The causal structure of microlocalized rough Einstein metrics

Transcription

1 Annals of Mathematics, 6 005, The causal structure of microlocalized rough Einstein metrics By Sergiu Klainerman and Igor Rodnianski Abstract This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations epressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. In this paper we develop the geometric analysis of the Eikonal equation for microlocalized rough Einstein metrics. This is a crucial step in the derivation of the decay estimates needed in the first paper.. Introduction This is the second in a series of three papers in which we initiate the study of very rough solutions of the Einstein vacuum equations. By very rough we mean solutions which cannot be dealt with by the classical techniques of energy estimates and Sobolev inequalities. In fact in this work we develop and take advantage of Strichartz-type estimates. The result, stated in our first paper [Kl-Ro], is in fact optimal with respect to the full potential of such estimates. We recall below our main result: Theorem. Main Theorem. et g be a classical solution of the Einstein equations R αβ g =0 epressed 3 relative to wave coordinates α, g α = g μg μν g ν α =0. To go beyond our result will require the development of bilinear techniques for the Einstein equations; see the discussion in the introduction to [Kl-Ro]. We denote by R αβ the Ricci curvature of g. 3 In wave coordinates the Einstein equations take the reduced form g αβ α β g μν = N μνg, g with N quadratic in the first derivatives g of the metric.

2 96 SERGIU KAINERMAN AND IGOR RODNIANSKI Assume that on the initial spacelike hyperplane Σ given by t = 0 =0, g αβ 0 H s Σ, t g αβ 0 H s Σ with denoting the gradient with respect to the space coordinates i, i =,, 3 and H s the standard Sobolev spaces. Also assume that g αβ 0 is a continuous orentz metric and sup =r g αβ 0 m αβ 0 as r, where = 3 i= i and m αβ is the Minkowski metric. Then 4 the time T of eistence depends in fact only on the size of the norm g μν 0 H s Σ = g μν 0 H s Σ + t g μν 0 H s Σ, for any fied s>. In [Kl-Ro] we have given a detailed proof of the theorem by relying heavily on a result, which we have called the Asymptotics Theorem, concerning the geometric properties of the causal structure of appropriately microlocalized rough Einstein metrics. This result, which is the focus of this paper, is of independent interest as it requires the development of new geometric and analytic methods to deal with characteristic surfaces of the Einstein metrics. More precisely we study the solutions, called optical functions, of the Eikonal equation 3 H αβ λ αu β u =0, associated to the family of regularized orentz metrics H λ, λ N, defined, starting with an H +ε Einstein metric g, by the formula 4 H λ = P <λ gλ t, λ where 5 P <λ is an operator which cuts off all the frequencies above 6 λ. The importance of the eikonal equation 3 in the study of solutions to wave equations on a background orentz metric H is well known. It is mainly used, in the geometric optics approimation, to construct parametrices associated to the corresponding linear operator H. In particular it has played a fundamental role in the recent works of Smith[Sm], Bahouri-Chemin [Ba-Ch], [Ba-Ch] and Tataru [Ta], [Ta] concerning rough solutions to linear and nonlinear wave equations. Their work relies indeed on parametrices defined with the help of specific families of optical functions corresponding to null 4 We assume however that T stays sufficiently small, e.g. T. This a purely technical assumption which one should be able to remove. 5 More precisely, for a given function of the spatial variables =,, 3, the ittlewood Paley projection P <λ f = μ< λ Pμf, Pμf = F χμ ξ ˆfξ with χ supported in the unit dyadic region ξ. 6 The definition of the projector P <λ in [Kl-Ro] was slightly different from the one we are using in this paper. There P <λ removed all the frequencies above M 0 λ for some sufficiently large constant M 0. It is clear that a simple rescaling can remedy this discrepancy.

3 ROUGH EINSTEIN METRICS 97 hyperplanes. In [Kl], [Kl-Ro], and also [Kl-Ro] which do not rely on specific parametrices, a special optical function, corresponding to null cones with vertices on a timelike geodesic, was used to construct an almost conformal Killing vectorfield. The main message of our paper is that optical functions associated to Einstein metrics, or microlocalized versions of them, have better properties. This fact was already recognized in [Ch-Kl] where the construction of an optical function normalized at infinity played a crucial role in the proof of the global nonlinear stability of the Minkowski space. A similar construction, based on two optical functions, can be found in [Kl-Ni]. Here, we take the use of the special structure of the Einstein equations one step further by deriving unepected regularity properties of optical functions which are essential in the proof of the Main Theorem. It was well known see [Ch-Kl], [Kl], [Kl-Ro] that the use of Codazzi equations combined with the Raychaudhuri equation for the trχ, the trace of null second fundamental form χ, leads to the improved estimate for the first angular derivatives of the traceless part of χ. A similar observation holds for another null component of the Hessian of the optical function, η. The role of the Raychaudhuri equation is taken by the transport equation for the mass aspect function μ. In this paper we show, using the structure of the curvature terms in the main equations, how to derive improved regularity estimates for the undifferentiated quantities ˆχ and η. In particular, in the case of the estimates for η we are led to introduce a new nonlocal quantity μ/ tied to μ via a Hodge system. The properties of the optical function are given in detail in the statement of the Asymptotics Theorem. We shall give a precise statement of it in Section after we introduce a few essential definitions. The paper is organized as follows: In Section we construct an optical function u, constant on null cones with vertices on a fied timelike geodesic, and describe our basic geometric entities associated to it. We define the surfaces, the canonical null pair, and the associated Ricci coefficients. This allows us to give a precise statement of our main result, the Asymptotic Theorem.5. In Section 3 we derive the structure equations for the Ricci coefficients. These equations are a coupled system of the transport and Codazzi equations and are fundamental for the proof of Theorem.5. In Section 4 we obtain some crucial properties of the components of the Riemann curvature tensor R αβγδ. The remaining sections are occupied with the proof of the Asymptotics Theorem. We give a detailed description of their content and the strategy of the proof in Section 5. The paper is essentially self-contained. From the first paper in this series [Kl-Ro] we only need the result of Proposition.4 Background Estimates which in any case can be easily derived from the the metric hypothesis 5, the

4 98 SERGIU KAINERMAN AND IGOR RODNIANSKI Ricci condition, and the definition 4. We do however rely on the following results: Isoperimetric and trace inequalities, see Proposition 6.6. Calderon-Zygmund type estimates, see Proposition 6.0. Theorem 8.. The proof of the important Theorem 8. is delayed to our third paper in the series [Kl-Ro]. The first two ingredients are standard modifications of the classical isoperimetric and Calderon-Zygmund estimates; see [Kl-Ro]. We recall our metric hypothesis referred in [Kl-Ro, ] as the bootstrap hypothesis on the components of g relative to our wave coordinates α. 5 Metric Hypothesis. for some fied γ>0. g [0,T ] H+γ + g [0,T ] B 0,. Geometric preliminaries We start by recalling the basic geometric constructions associated with a orentz metric H = H λ. Recall, see [Kl-Ro, ], that the parameters of the Σ t foliation are given by n, v, the induced metric h and the second fundamental form k ij, according to the decomposition, 6 H = n dt + h ij d i + v i dt d j + v j dt, with h ij the induced Riemannian metric on Σ t, n the lapse and v = v i i the shift of H. Denoting by T the unit, future oriented, normal to Σ t and k the second fundamental form k ij = D i T, j we find, 7 t = nt + v, t,v =0, k ij = T H ij = n t h ij v h ij with X denoting the ie derivative with respect to the vectorfield X. We also have the following; see [Kl-Ro,, 8]: 8 c ξ h ij ξ i ξ j c ξ, c n v h for some c>0. Also n, v. The time ais is defined as the integral curve of the forward unit normal T to the hypersurfaces Σ t. The point Γ t is the intersection between Γ and Σ t.

5 ROUGH EINSTEIN METRICS 99 Definition.. The optical function u is an outgoing solution of the Eikonal equation 9 H αβ α u β u =0 with initial conditions uγ t =t on the time ais. The level surfaces of u, denoted C u, are outgoing null cones with vertices on the time ais. Clearly, 0 T u = u h where h is the induced metric on Σ t, u h = 3 i= e iu relative to an orthonormal frame e i on Σ t. We denote by the surfaces of intersection between Σ t and C u. They play a fundamental role in our discussion. Definition. Canonical null pair. = b = T + N, =T = T N. Here = H αβ β u α is the geodesic null generator of C u, b is the lapse of the null foliation or shortly null lapse b =,T = T u, and N is the eterior unit normal, along Σ t, to the surfaces. Definition.3. A null frame e,e,e 3,e 4 at a point p consists, in addition to the null pair e 3 =,e 4 =, ofarbitrary orthonormal vectors e,e tangent to. All the estimates in this paper are in fact local and independent of the choice of a particular frame. We do not need to worry that these frames cannot be globally defined. Definition.4 Ricci coefficients. et e,e,e 3,e 4 be a null frame on as above. The following tensors on 3 χ AB = D A e 4,e B, χ AB = D A e 3,e B, η A = D 3e 4,e A, η A = D 4e 3,e A, ξ A = D 3e 3,e A are called the Ricci coefficients associated to our canonical null pair. We decompose χ and χ into their trace and traceless components. 4 trχ = H AB χ AB, trχ = H AB χ AB, 5 ˆχ AB = χ AB trχh AB, ˆχ AB = χ AB trχh AB.

6 00 SERGIU KAINERMAN AND IGOR RODNIANSKI We define s to be the affine parameter of, i.e. s = and s =0on the time ais Γ t. In [Kl-Ro], where n = we had s = t u. Such a simple relation does not hold in this case; we have instead, along any fied C u, 6 dt ds = n. We shall also introduce the area At, u of the -surface St, u and the radius rt, u defined by 7 A =4πr. Along a given C u we have 7 A t = ntrχ. S Therefore, along C u, 8 dr dt = r ntrχ where, given a function f, we denote by ft, u its average on.thus ft, u = 4πr f. The following Ricci equations can also be easily derived see [Kl-Ro]. They epress the covariant derivatives D of the null frame e A A=,,e 3,e 4 relative to itself. 9 D A e 4 = χ AB e B k AN e 4, D A e 3 = χ AB e B + k AN e 3, D 4 e 4 = k NN e 4, D 4 e 3 =η A e A + k NN e 3, D 3 e 4 =η A e A + k NN e 4, D 3 e 3 =ξ A e A k NN e 3, D 4 e A = D/ 4 e A + η A e 4, D 3 e A = D/ 3 e A + η A e 3 + ξ A e 4, D B e A = / B e A + χ AB e 3 + χ AB e 4 where, D/ 3, D/ 4 denote the projection on of D 3 and D 4, / denotes the induced covariant derivative on and, for every vector X tangent to Σ t, 0 k NX = k NX n X n. Thus k NN = k NN n Nn and k AN = k AN n A n. Also, χ AB = χ AB k AB, η A v = k AN, ξ A = k AN + n A n η A 7 This follows by writing the metric on in the form γ ABst, θ,θdθ a dθ B, relative to angular coordinates θ,θ, and its area At, u = γdθ dθ d. Thus, A = dt γab d dt γab γdθ dθ. On the other hand d ds γab =χab and = n. ds dt

7 ROUGH EINSTEIN METRICS 0 and, η A = b / A b + k AN. The formulas 9, and can be checked in precisely the same manner as in [Kl-Ro]. The only difference occurs because D T T no longer vanishes. We have in fact, relative to any orthonormal frame e i on Σ t, 3 D T T = n e i ne i. To check 3 observe that we can introduce new local coordinates i = i t, on Σ t which preserve the lapse n while making the shift V to vanish identically. Thus t = nt and therefore, for an arbitrary vectorfield X tangent to Σ t, we easily calculate, D T T,X = n X i D t t, i = n X i t, D t i = n X i t, D i t = n X i i t, t = n X i in =n Xn. Equations indicate that the only independent geometric quantities, besides n, v and k are trχ, ˆχ, η. We now state the main result of our paper giving the precise description of the Ricci coefficients. Note that a subset of these estimates was stated in Theorem 4.5 of [Kl-Ro]. Theorem.5. et g be an Einstein metric obeying the Metric Hypothesis 5 and H = H λ be the family of the regularized orentz metrics defined according to 4. Fi a sufficiently large value of the dyadic parameter λ and consider, corresponding to H = H λ, the optical function u defined above. et I 0 + be the future domain of the origin on Σ 0. Then for any ε 0 > 0, such that 5ε 0 <γwith γ from 5, the optical function u can be etended throughout the region I 0 + ] R 3 and there the Ricci coefficients trχ, ˆχ, and η [0,λ 8ε0 satisfy the following estimates: trχ 4 r + ˆχ t + η t t λ 3ε0, trχ r 5 q + ˆχ q + η q λ 3ε0, with q 4. In the estimate 8 the function r can be replaced with nt u. In addition, in the eterior region r t/, trχ 6 s t λ 4ε0, ˆχ t λ ε0 + Ht, η λ + λ ε0 t + λ ε Ht where the last estimate holds for an arbitrary positive ε, ε<ε 0. Also, there eist the following estimates for the derivatives of trχ:

8 0 SERGIU KAINERMAN AND IGOR RODNIANSKI 7 sup r t trχ r t + sup trχ r nt u t λ 3ε0, t 8 sup / trχ t r t + sup / trχ r t nt u t λ 3ε0. In addition, there are weak estimates of the form, sup /, 9 trχ λ u nt u C t for some large value of C. The inequalities indicate that the bounds hold with some universal constants including the constant B 0 from Null structure equations In the proof of Theorem.5 we rely on the system of equations satisfied by the Ricci coefficients χ, η. Below we write down our main structural equations. Their derivation proceeds in eactly the same way as in [Kl-Ro] see Propositions. and.3 from the formulas 9 above. Proposition 3.. The components trχ, ˆχ, η and the lapse b verify the following equations: b= b k NN, trχ+ trχ = ˆχ k NN trχ R 44, D/ 4 ˆχ AB + trχˆχ AB = k NN ˆχ AB ˆα AB, D/ 4 η A + trχη A = k BN + η B ˆχ AB trχk AN β A. Here ˆα AB = R 4A4B R 44δ AB and β A = R 4A34. Also, when 34 μ = trχ trχ k NN + n N n trχ, 8 which can be interpreted as transport equations along the null geodesics generated by. Indeed observe that if an S tangent tensorfield Π satisfies the homogeneous equation D/ 4Π = 0 then Π is parallel transported along null geodesics.

9 ROUGH EINSTEIN METRICS 03 there is the equality 35 μ+trχμ =η A η A / A trχ ˆχ AB / A η B +η A η B + k NN ˆχ AB +trχˆχ AB +ˆχ AC ˆχ CB +k AC χ CB + R B43A R 44 +k NN 4n N n trχ ˆχ k NN trχ R k NNtrχ + trχ +4 k NN ˆχ + R 44 trχ k AN η A n A n. n Nn + R k Nm kn m. Remark 3.. Equation 3 is known as the Raychaudhuri equation in the relativity literature; see e.g. [Ha-El]. Remark 3.3. Observe that our definition of μ differs from that in [Kl-Ro]. Indeed there we had, instead of μ, μ = trχ trχ 3 k NN trχ and the corresponding transport equation: 36 μ+trχ μ =η A η A / A trχ ˆχ AB / A η B +η A η B + k NN ˆχ AB +trχˆχ AB +ˆχ AC ˆχ CB +k AC χ CB + R B43A R 44 k NN trχ 3 k NN trχ +4 k NNtrχ +trχ +4 k NN ˆχ + R 44. We obtain 35 from 36 as follows: The second fundamental form k verifies the equation see formula.0.3a in [Ch-Kl], In particular, nt k ij = i j n + nr it jt k im k m j. nt k NN = Nn + nr NTNT k Nm k m N. Eploiting the definition of the ie derivative nt, we obtain T k NN +k N T,N= n Nn +R NTNT k Nm k m N. It then follows that k NN+ k NN k NN k AN = n Nn +R NTNT k Nm k m N.

10 04 SERGIU KAINERMAN AND IGOR RODNIANSKI Therefore, since + =T, =N, k NN n Nn = k NN n Nn +R NTNT + k Nm k m N +n N Nn n Nn. Recall that k NN = k NN n Nn and N N,e A = k AN η A.Thus k NN = k NN + n Nn +k AN η A n A n n Nn + R k Nm kn m. Therefore taking μ = trχ trχ k NN + n Nntrχ we derive the desired transport equation 35. Proposition 3.4. The epressions div / ˆχ A = / B ˆχ AB, div / η = / B η B and curl / η AB = / A η B / B η A verify the following equations: div / ˆχ A +ˆχ AB k BN = / A trχ + k ANtrχ R B4AB, div / η = μ +n Nntrχ η ˆχ k AB χ AB R B43A, curl / η = εab k AC ˆχ CB εab R B43A. We also have the Gauss equation, 40 K =ˆχ AB ˆχ AB trχtrχ + R ABAB. We add two useful commutation formulas. emma 3.5. et Π A be an m-covariant tensor tangent to the surfaces. Then, 4 / B D/ 4 Π A D/ 4 / B Π A = χ BC / C Π A n / B nd/ 4 Π A + i χ AiB k CN χ BC kain + R CAi4BΠ A..Č..Am. Also, for a scalar function f, 4 / N / A f / A / N f = 3 k AND 4 f η A + k AN D 3 f χ AB χ AB / B f.

11 ROUGH EINSTEIN METRICS 05 Proof. For simplicity we only provide the proof of the identity 4. The derivation of 4 is only slightly more involved see [Ch-Kl], [Kl-Ro]. We have / N / A f / A / N f =[N,e A ]f / N e A f =D N e A / N e A f D A Nf. Now using the identity N = e 4 e 3 and the Ricci equations 9 we can easily infer Special structure of the curvature tensor R In this section we describe some remarkable decompositions 9 of the curvature tensor of the metric H. Given a system of coordinates 0 α relative to which H is a nondegenerate orentz metric with bounded components H αβ, we define the coordinate dependent norm 43 H = ma γh αβ. α,β,γ A frame e a,e b,e c,e d is bounded, with respect to our given coordinate system, if all components of e a = e α a α are bounded. Consider an arbitrary bounded frame e a,e b,e c,e d and R abcd the components of the curvature tensor relative to it. Relative to any system of coordinates, 44 R abcd = e α a e β b eγ c e δ d αγh βδ + βδ H αγ βγ H αδ αδ H βγ. Using our given coordinates α we introduce the flat Minkowski metric m αβ = diag,,,. We denote by D the corresponding flat connection. Using D we define the following tensor: πx, Y, Z = D Z HX, Y. Thus in our local coordinates α, π αβγ = γ H αβ. Proposition 4.. Relative to an arbitrary bounded frame e a,e b,e c,e d there is the following decomposition: 45 R abcd = D a π bdc + D b π acd D a π bcd D b π dac + E abcd where the components of the tensor E are bounded pointwise by the square of the first derivatives of H. More precisely, since E = ma a,b,c,d E abcd ma α,β,γ,δ E αβγδ, 46 E H. 9 The results of this section apply to an arbitrary orentz metric H. 0 This applies to the original wave coordinates α.

12 06 SERGIU KAINERMAN AND IGOR RODNIANSKI Remark 4.. It will be clear from the proof below that we can interchange the indices a, c and b, d in the formula above and obtain similar decompositions. We show that each term appearing in 44 can be epressed in terms of a corresponding derivative of π plus terms of type E. Consider the term R = e α a e β b eγ c e δ d αδ H βγ. We show that it can be epressed in the form D a π bcd plus terms of type E. Indeed, D a π bcd = e a π bcd π Dabcd π bdacd π bcdad = e α a α e δ d eβ b eγ c δ H βγ π Dabcd π bdacd π bcdad = R + e α a α e δ d eβ b eγ c δ H βγ π Dabcd π bdacd π bcdad = R + e δ d eα a α e β b eγ c δ H βγ π Dabcd.... Now, π Dabcd = D d HD a e b,e c =e δ d D ae b β e γ c δ H βγ. Thus, D a π bcd = R + e δ d eγ c δ H βγ e α a α e β b D ae b β. On the other hand D a e b β = D a e b, μ H βμ Henceforth, we infer that, = e α a α e β b e b, D a μ H βμ e b, μ e α a α H βμ. R abcd = D aπ bcd + E abcd with E = e δ d eγ c δ H βγ eb, D a μ H βμ + e b, μ e α a α H βμ. Since D a μ can be epressed in terms of the first derivatives of H we conclude that E H as desired. The other terms in the formula 44 can be handled in precisely the same way. Remark 4.3. We will apply Proposition 4. to our metric H, wave coordinates α and our canonical null frames. We remark that our wave coordinates are nondegenerate relative to H, see 8, and any canonical null frame e 4 =T + N,e 3 =T N, e A is bounded relative to α. Corollary 4.4. Relative to an arbitrary frame e A on, 47 R ABCD = / A π BDC + / B π ACD / A π BCD / B π DAC + E ABCD Recall that D β μ =Γ γ βμ γ with Γ the standard Christoffel symbols of H.

13 ROUGH EINSTEIN METRICS 07 where E is an error term of the type, E H + χ H and π H. Corollary 4.5. There eist a scalar π, ans-tangent -tensor π AB and -form E A such that, the component R B4AB admits the decomposition Moreover, R B4AB = / A π + / B π AB + E A. π H, E H + χ H. Corollary 4.6. There eist an S-tangent vector π A and scalar E such that ε AB R AB34 = curl / π + E and π H E H + χ H. Corollary 4.7. There eist S-tangent vectors π A,π A and scalars E,E such that δ AB R A43B = div / π + R + R 34 + E, ε AB R A43B = curl / π + E, where R is the scalar curvature. Moreover, π, H, E, H + χ H. Proof. Observe that R AB = H μν R AμBν = R A3B4 R A4B3 δ CD R ACBD. Hence, since R A3B4 = R B4A3, we have δ AB R AB = δ AB R A4B3 δ AB δ CD R ACBD, and therefore, δ AB R A43B = δ AB R AB + δ AB δ CD R ACBD = R + R 34 + δ AB δ CD R ACBD. We now appeal to Corollary 4.4 and epress δ AB R A43B in the form δ AB R A43B = div / π + R + R 34 + E,

14 08 SERGIU KAINERMAN AND IGOR RODNIANSKI where π H E H + χ H. On the other hand since R A3B4 + R AB43 + R A43B R A3B4 R A4B3 = R AB43. Thus, = 0, we infer that ε AB R A43B = ε AB R AB43. In view of Corollary 4.6 we can therefore epress ε AB R A43B in the form curl / π + E. 5. Strategy of the proof of the Asymptotics Theorem In this section we describe the main ideas in the proof of the Asymptotics Theorem. Section 6. We start by making some primitive assumptions, which we refer to as Bootstrap assumptions. They concern the geometric properties of the C u and foliations. Based on these assumptions we derive further important properties, such as Sharp comparisons between the functions u, r and s. Isoperimetric and Sobolev inequalities on. Trace inequality; restriction of functions in H Σ t to. Transport lemma Elliptic estimates on Hodge systems. Section 7. We recall the background estimates on H = H λ proved in [Kl-Ro]. We establish further estimates of H related to the surfaces and null hypersurfaces C u. q estimates for H and RicH. Energy estimates on C u. Statement of the estimate for the derivatives of Ric 44 H. 3 Section 8. Using the bootstrap assumptions and the results of Sections 6 and 7 we provide a detailed proof of the Asymptotics Theorem.

15 ROUGH EINSTEIN METRICS Bootstrap assumptions and Basic Consequences Throughout this section we shall use only the following background property, see Proposition.4 in [Kl-Ro], of the metric H in [0,t ] R 3 : 48 H t λ 4ε0. By the Hölder inequality we also have, 49 H t λ 8ε0. The maimal time t verifies the estimate t λ 8ε Bootstrap assumptions. We start by constructing the outgoing null geodesics originating from the ais Γ t, t [0,t ]. The geodesics emanating from the same points Γ t form the null cones C u. We define Ω [0,t ] R 3 to be the largest set properly foliated by the null cones C u with the following properties: A Any point in Ω lies on a unique outgoing null geodesic segment initiated from Γ t and contained in Ω. A Along any fied C u, r s ass 0. Here s denotes the affine parameter along C u, i.e. s =ands Γt = 0. Recall also that r = rt, u denotes the radius of = C u Σ t. Moreover, the following bootstrap assumptions are satisfied for some q>, sufficiently close to : B trχ r t λ ε0, ˆχ t λ ε0, η t λ ε0, B trχ r q λ ε0, ˆχ q λ ε0, η q λ ε0. Remark 6.. It is straightforward to check that B and B are verified in a small neighborhood of the time ais Γ t. Indeed for each fied λ our metrics H λ are smooth and therefore we can find a sufficiently small neighborhood, whose size possibly depends on λ, where the assumptions B and B hold. Remark 6.3. We shall often have to estimate functions f in Ω which verify equations of the form df ds = F with f = f 0 on the ais Γ t. According to A we can epress the value of f at every point P Ω by the formula, fp =f 0 P 0 + F with γ the unique null geodesic in Ω connecting the point P with the time ais Γ t and P 0 = γ Γ t. For convenience we shall rewrite this formula, relative γ

16 0 SERGIU KAINERMAN AND IGOR RODNIANSKI to the affine parameter s in the form fs =f0 + s 0 F s ds. It will be clear from the contet that the integral with respect to s denotes the integral along a corresponding null geodesic γ Comparison results. We start with a simple comparison between the affine parameter s and nt u. emma 6.5. In the region Ω s t u, i.e., s t u and t u s. 50 Proof. Observe that dt ds = t =T t =n and, since u Γt = t, t u = γ n = s 0 n s ds. Thus, since n is bounded uniformly from below and above, we infer that s and t u are comparable, i.e. s t u. In particular s λ 4ε0 everywhere in Ω. Remark 6.6. The formula ds dt = n along γ together with the uniform boundedness of n, used in emma 6.5 above, allows us to estimate integrals along the null geodesics γ as follows: s s F = F s ds = F ts,s ds γ 0 = 0 t 0 nf t,s t dt F t. We shall make a frequent use of this remark and refine the comparison between s and t u. emma 6.7. In the region Ω, nt u =s +Oλ 4ε0. In [Kl-Ro] we had in fact n = and s = t u. In our contet this is no longer true due to the nontriviality of the lapse function n.

17 ROUGH EINSTEIN METRICS Proof. Consider U = nt u s and proceed as in the lemma above by noticing that du ds = 0. Therefore, d ds U = d nt u s = n nnt u ds = n ns + n n nt u s. Integrating from the ais Γ t we find, 5 Us = s n nds + γ γ Us n nds where γ is the null geodesic starting on the ais Γ t and passing through a point P 0 corresponding to the value s. By Gronwall we find, Us s 0 s n n ds ep s 0 n n ds. According to Remark 6.6, s 0 n n H t. We can now make use of the inequality 49 and infer that nt u =s +Oλ 8ε0. emma 6.8. The null lapse function b, see Definition., satisfies the estimate 5 throughout the region Ω. bs ns λ 8ε0 Proof. Integrating the transport equation 30, b = b k NN, along the null geodesic γs, we infer that, s bs =b0 ep k NN. 0 Since k NN H, the condition 49 gives s 0 k NN λ 8ε0. According to our definition b = T u and u Γt = t. Thus b 0 = T t =n 0 and therefore, bs n0 λ 8ε0. To finish the proof it only remains to observe that ns n0 γ n. λ 8ε0 Recall that the Hardy-ittlewood maimal function 3 Mft offt is defined by t Mft = sup fτ dτ, t 0 t t 0 t 0 3 restricted to the interval [0,t ]

18 SERGIU KAINERMAN AND IGOR RODNIANSKI and that, for any <p<. Mf p t f p t emma 6.9. et a be a solution of the transport equation a =F. Then for any point P Ω Σ t γ, where γ is the null geodesic beginning on the ais Γ t at the point P 0 Σ t0 and terminating at the point P, 53 ap ap 0 sm F t where s is the value of the affine parameter of γ corresponding to P. Proof. Integrating the equation a = da ds = F along γ we obtain t ap ap 0 = F F Σ τ dτ t t 0 M F t. γ t 0 It remains to observe that t t 0 = t u and that according to emma 6.5, t u s. Using emma 6.9 we can now refine the conclusions of emmas 6.8, 6.7. Corollary b = n + so M Ht, 55 nt u =s + s O M Ht, 56 nt u M Ht, s 57 nt u s t λ 4ε0 where M Ht is the maimal function of Ht. Proof. The proof of 54 is straightforward since b n = b k NN n. Now observe that the right-hand side b k NN +n H and b n Γt =0. Since, according to emma 6.7, nt u s, the equation nt u s = n nnt u can be written in the form d ds nt u s s H. Thus with the help of emma 6.9 we obtain nt u s s M H. The inequality 56 is an immediate consequence of 55 and emma 6.7. The estimate 57 follows from 56, 48, and the estimate for the Hardy- ittlewood maimal function.

19 ROUGH EINSTEIN METRICS 3 We shall now compare the values of the parameters s and r = 4π A at a point P. emma 6.. The identity r = s +Oλ 6ε0 holds throughout the region Ω. In particular this implies that πs At, u 8πs with At, u the area of. Proof. Similarly to 8, we have r = r trχ = trχ. 8πr Using the identity A =4πr, we obtain dr ds =+ trχ 58. 8πr r Integrating along the null geodesic γ passing through the point P = P s 4 we have rp s 59 γ r trχ r 4π γ r trχ r γ r s trχ r + γ s trχ r. Thus by Gronwall, and the bootstrap estimate B, trχ λ 4ε0 trχ r r t we infer that, r s sλ 6ε0. t λ 6ε0 Having established that r s we shall now derive more refined comparison estimates involving trχ s and its iterated maimal functions. These will be needed later on in Section 9.6 where trχ s rather than trχ r appears naturally. 60 Corollary 6.. r s s M 3 trχ s, 6 r s s 3 trχ s t. 4 Observe that according to A, r s 0ass 0 along C u.

20 4 SERGIU KAINERMAN AND IGOR RODNIANSKI Here, M k is the k th maimal function. Moreover, trχ r trχ s + M3 trχ 6 s trχ r trχ 63 t s, t trχ r +r q trχ 64 q s, t r nt u trχ 65 + λ 4ε0. s t t, Proof. We write the transport equation for r in the following form: r = trχ πr s 8πr s. Differentiating s 67 = s we obtain s trχ s = trχ + s s s. Furthermore, s = s trχ. s Since s r 0asr 0, we have s 8π. Using emmas 6. and 6.9 we infer that =8π + sm s trχ s Integrating 67 and using emma 6.9 once more we obtain s =8πs + s M trχ + s M s trχ s Again, according to emma 6., r s. Thus by 66 r = s r + trχ + sm trχ 8πr s s or, equivalently, r =s + trχ + rsm trχ 4π s s...

21 ROUGH EINSTEIN METRICS 5 Integrating with the help of emma 6.9 we infer that, r = s + s 3 M 3 trχ + s 3 M s trχ s. It then follows that 68 r = s + s M 3 trχ s. Observe that if during each integration along γ we used Hölder inequality instead of the bounds involving maimal functions, we would have the estimate r = s + s 3 trχ 69. s t This estimate can be used effectively to compare r and s on a single surface while 68 works well with the norms involving integration in time. Thus, we infer from from 68 that r s M3 trχ 70 s, r s trχ 7. s t t In addition, 69 implies that r 7 s q r q trχ s t. Inequalities 6 64 follow from the identity trχ r =trχ s + r s and Finally, 65 follows from 70 and 57. Remark 6.3. Observe that equation 58 and emma 6.9 also give the estimate r s s M trχ t. r Thus with the help of the bootstrap assumption B and the estimate for the maimal function we infer that, trχ s trχ t r + t r 73 s t trχ λ ε0. r t

22 6 SERGIU KAINERMAN AND IGOR RODNIANSKI Moreover, since r s, equation 58, Hölder inequality and the bootstrap assumption B also imply that r s r q trχ r λ ε0 sr q. q γ Using the bootstrap assumption B once again we infer that trχ s q trχ r + q r 74 s q λ ε0 + λ ε0 r q q λ ε0. Estimates 74, 73 indicate that the bootstrap assumptions B, B also hold for trχ s Isoperimetric, Sobolev inequalities and the transport lemma. We consider now the foliation induced by on Σ t Ω. Relative to this foliation the induced metric h on Σ t takes the form h = b du + γ AB dφ A dφ B where φ A are local coordinates on S. We state below a proposition concerning the trace and isoperimetric inequalities on Σ t Ω. The proposition requires a very weak assumption on the metric h; in fact we only need 75 sup r ε 3 +ε h Σ t Λ 0 Ω for some large constant Λ 0 > 0 and an arbitrarily small ε>0. In this and the following subsection we shall assume a slightly stronger property that 76 sup r ε +ε H Σ t Λ 0. Ω Remark 6.5. The assumption 76 is easily satisfied by our families of metrics H = H λ ; see Remark 7.. Proposition 6.6. et be a fied surface in Σ t Ω with N the eterior unit normal on Σ t. Under the assumption 76 the following estimates hold true with constants independent of : i For any smooth function f : R, the following isoperimetric inequality holds: f /f + r 77 f.

23 ROUGH EINSTEIN METRICS 7 ii The following Sobolev inequality holds on : for any δ 0, and p from the interval p, ], δp sup f r εp p+δp 78 p+δp /f + r f [ ] δ /f p + r p f p p+δp. iii Consider an arbitrary function f :Σ t R such that f H +ε R 3. The following trace inequality holds true: 79 f +ε f Σ t + ε f Σ t. More generally, for any q [, 80 f q 3 q +ε f Σ t + 3 q ε f Σ t. Also, consider the region Ω 4 r, r = 4 r ρ r ρ : where r = rt, u, then, 8 f Nf Ω 4 r,r f Ω 4 r,r + r f Ω 4 r,r. Proof. See [Kl-Ro]. Finally we state below, emma 6.7 The transport lemma. et Π A be an S-tangent tensorfield verifying the following transport equation with σ>0: D/ 4 Π A + σtrχπ A = F A. Assume that the point t, =t, s, ω belongs to the domain Ω.IfΠ satisfies the initial condition s σ Π A s 0 as s 0, then 8 Πt, 4 F t. In addition, if σ q and Π satisfies the initial condition rσ q Π q 0 as r 0, then on each surface Ω, 83 Π q rt σ q t u rt σ q F q S t,u dt. Finally, if Π is a solution of the transport equation D/ 4 Π A + σtrχπ A = r F A, verifying the initial condition s σ Π A s 0 with some σ>, then 84 Πt, 4M F t.

24 8 SERGIU KAINERMAN AND IGOR RODNIANSKI Proof. The proof of 8 and 83 is straightforward. For a similar version see emma 5. in [Kl-Ro]. Estimate 84 can be proved in the same manner as 53 of emma Elliptic estimates. Net we establish a proposition concerning the estimates of Hodge systems on the surfaces. They are similar to the estimates of emma 5.5 in [Kl-Ro]. We need however to make an important modification based on Corollary 4.4. Proposition 6.9. et ξ be an m+ covariant, totally symmetric tensor, a solution of the Hodge system on the surface Ω ; then div / ξ = F, curl / ξ = G, trξ =0. Then ξ obeys the estimate /ξ + m + 85 r ξ { F + G }. Proof. Using the standard Hodge theory, see Theorem 5.4 in [Kl-Ro] or Chapter in [Ch-Kl], we have 86 /ξ +m +K ξ = { F + G }. The Gauss curvature K of the -surface can be epressed as K = 4 trχ + trχtrk + ˆχ ˆχ + R ABAB. Thus it follows from Corollary 4.4 that K r = / A Π A + E where the tensor Π and the error term E, relative to the standard coordinates α, obey the pointwise estimates Π H and E H + ˆχ + χ H. Then, 87 /ξ + m + r ξ { F + G +m + / A Π A + E ξ }. Integrating the term / A Π A ξ by parts we obtain for all sufficiently large p, = p + q, / A Π A ξ = Π A / A ξ ξ /ξ ξ p Π q.

25 ROUGH EINSTEIN METRICS 9 The isoperimetric inequality implies that for p< ξ p r p /ξ + r ξ. We also deduce from the trace inequality that Π q H q 3 + +ε q H Σ t ε q H Σ t. Thus the smallness condition r q 3 + +ε q H Σ t Λ 0 ensures that we can absorb the term m + / A Π A ξ on the left-hand side of 87. For large p the above condition coincides with 75. It remains to estimate E ξ. The most dangerous term is ˆχ ξ. Applying the Hölder inequality we infer that, ˆχ ξ ξ p ˆχ q S. t,u Using the isoperimetric inequality once more, we conclude that we need a smallness condition on r q ˆχ q for some q>. This is guaranteed by our bootstrap assumption B. We shall net formulate versions of the Calderon-Zygmund theorem for the above types of Hodge systems; see also [B-W]. The proof is a straightforward modification of the standard approach. Proposition 6.0. et ξ be a covariant, traceless, symmetric tensor, verifying the Hodge system on the surface Ω, div / ξ = /ν+ e for some scalar ν and -form e. Then, 88 ξ q ν q + e p where p = + q. Also, 5 89 ξ ν log + r /ν +r p e p for any p>, where log + z = log + z. Similar estimates hold in the case when ξ is a -form verifying the Hodge system div / ξ = div / ν + e, curl / ξ = curl / ν + e for some -forms ν =ν,ν and scalars e =e,e. 5 The term r /ν can in fact be replaced by r /ν r for r>.

26 0 SERGIU KAINERMAN AND IGOR RODNIANSKI 7. Properties of the metric H and its curvature tensor R 7.. Background estimates. We start by recalling the background estimates on the family of the orentz metrics H = H λ proved in [Kl-Ro]; see Proposition.4. The metric H admits the canonical decomposition H = n dt + h ij d i + v i dt d j + v j dt and satisfies the following estimates on the time interval [0,t ] with t λ 8ε0 : c ξ h ij ξ i ξ j c ξ, n v h c>0, n, v c +m H, [0,t ] λ 8ε0 m 0 +m H [0,t ] 4ε0, m 0 +m H [0,t ] λ 4ε0, m 0 +m H [0,t ] λ m, for m +4ε 0 +m H [0,t ] λ 4ε0, for +4ε 0 m m H αβ α β H, m 0 [0,t ] λ 8ε0 m Rαβ H λ, m 0 m R αβ H, [0,t ] λ 8ε0 m 0. Remark 7.. The inequality 9 with m = 0 is consistent with the property 48, which we have used throughout Section 6. Moreover, since in the region Ω the radius r of the surfaces does not eceed λ 8ε0, we have, according to 94, r ε +ε H [0,t ] λ 4ε0ε λ ε λ ε. This verifies condition q estimates. The trace inequality 80 of Proposition 6.6 allows us to derive the q estimates on the metric H from 94. Proposition 7.4. For any q in the interval q 4 99 H q λ q 8 q ε0. In addition, 00 RicH p λ p 8 p ε0 for p [, ].

27 ROUGH EINSTEIN METRICS Proof. Since q 4, by the Hölder inequality, H q r q H 4 λ q 8ε0 H 4. Using the trace estimate 80 we infer that H q λ q 8ε0 H Ḣ R 3 λ q 8 q ε0 where we have used H Ḣ R 3 λ from 94. The inequality 00 follows similarly from the trace theorem and Energy estimates on C u. In this subsection we shall derive energy estimates, along the null hypersurfaces C u, for tangential derivatives of the first derivatives of the rescaled metric t Gt, =g λ, 0. λ Recall that the original space time Einstein metric g verifies R μν g = 0. In addition, since our coordinates α satisfy the wave coordinate condition, the metric g satisfies the quasilinear wave equation 0 g αβ α β g μν = N μν g, g. We have also defined the truncated metric g <λ = μ< λ P μg and, by rescaling, our background metric t Ht, =g <λ λ,. λ Similarly, for a dyadic μ we can define G μ t, =P μλ g t λ,. λ Observe that H has frequencies and G μ is localized to the frequencies of size μ which cannot fall below. We now formulate a basic energy estimate on the null cones C u for H and G μ. Definition 7.6. Given a scalar function F in Ω we denote by D F the C u tangential derivatives of F. More precisely, D F = /F,F. We shall use this notation for the components of the metrics H and G relative to our fied system of coordinates. We also use this notation applied to all components of the derivatives H and G. Thus D H = α,β,γ D γ H αβ. Proposition 7.7. The following estimates hold in the region Ω : 03 D H C u λ, D H C u λ.

28 SERGIU KAINERMAN AND IGOR RODNIANSKI In addition, for the functions G μ defined above, 04 D G μ C u μ 4ε0 λ 4ε0, D G μ C u ma{μ 4ε0 λ 4ε0,μ 4ε0 λ 4ε0 }. Proof of Proposition 7.7. Metric g is a H +γ solution of the Einstein equation. Thus after rescaling and taking into account γ>5ε 0, we infer that in addition to the estimates 9 96 for H, we also have 05 +m G μ t λ 4ε0 μ m 4ε0, for m =0,. We shall make use of the rescaled version of emma 8.9 in [Kl-Ro] to derive the equations for H and G μ. 06 H αβ α β H = F, H αβ α β G μ = F μ, with the right-hand sides F, F μ obeying the estimates 07 F t λ, F t λ, 08 F μ t λ μ 4ε0 4ε0, F μ t λ μ 4ε0 4ε0. We shall use the generalized energy identity with the vectorfield T in the region M t0,t,u bounded by the cone C u and the time slices Σ t0,σ t intersecting C u. The vectorfield is orthogonal, in the sense of the orentzian metric H, to the cone C u.thus Q[H]T,+ Q[H]T,T= Q[H]T,T C u Σ t0 Σ t0 Q αβ [H] T π αβ + FTH with the energy-momentum tensor M t0,t,u Q[f] αβ = α f β f H αβ ν f ν f and the deformation tensor T π αβ = T H of the vectorfield T. A similar identity also holds for G μ. According to 7 and 3 the components of the deformation tensor T π can be described as follows: T π ij = k ij, T π i0 = n i n, T π 00 =0. Thus the deformation tensor T π H, and by 9 obeys the estimate 09 Observe that T π t λ 4ε0. Q[H]T,= H + /H = D H, Q[H]T,T= TH + H = H.

29 ROUGH EINSTEIN METRICS 3 In addition, Q αβ f f. Thus, using 94, 07, and 09, we obtain D H C u H +4 Σ t0 T π H + F H M t0,t,u H t + T π t H t + F t H t λ. Similarly, D G μ C u G μ +4 Σ t0 M t0,t,u T π G μ + F μ G μ G μ t + T π t Gμ t + F μ t Gμ t ma{μ 8ε0 λ 8ε0,μ 8ε0 λ 8ε0 }. To get the estimates for D H and D G μ we differentiate the equations 06. Commuting the derivative with the metric H we obtain, H αβ α β H = F + H αβ α β H = F, H αβ α β G μ = F μ + H αβ α β G μ = Fμ. Using 07, 08 and the inequality H t λ 4ε0 of 9, we infer that F t λ, F μ t μ 4ε0 λ 4ε0. Thus using the generalized energy identity for H and G μ we will have Also, C u D H H t + T π t H t + F t H t λ. C u D G μ G μ t + T π t G μ t + F μ t G μ t μ 8ε0 λ 8ε0. The following result can be deduced from Propositions 7.7 and 4.. Corollary 7.8. Any component of the curvature R abcd =Re a,e b,e c,e d, of the metric H, with vectorfields e a,e b,e c varying between, e A,A =,, obeys the energy estimates on C u : In particular, R C u := A,B,C,D λ. R abcd C u λ. R ABCD C u + R ABC4 C u + R B43A C u

30 4 SERGIU KAINERMAN AND IGOR RODNIANSKI 8. A remarkable property of R 44 While the spacetime metric g verifies the Einstein equations R μν g =0 this is certainly not true for the effective metric H = H λ. This could create serious problems in the proof of the asymptotics theorem as the Ricci curvature appears as a source term in the null structure equations. We have already established an improved estimate for RicH in t, see 98. This was done by comparing R μν H with R μν G = 0 where G = gλ t, λ is the rescaled Einstein metric. We need however a stronger estimate involving the derivatives of R 44 H along the null cones C u. To establish such an estimate we encounter an additional difficulty: the null cones C u have been constructed relative to the approimate metric H. This leads to significant differences between the C u energy estimates for the second derivatives of H, see 03 and the corresponding ones 6 for G; see 04 in Proposition 7.7. Using however the specific structure of the component R 44 relative to the wave coordinates we can overcome this difficulty and prove the following: Theorem 8.. On any null hypersurface C u, 0 t u R 44 H S τ,udτ λ. Proof. The proof of the theorem requires a rather long and tedious argument which we present in our paper [Kl-Ro]. 9. Asymptotics Theorem We start by recalling already established estimates for the metric related quantities which play a crucial role in what follows. 3 H t λ 4ε0, H q λ 8 q q ε0 for q 4, RicH t, λ 8ε0 4 RicH p λ 8 p p ε0 for p, 5 D H C u λ, s R 44 λ ε0, R C u λ 6 The estimates for the second derivatives of the higher frequencies of G do in fact diverge badly.

31 ROUGH EINSTEIN METRICS 5 where R C u := A,B,C,D R ABCD C u + R ABC4 C u + R B43A C u. Note that some of the above estimates hold only throughout the region Ω. Theorem 9.. Throughout the region Ω the quantities trχ r, ˆχ, and η satisfy the following estimates: trχ 8 r + ˆχ t + η t t λ 3ε0, trχ 9 r + ˆχ q + η q λ 3ε0. q In the estimate 8 function r can be replaced with nt u. Also, the corresponding t estimate follows by Hölder inequality: trχ λ 3ε0, nt u trχ 0 λ 3ε0. nt u t In addition, in the eterior region r t/, t trχ s t λ 4ε0, ˆχ t λ ε0 + Ht, sup r t η λ + λ ε0 t + λ ε Ht where the last estimate holds for an arbitrary positive ε, ε<ε 0. There are now the following estimates for the derivatives of trχ: trχ r t 3 sup / trχ t r t + sup trχ r t + sup / trχ r t nt u t λ 3ε0, nt u t λ 3ε0. In addition, there are have weak estimates of the form, sup /, 4 trχ λ u nt u C t for some large value of C.

32 6 SERGIU KAINERMAN AND IGOR RODNIANSKI Corollary 9.. The estimates of Theorem 9. can be etended to the whole region I 0 + [0,t ] R 3, where I 0 + is the future domain of the origin on Σ 0. Remark 9.3. The proof of Corollary 9. requires an etension argument. The estimates of the Asymptotics Theorem, which are uniform with respect to the bootstrap region Ω, provide very good control of the foliations C u and. By the standard continuity argument this allows us to show that the estimates, in fact, hold in the maimal domain allowed by the background estimates 7 on the metric H, I 0 + [0,t ] R 3. Remark 9.4. Observe also that we can etend the results of 8 to a slightly larger domain I + [0,t ] R 3. This is in fact needed to derive the first derivative estimates 3, in I 0 + [0,t ] R 3, whose proof depends on Theorem 8.. That theorem, to be proved in [Kl-Ro], requires indeed the estimates for Θ, see definition below, in a slightly larger domain. The estimates for Θ however, i.e. 8, are independent of Theorem 8.. Proof. To simplify our calculations we start with the following definition. Definition 9.5. Θ= trχ r + trχ 5 s + ˆχ + η + H. In our calculations below we shall often us the notation Θ but mean in fact OΘ. In view of our bootstrap assumptions B, B see Section 6., Remark 6.3, as well as the estimates, for H we can freely make use of the following: 6 Θ t λ ε0, Θ q λ ε0 inside the bootstrap region Ω Estimates for trχ, ˆχ. We start with estimates 8 for trχ. Observe that in view of Corollary 6. it suffices to prove the desired estimates for trχ s. Writing y = trχ s we have, 7 y+trχy = R 44 s k NN +Θ. Applying the transport emma 6.7 we infer that at any point P Ω, s yp s R 44 + s H +Θ γ

33 ROUGH EINSTEIN METRICS 7 where γ is the outgoing null geodesic starting on the time ais Γ t, passing through P, and s is the corresponding value of the affine parameter s. Therefore, yp R 44 t + H + Θ s t and, in view of 6 and 3, 8 yp λ 4ε0 + λ 4ε0 + s γ γ H. In the eterior region s t, using the condition, we infer that, trχ 9 s t λ 4ε0, which proves. On the other hand, see also the proof of emma 6.9, 8 leads to a global estimate, trχ 30 λ 4ε0 + M Ht s where M H is the maimal function of Ht. The estimates 30 and together with the corresponding maimal function estimates readily imply that trχ s λ 4ε0 + M Ht t λ 4ε0 + H t λ 4ε0. t On the other hand, using the comparison results between r and s, see Section 6.3., s λ 8ε0 λ, and the Hölder inequalities trχ s r q y q λ q λ 4ε 0 + s q s H t λ q λ 4ε 0 λ 4ε0 provided that q>is chosen sufficiently close to. Using the comparison results between r and s of Corollary 6. we infer that, trχ 3 r λ 4ε0, t trχ 3 r λ 4ε0, q as desired in 8 and 9. Finally, 0 follows from 57 of Corollary 6.0. We shall now estimate ˆχ from the Codazzi equations 37, 33 div / ˆχ A +ˆχ AB k BN = / A trχ + k ANtrχ R B4AB.

34 8 SERGIU KAINERMAN AND IGOR RODNIANSKI Taking advantage of Corollary 4.5, with a different error term E, we rewrite 33 in the form, div / ˆχ A = / A trχ 34 + / r A π + / B π AB + E with π and E obeying pointwise estimates π H, E Θ H + r H. We shall now take advantage of the elliptic estimate of Proposition 6.0 and write ˆχ λ ε trχ r 35 with q>. +λ ε π + r q E q Remark 9.7. In the application of the elliptic estimate 89 in the derivation of 35 we need some rough estimates for / trχ of the type r / trχ λ C for some large constant C>0. These weak estimates, consistent with 4, are a lot easier to derive and can be obtained directly from the transport equations 3, 3 for trχ and ˆχ. We refer the reader to our paper [Kl-Ro] for more details. Therefore, choosing q = + ε for sufficiently small ε > 0, and using the bootstrap assumptions B as well as assumptions we infer that ˆχ λ ε trχ 36 r + λ ε H Θ qst,u H + r + q H +r q λ ε trχ r + H. Now we observe that the desired pointwise estimate in the eterior region r t/ follows from 9 and the estimate r s s λ λ ε0 ε0 t. Thus 37 ˆχ t λ ε0 + H. We can also add a global estimate following from Corollary 6. 7 and ˆχ λ 4ε0 + Ht+M 4 Ht. 7 Namely, the inequality trχ r M3 trχ s.

35 ROUGH EINSTEIN METRICS 9 Now squaring and integrating 36 in time we infer from and the just proved estimate 3 for trχ r that ˆχ t trχ λε 39 r t + H t λ 3ε0, which is the estimate claimed in 8 of Theorem 9.. On the other hand, application of the elliptic estimate 88 of Proposition 6.0 to the equation 34 yields the following: ˆχ q trχ r q + H q + E p for some q, p = + q. Choosing q =+ε as in bootstrap assumption B we infer, with the help of the estimate 3 for trχ r and, that ˆχ q λ 4ε0 + Θ H p + r H p λ 4ε0 + H Θ q + H q λ 4ε Estimates for η. We start with the Hodge system 38, 39: div / η= μ + k NN trχ η ˆχ k AB χ AB δab R A43B, curl / η= εab k AC ˆχ CB εab R A43B with μ defined as in 34, μ = trχ trχ k NN +n N n trχ, satisfying the transport equation 35, 40 μ+trχμ =η A η A / A trχ ˆχ AB / A η B +η A η B + k NN ˆχ AB +trχˆχ AB +ˆχ AC ˆχ CB +k AC χ CB + R B43A R 44 +k NN 4n N n trχ ˆχ k NN trχ R k NNtrχ + trχ +4 k NN ˆχ + R 44 trχ k AN η A n A n n Nn + R k Nm kn m. Observe that in view of Corollary 4.7 we can rewrite our div-curl system for η as follows: 4 div / η= div / π + μ + k NN trχ η ˆχ k AB χ AB w + E, curl / η= curl / π + εab k AC ˆχ CB + E

36 30 SERGIU KAINERMAN AND IGOR RODNIANSKI where w =R + R 34 and π, H E, H + χ H. Remark 9.9. We would like to treat the system formed by the transport equation 40 coupled with the elliptic system 4 in the same manner as we have dealt with the system for trχ and ˆχ. Indeed the Hodge system 4 is similar to the Hodge system 33. The transport equation for μ differs however significantly from the transport equation 7 for trχ. Indeed the only curvature term on the right-hand side of 7 is R 44 while the righthand side of 40 ehibits the far more dangerous term R 44. In what follows we shall get around this difficulty by introducing a new covector μ/ through a Hodge system on the surfaces. Using once more the special structure of the Einstein equations we shall derive a new transport equation for μ/ whose right-hand side ehibits only terms depending on RicH and favorable components of the curvature tensor. We define an auiliary S-tangent co-vector μ/ A as a solution of the Hodge system 4 43 div / μ/ = μ w, curl / μ/ =0 with w = R 43 + R. We now prove the following Proposition 9.0. The covector μ/ verifies the following: div / D/ 4 μ/ + trχμ/ ˆχ μ/ = H D/ 4 μ/ + / A R A4 + r π A +Θ Θ r 3R 34 +R+ΘRic +ΘR +Θ D H +Θ Θ Θ+ r Θ Θ+ r H, curl / D/ 4 μ/ + trχμ/ ˆχ μ/ = H D/ 4 μ/ + / Θ Θ + R Θ + Θ Θ+Θ Θ Θ. r The covector μ/ verifies the following estimates μ/ λ + M H, μ/ q λ 3ε0.

37 ROUGH EINSTEIN METRICS 3 Proof of part of Proposition 9.0. Remark 9.. For convenience we etend our bootstrap assumptions B and B to include μ/. Thus, throughout the proof below, we redefine Θ, see 5, as follows: Θ=O trχ r + trχ 46 s + ˆχ + η + H + μ/. This is justified since our stated estimates are stronger than B and B for μ/. Assuming the first part of Proposition 9.0 we now derive the estimates of part and start by applying the elliptic estimates of Proposition 6.0 to the Hodge system of Proposition 9.0. Thus for some q>, with M = D/ 4 μ/ + trχμ/ ˆχ μ/, we have, M H q M + λ RicH ε + Θ S + t,u r H + r q ΘR q + Θ / H q + ΘRicH q + r RicH q + Θ 3 q + r Θ q + r H q Remark 9.. As in the case of the estimates for ˆχ, the use of the elliptic estimates 89 of Proposition 6.0 for the Hodge system satisfied by the quantity M requires rough estimates of the type r / R A4 + /π + rθ / Θ q λ C for some q>. The estimate for the derivatives of the Ricci curvature and the metric H are contained in our background estimates In addition to trχ and ˆχ, for which we have already outlined the procedure of obtaining such weak estimates, the quantity Θ contains η and μ/. Once again, we can use the transport equation 33 for η and the Hodge system 4, 43 combined with the transport equation 40 for μ to handle these terms. Taking q sufficiently close to q =, using the bootstrap assumption, Θ q λ ε0, and the estimate H q λ ε0 < / we.