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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.

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Notes

  1. The original proof in [5] relied on representation formulas, following an approach pioneered by Sobolev, see [38].

  2. Based only on energy estimates and classical Sobolev inequalities.

  3. The original proof in [12, 14] actually requires one more derivative for the uniqueness. The fact that uniqueness holds at the same level of regularity than the existence has been obtained in [33].

  4. That is any past directed, in-extendable causal curve in \({\mathcal M}\) intersects \(\Sigma _0\).

  5. Recall that the entire theory of shock waves for 1+1 systems of conservation laws is based on BV norms, which are critical with respect to the scaling of the equations. Note also that these BV norms are not, typically, conserved and that Glimm’s famous existence result [13] can be interpreted as a global well posedness result for initial data with small BV norms.

  6. Note that the dimension here is \(n=3\).

  7. As we shall see, from the precise theorem stated below, other weaker conditions, such as a lower bound on the volume radius, are needed.

  8. In other words we believe that no well-posedness results can be established for general initial data in \(H^s\), with \(s<2\).

  9. Note that no such result, i.e. well-posedness for \(s=2\), is presently known for either scalar equations of the form (1.5) or systems of the form (1.3).

  10. This corresponds precisely to the \(s=2\) exponent in the case of the Einstein-vacuum equations.

  11. Maximal foliation together with spatial harmonic coordinates on the leaves of the foliation would be the coordinate condition closest in spirit to the Coulomb gauge.

  12. We would like to thank L. Anderson for pointing out to us the possibility of using such a formalism as a potential bridge to [16].

  13. Note also that additional error terms are generated by projecting the equations on the components of the frame.

  14. Assuming that the initial has more regularity so that the right-hand side of (2.8) makes sense.

  15. Here and in what follows the notations \(R, {\mathbf {R}}\) will stand for the Riemann curvature tensors of \(\Sigma _t\) and \({\mathcal M}\), while \(Ric\), \({\mathbf {Ric}}\) and \(R_{scal}, {\mathbf {R}}_{scal}\) will denote the corresponding Ricci and scalar curvatures.

  16. Since in what follows there is no danger to confuse the Ricci curvature \(\text{ Ric }\) with the scalar curvature \(R\) we use the short hand \(R\) to denote the full curvature tensor \(\text{ Ric }\).

  17. We follow the proof of the bilinear estimates outlined in [21] which differs substantially from that of [15] and is reminiscent of the null frame space strategy used by Tataru in his fundamental paper [48].

  18. (2.19) actually corresponds to the representation formula for a half-wave. The full representation formula corresponds to the sum of two half-waves (see Sect. 10)

  19. Here \(f\) is in fact at the level of the Fourier transform of the initial data and the norm \(\Vert \lambda f\Vert _{L^2({\mathbb R}^3) } \) corresponds, roughly, to the \(H^1\) norm of the data.

  20. In the flat Minkowski space \({ \, \, ^{\omega } u}(t,x)=t\pm x\cdot \omega \).

  21. Such as contractions between the Riemann curvature tensor and derivatives of solutions of scalar wave equations.

  22. Classically, this requires, at the very least, the control of \({\mathbf {R}}\) in \(L^\infty \).

  23. Taking into account the different behavior in tangential and transversal directions with respect to the level surfaces of \({ \, \, ^{\omega } u}\).

  24. In the time symmetric case \(k=0\), this is exactly the minimal surface equation.

  25. Note in particular that the corresponding estimate in the flat case is sharp.

  26. Note that the procedure we describe would prove not only (2.24) but the full range of mixed Strichartz estimates.

  27. The regularity (2.25) is necessary to make sense of the change of variables involved in the stationary phase method.

  28. We refer to the approach based on the overlap estimates for wave packets derived in [36] and [37] in the context of Strichartz estimates respectively for \(C^{1,1}\) and \(H^{2+\epsilon }\) metrics. Note however that our approach does not require a wave packet decomposition.

  29. Recall that \(\text{ tr }k=0\).

  30. Note that our smallness assumptions on \(\tilde{A}\) make the proof of the Lemma much simpler than the original result of Uhlenbeck.

  31. In the context of this paper the lemma is applied to the purely spatial part of \({\mathbf {A}}\) i.e. \( (A_i)_{jk}\).

  32. such that the corresponding connection \({\mathbf {A}}\) verify the Coulomb gauge condition (3.28) and the estimates of proposition 4.4.

  33. Strictly speaking Uhlenbeck’s lemma takes care of the purely spatial part of \(A\); one needs to also use the constraint equation for \(k\) to derive (3.29).

  34. Using the lapse equation \(\Delta n=n|k|^2\) and \(k, \nabla k\in L^{\infty }_tL^{2}(\Sigma _t)\), see (5.7), together with the Sobolev embedding (6.5) we only deduce \(k\in L^{\infty }_tL^{6}(\Sigma _t)\) from which \(\Delta n\in L^{\infty }_tL^{3}(\Sigma _t)\). This would yield \(\nabla ^2n\in L^{\infty }_tL^{3}(\Sigma _t)\), and thus \(\nabla n\) misses to be in \(L^\infty ({\mathcal M})\) by a log divergence. However, one can overcome this loss by exploiting the Besov improvement with respect to the Sobolev embedding (6.5). We refer the reader to section 4.4 in [44] for the details.

  35. i.e. it satisfies assumptions (5.1) and (5.2).

  36. In fact, we need to restrict \(s\) to an open interval of \([0,1]\) for which the bootstrap assumptions (5.3) (5.4) on \({\mathbf {R}}\) hold.

  37. We ignore in the estimates below the lower order term \(\Vert {\partial }B \Vert _{L^2({\mathcal H}_u)}\) which is easier to handle.

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Acknowledgments

This work would be inconceivable without the extraordinary advancements made on nonlinear wave equations in the last twenty years in which so many have participated. We would like to single out the contributions of those who have affected this work in a more direct fashion, either through their papers or through relevant discussions, in various stages of its long gestation. D. Christodoulou’s seminal work [8] on the weak cosmic censorship conjecture had a direct motivating role on our program, starting with a series of papers between the first author and M. Machedon, in which spacetime bilinear estimates were first introduced and used to take advantage of the null structure of geometric semilinear equations such as Wave Maps and Yang–Mills. The works of Bahouri–Chemin [2, 3] and D.Tataru [50] were the first to go below the classical Sobolev exponent \(s=5/2\), for any quasilinear system in higher dimensions. This was, at the time, a major psychological and technical breakthrough which opened the way for future developments. Another major breakthrough of the period, with direct influence on our approach to bilinear estimates in curved spacetimes, is D. Tataru’s work [48] on critical well posedness for Wave Maps, in which null frame spaces were first introduced. His joint work with H. Smith [37] which, together with [20], is the first to reach optimal well-posedness without bilinear estimates, has also influenced our approach on parametrices and Strichartz estimates. The authors would also like to acknowledge fruitful conversations with L. Anderson, and J. Sterbenz.

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Correspondence to Sergiu Klainerman.

Additional information

S. Klainerman and I. Rodnianski are supported by the FRG grant NSF DMS-1065710. J. Szeftel is supported by the project ERC 291214 BLOWDISOL.

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:

  1. (i)

    Partition of unity

    $$\begin{aligned} \sum _jQ_j=I. \end{aligned}$$
    (15.1)
  2. (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)
  3. (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

    $$\begin{aligned}\Vert Q_j \nabla F\Vert _{L^2(\Sigma _t)}\lesssim 2^j \Vert F\Vert _{L^2(\Sigma _t)}\end{aligned}$$
  4. (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.

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Klainerman, S., Rodnianski, I. & Szeftel, J. The bounded \(L^2\) curvature conjecture. Invent. math. 202, 91–216 (2015). https://doi.org/10.1007/s00222-014-0567-3

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