{6}{5}}(U)}+||{\hbox {curl}}X||_{L^{\frac{6}{5}}(U)}+||g^{ij}-\delta ^{ij}||_{L^\infty (U)}||X||_{L^2(U)}\\&\qquad +||\dot{{\partial }}\varrho ||_{L^{\frac{3}{2}}(U)}||X||_{L^6(U)}\\&\quad \lesssim ||{\text{ div }}X||_{L^{\frac{6}{5}}(U)}+||{\hbox {curl}}X||_{L^{\frac{6}{5}}(U)}+\delta ||X||_{L^2(U)}+C(\delta )||X||_{L^6(U)}\\ \end{aligned}$$

We then sum the contributions of the covering of $\Sigma _0$ by harmonic coordinate patches $U$ satisfying (6.2) together with a partition of unity $(\varrho _U)$ subordinate to the covering. Eventually increasing $C(\delta )$, we obtain

$$\begin{aligned} ||X||_{L^2(\Sigma _0)} \!&\lesssim \! ||{\text{ div }}X||_{L^{\frac{6}{5}}(\Sigma _0)}\!+\!||{\hbox {curl}}X||_{L^{\frac{6}{5}}(\Sigma _0)}\!+\!\delta ||X||_{L^2(\Sigma _0)}\!+\!C(\delta )||X||_{L^6(\Sigma _0)}. \end{aligned}$$

Recall from Lemma 6.2 that we have the freedom of choice for $\delta >0$. By choosing $\delta >0$ small enough, we finally obtain

$$\begin{aligned} ||X||_{L^2(\Sigma _0)}&\lesssim ||{\text{ div }}X||_{L^{\frac{6}{5}}(\Sigma _0)}+||{\hbox {curl}}X||_{L^{\frac{6}{5}}(\Sigma _0)}+||X||_{L^6(\Sigma _0)}. \end{aligned}$$

This concludes the proof of Lemma 4.2.

Appendix B: Proof of (6.15)

The goal of this appendix is to prove (6.15). We first introduce Littlewood-Paley projections on $\Sigma _t$. These were constructed in [44] (see section 3.6 in that paper) using the heat flow on $\Sigma _t$. We recall below their main properties:

Proposition 15.1

(Main properties of the LP $Q_j$ [44]) Let $F$ a tensor on $\Sigma _t$. The LP-projections $Q_j$ on $\Sigma _t$ verify the following properties:

• (i) Partition of unity

  $$\begin{aligned} \sum _jQ_j=I. \end{aligned}$$

  (15.1)

• (ii) $L^p$ -boundedness    For any $1\le p\le \infty$, and any interval $I\subset \mathbb {Z}$,

  $$\begin{aligned} \Vert Q_IF\Vert _{L^p(\Sigma _t)}\lesssim \Vert F\Vert _{L^p(\Sigma _t)} \end{aligned}$$

  (15.2)

• (iii) Finite band property   For any $1\le p\le \infty$.

  $$\begin{aligned} \begin{array}{lll} \Vert \Delta Q_j F\Vert _{L^p(\Sigma _t)}&{}\lesssim &{} 2^{2j} \Vert F\Vert _{L^p(\Sigma _t)}\\ \Vert Q_jF\Vert _{L^p(\Sigma _t)} &{}\lesssim &{} 2^{-2j} \Vert \Delta F \Vert _{L^p(\Sigma _t)}. \end{array} \end{aligned}$$

  (15.3)

  In addition, the $L^2$ estimates

  $$\begin{aligned} \begin{array}{lll} \Vert \nabla Q_j F\Vert _{L^2(\Sigma _t)}&{}\lesssim &{} 2^{j} \Vert F\Vert _{L^2(\Sigma _t)}\\ \Vert Q_jF\Vert _{L^2(\Sigma _t)} &{}\lesssim &{} 2^{-j} \Vert \nabla F \Vert _{L^2(\Sigma _t)} \end{array} \end{aligned}$$

  (15.4)

  hold together with the dual estimate

  $$\begin{aligned}\Vert Q_j \nabla F\Vert _{L^2(\Sigma _t)}\lesssim 2^j \Vert F\Vert _{L^2(\Sigma _t)}\end{aligned}$$

• (iv) Bernstein inequality   For any $2\le p\le +\infty$ and $j\in \mathbb {Z}$

  $$\begin{aligned}\Vert Q_j F\Vert _{L^p(\Sigma _t)}\lesssim 2^{\frac{3}{2}(1-\frac{2}{p})j} \Vert F\Vert _{L^2(\Sigma _t)}\end{aligned}$$

  together with the dual estimates

  $$\begin{aligned}\Vert Q_j F\Vert _{L^2(\Sigma _t)}\lesssim 2^{\frac{3}{2}(1-\frac{2}{p})j} \Vert F\Vert _{L^{p'}(\Sigma _t)}\end{aligned}$$

We now rely on Proposition 15.1 to prove (6.15). Using Proposition 15.1, we have for any scalar function $v$ on $\Sigma _t$:

$$\begin{aligned} ||(-\Delta )^{-1}v||_{L^{\infty }(\Sigma _t)}&\lesssim \sum _{j\in \mathbb {Z}}||Q_j(-\Delta )^{-1}v||_{L^{\infty }(\Sigma _t)}\\&\lesssim \sum _{j\in \mathbb {Z}}2^{\frac{3j}{2}}||Q_j(-\Delta )^{-1}f||_{L^{2}(\Sigma _t)}\\&\lesssim \sum _{j\in \mathbb {Z}}2^{-\frac{j}{2}}||Q_jf||_{L^{2}(\Sigma _t)}\\&\lesssim \left( \sum _{j\ge 0}2^{-\frac{j}{14}}\right) ||f||_{L^{\frac{14}{9}}(\Sigma _t)}+\left( \sum _{j< 0}2^{\frac{j}{13}}\right) ||f||_{L^{\frac{13}{9}}(\Sigma _t)}\\&\lesssim ||f||_{L^{\frac{14}{9}}(\Sigma _t)}+||f||_{L^{\frac{13}{9}}(\Sigma _t)}. \end{aligned}$$

This concludes the proof of (6.15).

Appendix C: Proof of Lemma 9.2

Recall that $3<p<+\infty$ and $v$ is the solution of

$$\begin{aligned}\Delta v=f.\end{aligned}$$

In view of Lemma 9.1, we have

$$\begin{aligned} ||v||_{L^p(\Sigma _t)}\lesssim ||f||_{L^{\frac{3p}{2p+3}}(\Sigma _t)}. \end{aligned}$$

(16.1)

Next, we need to derive an estimate for ${\partial }v$. We proceed as in the proof of Lemma 4.2. Using the harmonic coordinate system on $\Sigma _t$ of Lemma 6.2, we have

$$\begin{aligned} \Delta v&= \frac{1}{\sqrt{|g|}}\dot{{\partial }}_i(\sqrt{|g|}g^{ij}\dot{{\partial }}_jv)\\&= \dot{\Delta }v+\frac{1}{\sqrt{|g|}}\dot{{\partial }}_i\Big (\sqrt{|g|}(g^{ij}-\delta ^{ij})\dot{{\partial }}_jv+(\sqrt{|g|}-1)\delta ^{ij}\dot{{\partial }}_jv\Big ) \end{aligned}$$

This yields

$$\begin{aligned} \dot{\Delta }v&= f -\frac{1}{\sqrt{|g|}}\dot{{\partial }}_i\Big (\sqrt{|g|}(g^{ij}-\delta ^{ij})\dot{{\partial }}_jv+(\sqrt{|g|}-1)\delta ^{ij}\dot{{\partial }}_jv\Big ). \end{aligned}$$

Let $\varrho$ a smooth cut-off function localized in the coordinate patch $U$. Then, we have

$$\begin{aligned}&\dot{\Delta }(\varrho v)\nonumber \\ \nonumber&\quad = \varrho f -\frac{\varrho }{\sqrt{|g|}}\dot{{\partial }}_i\Big (\sqrt{|g|}(g^{ij}-\delta ^{ij})\dot{{\partial }}_jv+(\sqrt{|g|}-1)\delta ^{ij}\dot{{\partial }}_jv\Big )\\ \nonumber&\quad \qquad -2\dot{\nabla }\varrho \cdot \dot{\nabla }v-\dot{\Delta }\varrho v\\ \nonumber&\quad = \varrho f -\dot{{\partial }}_i\left( \varrho (g^{ij}-\delta ^{ij})\dot{{\partial }}_jv+\varrho \left( 1-\frac{1}{\sqrt{|g|}}\right) \delta ^{ij}\dot{{\partial }}_jv\right) \\ \nonumber&\quad \quad -2\dot{\nabla }\varrho \cdot \dot{\nabla }v-\dot{\Delta }\varrho v\\ \nonumber&\quad = \varrho f -\dot{{\partial }}_i\left( \varrho (g^{ij}-\delta ^{ij})\dot{{\partial }}_jv+\varrho \left( 1-\frac{1}{\sqrt{|g|}}\right) \delta ^{ij}\dot{{\partial }}_jv\right) \\&\quad \quad -2\dot{{\text{ div }}}(v\dot{\nabla }\varrho )+\dot{\Delta }\varrho v. \end{aligned}$$

(16.2)

Now, we use the following standard elliptic estimates on ${\mathbb R}^3$ for any $3<p<+\infty$:

$$\begin{aligned} (-\dot{\Delta })^{-1}\dot{\partial }_i\in {\mathcal L}\left( L^{\frac{3p}{2p+3}}({\mathbb R}^3), L^{\frac{3p}{p+3}}({\mathbb R}^3)\right) ,\quad (-\dot{\Delta })^{-1}\dot{\partial }^2_{ij}\in {\mathcal L}\left( L^{\frac{3p}{p+3}}({\mathbb R}^3)\right) . \end{aligned}$$

Together with (16.2) and our assumptions on the harmonic coordinates (6.2) (6.3), this yields in a coordinate Patch $U$:

$$\begin{aligned}&||{\partial }(\varrho v)||_{L^\frac{3p}{p+3}(U)}\\&\quad \lesssim ||f||_{L^\frac{3p}{2p+3}(U)}+||(g^{ij}-\delta ^{ij})\dot{{\partial }}_jv||_{L^\frac{3p}{p+3}(U)}\\&\qquad +||(\sqrt{|g|}-1)\dot{{\partial }}_jv||_{L^\frac{3p}{p+3}(U)}+||\dot{\nabla }\rho v||_{L^\frac{3p}{p+3}(U)}+ ||\dot{\Delta }\varrho v||_{L^\frac{3p}{2p+3}(U)}\\&\quad \lesssim ||f||_{L^\frac{3p}{2p+3}(U)}+||g^{ij}-\delta ^{ij}||_{L^\infty (U)}||\dot{{\partial }}_jv||_{L^\frac{3p}{p+3}(U)}+(||\dot{\nabla }\varrho ||_{L^3(U)}\\&\qquad +||\dot{\Delta }\varrho ||_{L^{\frac{3}{2}}(U)})||v||_{L^p(U)}\\&\quad \lesssim ||f||_{L^\frac{3p}{2p+3}(U)}+\delta ||{\partial }v||_{L^\frac{3p}{p+3}(U)}+C(\delta )||v||_{L^p(U)}. \end{aligned}$$

We then sum the contributions of the covering of $\Sigma _t$ by harmonic coordinate patches $U$ satisfying (6.2) together with a partition of unity $(\varrho _U)$ subordinate to the covering. Eventually increasing $C(\delta )$, we obtain

$$\begin{aligned} ||{\partial }v||_{L^\frac{3p}{p+3}(\Sigma _t)}&\lesssim ||f||_{L^\frac{3p}{2p+3}(\Sigma _t)}+\delta ||{\partial }v||_{L^\frac{3p}{p+3}(\Sigma _t)}+C(\delta )||v||_{L^p(\Sigma _t)}. \end{aligned}$$

Recall from Lemma 6.2 that we have the freedom of choice for $\delta >0$. By choosing $\delta >0$ small enough, we finally obtain

$$\begin{aligned} ||{\partial }v||_{L^\frac{3p}{p+3}(\Sigma _t)}&\lesssim ||f||_{L^\frac{3p}{2p+3}(\Sigma _t)}+||v||_{L^p(\Sigma _t)} \end{aligned}$$

which together with (16.1) yields

$$\begin{aligned} ||v||_{L^p(\Sigma _t)}+||{\partial }v||_{L^\frac{3p}{p+3}(\Sigma _t)}&\lesssim ||f||_{L^\frac{3p}{2p+3}(\Sigma _t)}. \end{aligned}$$

This concludes the proof of Lemma 9.2.

Appendix D. Proof of Lemma 9.3

Recall that $3<p<+\infty$ and $v$ is the solution of

$$\begin{aligned}\Delta v={\partial }f.\end{aligned}$$

In view of Lemma 9.2, we have

$$\begin{aligned}{\partial }(-\Delta )^{-1}\in {\mathcal L}(L^{\frac{3q}{2q+3}}(\Sigma _t), \quad L^{\frac{3q}{q+3}}(\Sigma _t))\quad \mathrm{ for any }3<q<+\infty .\end{aligned}$$

Taking the dual, we infer

$$\begin{aligned}(-\Delta )^{-1}{\partial }\in {\mathcal L}(L^{\frac{3q}{2q-3}}(\Sigma _t), \quad L^{\frac{3q}{q-3}}(\Sigma _t))\quad \mathrm{ for any }3<q<+\infty .\end{aligned}$$

In particular, choosing

$$\begin{aligned}q=\frac{3p}{p-3}\in (3,+\infty )\end{aligned}$$

we obtain

$$\begin{aligned}(-\Delta ^{-1}){\partial }\in {\mathcal L}(L^{\frac{3p}{p+3}}(\Sigma _t),\quad L^p(\Sigma _t))\quad \mathrm{ for any }3<q<+\infty .\end{aligned}$$

Since

$$\begin{aligned}v=-(-\Delta )^{-1}{\partial }f,\end{aligned}$$

we deduce

$$\begin{aligned} ||v||_{L^p(\Sigma _t)}\lesssim ||f||_{L^{\frac{3p}{p+3}}(\Sigma _t)}. \end{aligned}$$

(17.1)

Next, we need to derive an estimate for ${\partial }v$. We proceed as in the proof of Lemma 9.2. We have the harmonic coordinate system on $\Sigma _t$ of Lemma 6.2

$$\begin{aligned} \dot{\Delta }v&= {\partial }f -\frac{1}{\sqrt{|g|}}\dot{{\partial }}_i\left( \sqrt{|g|}(g^{ij}-\delta ^{ij})\dot{{\partial }}_jv+(\sqrt{|g|}-1)\delta ^{ij}\dot{{\partial }}_jv\right) . \end{aligned}$$

Let $\varrho$ a smooth cut-off function localized in the coordinate patch $U$. Then, we have

$$\begin{aligned}&\dot{\Delta }(\varrho v)\\ \nonumber&= \varrho {\partial }f -\frac{\varrho }{\sqrt{|g|}}\dot{{\partial }}_i\Big (\sqrt{|g|}(g^{ij}-\delta ^{ij})\dot{{\partial }}_jv+(\sqrt{|g|}-1)\delta ^{ij}\dot{{\partial }}_jv\Big )\\ \nonumber&\quad -2\dot{\nabla }\varrho \cdot \dot{\nabla }v-\dot{\Delta }\varrho v\\ \nonumber&= {\partial }(\varrho f)-{\partial }\varrho f -\dot{{\partial }}_i\left( \varrho (g^{ij}-\delta ^{ij})\dot{{\partial }}_jv+\varrho \left( 1-\frac{1}{\sqrt{|g|}}\right) \delta ^{ij}\dot{{\partial }}_jv\right) \\ \nonumber&\quad -2\dot{{\text{ div }}}(v\dot{\nabla }\varrho )+\dot{\Delta }\varrho v. \end{aligned}$$

(17.2)

Now, we use the following standard elliptic estimates on ${\mathbb R}^3$ for any $3<p<+\infty$:

$$\begin{aligned} (-\dot{\Delta })^{-1}\dot{\partial }_i\in {\mathcal L}(L^{\frac{3p}{2p+3}}({\mathbb R}^3), L^{\frac{3p}{p+3}}({\mathbb R}^3)),\,\, (-\dot{\Delta })^{-1}\dot{\partial }^2_{ij}\in {\mathcal L}(L^{\frac{3p}{p+3}}({\mathbb R}^3)). \end{aligned}$$

Together with (17.2) and our assumptions on the harmonic coordinates (6.2) (6.3), this yields in a coordinate Patch $U$:

$$\begin{aligned}&||{\partial }(\varrho v)||_{L^\frac{3p}{p+3}(U)}\\&\quad \lesssim ||\varrho f||_{L^\frac{3p}{p+3}(U)}+||{\partial }\varrho f||_{L^\frac{3p}{2p+3}(U)}\\&\quad \quad +||(g^{ij}-\delta ^{ij})\dot{{\partial }}_jv||_{L^\frac{3p}{p+3}(U)}+||(\sqrt{|g|}-1)\dot{{\partial }}_jv||_{L^\frac{3p}{p+3}(U)}\\&\quad \quad +||\dot{\nabla }\rho v||_{L^\frac{3p}{p+3}(U)}+ ||\dot{\Delta }\varrho v||_{L^\frac{3p}{2p+3}(U)}\\&\quad \lesssim ||f||_{L^\frac{3p}{p+3}(U)}+||g^{ij}-\delta ^{ij}||_{L^\infty (U)}||\dot{{\partial }}_jv||_{L^\frac{3p}{p+3}(U)}\\&\qquad +(||\dot{\nabla }\varrho ||_{L^3(U)}+||\dot{\Delta }\varrho ||_{L^{\frac{3}{2}}(U)})||v||_{L^p(U)}\\&\quad \lesssim ||f||_{L^\frac{3p}{p+3}(U)}+\delta ||{\partial }v||_{L^\frac{3p}{p+3}(U)}+C(\delta )||v||_{L^p(U)}. \end{aligned}$$

We then sum the contributions of the covering of $\Sigma _t$ by harmonic coordinate patches $U$ satisfying (6.2) together with a partition of unity $(\varrho _U)$ subordinate to the covering. Eventually increasing $C(\delta )$, we obtain

$$\begin{aligned} ||{\partial }v||_{L^\frac{3p}{p+3}(\Sigma _t)}&\lesssim ||f||_{L^\frac{3p}{p+3}(\Sigma _t)}+\delta ||{\partial }v||_{L^\frac{3p}{p+3}(\Sigma _t)}+C(\delta )||v||_{L^p(\Sigma _t)}. \end{aligned}$$

Recall from Lemma 6.2 that we have the freedom of choice for $\delta >0$. By choosing $\delta >0$ small enough, we finally obtain

$$\begin{aligned} ||{\partial }v||_{L^\frac{3p}{p+3}(\Sigma _t)}&\lesssim ||f||_{L^\frac{3p}{p+3}(\Sigma _t)}+||v||_{L^p(\Sigma _t)} \end{aligned}$$

which together with (17.1) yields

$$\begin{aligned} ||v||_{L^p(\Sigma _t)}+||{\partial }v||_{L^\frac{3p}{p+3}(\Sigma _t)}&\lesssim ||f||_{L^\frac{p}{p+3}(\Sigma _t)}. \end{aligned}$$

This concludes the proof of Lemma 9.3.

Appendix E. Proof of Lemma 7.1

The goal of this appendix is to prove Lemma 7.1. The commutation formula (7.1) has already been proved at the beginning of Sect. 7. Thus, it only remains to prove the commutation formula (7.2). Recalling (3.25),

$$\begin{aligned} \square \phi =-{\partial }_0({\partial }_0\phi )+\Delta \phi +n^{-1}\nabla n\cdot \nabla \phi , \end{aligned}$$

Thus, we have:

$$\begin{aligned}{}[\square ,\Delta ]\phi&= [-{\partial }_0{\partial }_0+n^{-1}\nabla n\cdot \nabla +\Delta ,\Delta ]\phi \nonumber \\&= -[{\partial }_0{\partial }_0,\Delta ]\phi +[n^{-1}\nabla n\cdot \nabla ,\Delta ]\phi . \end{aligned}$$

(18.1)

We thus have to calculate the commutators $[{\partial }^2_0,\Delta ]\phi$ and $[n^{-1}\nabla n\cdot \nabla ,\Delta ]\phi$. For any tensor $U$ tangent to $\Sigma _t$, we denote by $\nabla _0U$ the projection of ${\mathbf {D}}_0U$ to $\Sigma _t$. We have the following commutator formula for any vectorfield $U$ tangent to $\Sigma _t$:

$$\begin{aligned}&[\nabla _b, \nabla _0]U_a=k_{bc}\nabla _cU_a-n^{-1}\nabla _bn \nabla _0U_a\nonumber \\&\quad +(n^{-1}k_{ab}\nabla _cn-n^{-1}k_{bc}\nabla _an+{\mathbf {R}}_{0abc})U_c, \end{aligned}$$

(18.2)

while for a scalar $\phi$, the commutator formula reduces to:

$$\begin{aligned}{}[\nabla _b, \nabla _0]\phi =k_{bc}\nabla _c\phi -n^{-1}\nabla _bn {\partial }_0\phi . \end{aligned}$$

(18.3)

Using the commutator formulas (18.2) and (18.3) and the fact that $[{\partial }_0,\Delta ]\phi =[\nabla _0,\nabla ^a]\nabla _a\phi +\nabla ^a[\nabla _0,\nabla _a]\phi$, we obtain:

$$\begin{aligned}&[{\partial }_0,\Delta ]\phi =-2k^{ab}\nabla _a\nabla _b\phi +2n^{-1}\nabla _bn\nabla _b({\partial }_0\phi )\nonumber \\&\quad +n^{-1}\Delta n{\partial }_0\phi -2n^{-1}\nabla _an k^{ab}\nabla _b\phi , \end{aligned}$$

(18.4)

where we used the constraint equation (2.2) and the fact that, in view of the Einstein equations and the symmetries of ${\mathbf {R}}$, we have:

$$\begin{aligned} {\mathbf {g}}^{ab}{\mathbf {R}}_{0abc}=0. \end{aligned}$$

Differentiating the commutator formula (18.4) with respect to ${\partial }_0$ and using the commutator formulas (18.2) and (18.3), we obtain:

$$\begin{aligned}&{\partial }_0([{\partial }_0,\Delta ]\phi )\\&\quad =-2k^{ab}\nabla _a\nabla _b({\partial }_0\phi )+2n^{-1}\nabla _bn\nabla _b({\partial }_0({\partial }_0\phi ))\\&\qquad +(-2\nabla _0k^{ab}+4k^{ac}k_c\,^b)\nabla _a\nabla _b\phi \\&\qquad +(2n^{-1}\nabla _b({\partial }_0n)-10k^{ab}n^{-1}\nabla _an)\nabla _b({\partial }_0\phi )\\&\qquad +(n^{-1}\Delta n+2n^{-2}|\nabla n|^2){\partial }_0({\partial }_0\phi )\\&\qquad +(2k^{ac}{\mathbf {R}}_{0acb}+2k^{ac}\nabla _ck_{ab}-2n^{-1}\nabla _an\nabla _0k^{ab}+2k^{ab}n^{-1}\nabla _a({\partial }_0n)\\&\qquad +4k^{ac}k_{cb}n^{-1}\nabla _an\\&\qquad +2|k|^2n^{-1}\nabla _bn-2k^{ab}n^{-2}\nabla _an{\partial }_0n)\nabla _b\phi \\&\qquad +(n^{-1}\Delta ({\partial }_0n)-4k^{ab}n^{-1}\nabla _a\nabla _bn+2n^{-2}\nabla _bn\nabla _b({\partial }_0n)){\partial }_0\phi . \end{aligned}$$

Together with the commutator formula (18.4) applied to ${\partial }_0\phi$, we obtain:

$$\begin{aligned}&[{\partial }_0{\partial }_0,\Delta ]\phi \nonumber \\ \nonumber&\quad =[{\partial }_0,\Delta ]{\partial }_0\phi +{\partial }_0([{\partial }_0,\Delta ]\phi )\\ \nonumber&\quad =-4k^{ab}\nabla _a\nabla _b({\partial }_0\phi )+4n^{-1}\nabla _bn\nabla _b({\partial }_0({\partial }_0\phi ))\\ \nonumber&\qquad +(-2\nabla _0k^{ab}+4k^{ac}k_c\,^b)\nabla _a\nabla _b\phi \\ \nonumber&\qquad +(2n^{-1}\nabla _b({\partial }_0n)-12k^{ab}n^{-1}\nabla _an)\nabla _b({\partial }_0\phi )\\ \nonumber&\qquad +(2n^{-1}\Delta n+2n^{-2}|\nabla n|^2){\partial }_0({\partial }_0\phi )\\ \nonumber&\qquad +(2k^{ac}{\mathbf {R}}_{0acb}+2k^{ac}\nabla _ck_{ab}-2n^{-1}\nabla _an\nabla _0k^{ab}+2k^{ab}n^{-1}\nabla _a({\partial }_0n)\\ \nonumber&\qquad +4k^{ac}k_{cb}n^{-1}\nabla _an\\ \nonumber&\qquad +2|k|^2n^{-1}\nabla _bn-2k^{ab}n^{-2}\nabla _an{\partial }_0n)\nabla _b\phi \\&\qquad +(n^{-1}\Delta ({\partial }_0n)-4k^{ab}n^{-1}\nabla _a\nabla _bn+2n^{-2}\nabla _bn\nabla _b({\partial }_0n)){\partial }_0\phi . \end{aligned}$$

(18.5)

We also compute the commutator $[n^{-1}\nabla n\nabla ,\Delta ]\phi$:

$$\begin{aligned}&[n^{-1}\nabla n\nabla ,\Delta ]\phi \\&\quad = -\Delta (n^{-1}\nabla _bn)\nabla _b\phi -\nabla _a(n^{-1}\nabla _bn)\nabla _a\nabla _b\phi +n^{-1}\nabla _bn[\nabla _b,\Delta ]\phi \\&\quad = -n^{-1}\nabla _b(\Delta n)\nabla _b\phi -n^{-1}[\Delta ,\nabla _b]n\nabla _b\phi +n^{-2}\nabla _an\nabla _a\nabla _bn\nabla _b\phi \\&\qquad +n^{-2}\nabla _bn\nabla _an\nabla _a\nabla _b\phi -n^{-1}\nabla _a\nabla _bn\nabla _a\nabla _b\phi +n^{-1}\nabla _bn[\nabla _b,\Delta ]\phi . \end{aligned}$$

Now, we have the following commutator formula:

$$\begin{aligned}{}[\nabla _b,\Delta ]\phi =R_{b}\,^{c} \nabla _c\phi = ({\mathbf {R}}_{b00}\,^{c}+k_{bd}k^{dc} )\nabla _c\phi , \end{aligned}$$

(18.6)

where we used the Gauss equation for $R$, the Einstein equations for ${\mathbf {R}}$ and the maximal foliation assumption. Thus, we obtain:

$$\begin{aligned}&[n^{-1}\nabla n\nabla ,\Delta ]\phi \\ \nonumber&= (-n^{-1}\nabla _a\nabla _bn+n^{-2}\nabla _bn\nabla _an)\nabla _a\nabla _b\phi +(-n^{-1}\nabla _b(\Delta n)+n^{-2}\nabla _an\nabla _a\nabla _bn\\ \nonumber&+2({\mathbf {R}}_{b00a}+k_{ba}k_a\,^c)n^{-1}\nabla _an)\nabla _b\phi . \end{aligned}$$

(18.7)

Finally, (18.1), (18.5) and (18.7) yield:

$$\begin{aligned}&[{\partial }_0{\partial }_0,\Delta ]\phi \\ \nonumber&= [{\partial }_0,\Delta ]{\partial }_0\phi +{\partial }_0([{\partial }_0,\Delta ]\phi )\\ \nonumber&= -4k^{ab}\nabla _a\nabla _b({\partial }_0\phi )+4n^{-1}\nabla _bn\nabla _b({\partial }_0({\partial }_0\phi ))\\ \nonumber&\quad +(-2\nabla _0k^{ab}+4k^{ac}k_c\,^b-n^{-1}\nabla _a\nabla _bn+n^{-2}\nabla _bn\nabla _an)\nabla _a\nabla _b\phi \\ \nonumber&\quad +(2n^{-1}\nabla _b({\partial }_0n)-12k^{ab}n^{-1}\nabla _an)\nabla _b({\partial }_0\phi )\\ \nonumber&\quad +(2n^{-1}\Delta n+2n^{-2}|\nabla n|^2){\partial }_0({\partial }_0\phi )\\ \nonumber&\quad +(2k^{ac}{\mathbf {R}}_{0acb}+2k^{ac}\nabla _ck_{ab}-2n^{-1}\nabla _an\nabla _0k^{ab}+2k^{ab}n^{-1}\nabla _a({\partial }_0n)\\ \nonumber&\quad +4k^{ac}k_{cb}n^{-1}\nabla _an\\ \nonumber&\quad +2|k|^2n^{-1}\nabla _bn-2k^{ab}n^{-2}\nabla _an{\partial }_0n-n^{-1}\nabla _b(\Delta n)+n^{-2}\nabla _an\nabla _a\nabla _bn\\&\quad +2({\mathbf {R}}_{b00a}+k_{ba}k_a\,^c)n^{-1}\nabla _an)\nabla _b\phi \\&\quad +(n^{-1}\Delta ({\partial }_0n)-4k^{ab}n^{-1}\nabla _a\nabla _bn+2n^{-2}\nabla _bn\nabla _b({\partial }_0n)){\partial }_0\phi , \end{aligned}$$

from which (7.2) easily follows. This concludes the proof of Lemma 7.1.