The resolution of the bounded L2 curvature conjecture in general relativity
Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel

Abstract. This paper reports on the recent proof of the bounded L2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the
Einstein-vacuum equations depends only on the L2 -norm of the curvature and a lower
bound of the volume radius of the corresponding initial data set.
Mathematics Subject Classification (2010). Primary 83C05; Secondary 35Q75,
58J45.
Keywords. Einstein equations, Cauchy problem, rough solutions, null structure, bilinear
estimates.

1. Introduction
This paper reports on the recent proof of the bounded L2 curvature conjecture.
More precisely we show that the time of existence of a classical solution to the
Einstein-vacuum equations depends only on the L2 -norm of the curvature and alower bound of the volume radius of the corresponding initial data set.
The entire proof of the conjecture is contained in the sequence of papers [20][33] [34] [35] [36] [37].

1.1. Initial value problem for the Einstein vacuum equations.
We consider the Einstein vacuum equations (EVE),
Ricαβ = 0(1)

where Ricαβ denotes the Ricci curvature tensor of a four dimensional Lorentzianspace time (M, g). (1) corresponds to an evolution problem. An initial data set
consists of a three dimensional manifold Σ0 together with a Riemannian metricg and a symmetric 2-tensor k on Σ0 . For a given initial data set (Σ0 , g, k), theCauchy problem consists in finding a metric g satisfying (1) and an embeddingof Σ0 in M such that the metric induced by g on Σ0 coincides with g and the
2-tensor k is the second fundamental form of the embedding.

2

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel

Remark 1.1. Since physically one should not be able to distinguished between
different coordinates systems, a solution of the Cauchy problem can be unique only
modulo a diffeomorphism.
The equations (1) are overdetermined and the initial data set (Σ0 , g, k) has tosatisfy the following compatibility conditions known as the constraint equations
 j∇ kij − ∇i trk = 0,
(2)
Rscal − |k|2 + (trk)2 = 0,
where the covariant derivative ∇ is defined with respect to the metric g, Rscal is
the scalar curvature of g, and trk is the trace of k with respect to the metric g.
In this paper we restrict ourselves to asymptotically flat initial data sets with
one end. More precisely, (Σ0 , g, k) is such that Σ0 minus a compact set is diffeomorphic to R3 minus a ball, with gij − δij and kij satisfying a suitable rate of fall
off at infinity in this coordinates system.
The first local existence and uniqueness result for (EVE) was established by
Y.C. Bruhat, see [3], with the help of wave coordinates which allowed her to cast
the Einstein vacuum equations in the form of a system of nonlinear wave equations
to which one can apply1 the standard theory of nonlinear hyperbolic systems. The
optimal, classical2 result states the following.
Theorem 1.2 (Classical local existence [7] [8]). Let (Σ0 , g, k) be an initial data set
for the Einstein vacuum equations (1). Assume that Σ0 can be covered by a locallyfinite system of coordinate charts, related to each other by C 1 diffeomorphisms,s−1ssuch that (g, k) ∈ Hloc
(Σ0 ) × Hloc
(Σ0 ) with s > 52 . Then there exists a unique3(up to an isometry) globally hyperbolic development (M, g), verifying (1), for which
Σ0 is a Cauchy hypersurface4 .
Our goal will be to lower the assumptions of the previous theorem on theregularity of the initial data set. To motivate our result, let us first emphasizein the next section why pushing for rough solutions is a main theme in nonlinear
evolution PDEs.

1.2. The quest for rough solutions in nonlinear evolution PDEs.
1.2.1. First examples. To illustrate the role played by rough solutions in nonlinear evolution PDEs, let us consider a nonlinear evolution equation possessing
a conserved quantity which is positive definite and in particular controls a norm
1 The original proof in [3] relied on representation formulas, following an approach pioneeredby Sobolev, see [29].
2 Based only on energy estimates and classical Sobolev inequalities.
3 The original proof in [7], [8] actually requires one more derivative for the uniqueness. The
fact that uniqueness holds at the same level of regularity than the existence has been obtainedin [25].
4 That is any past directed, in-extendable causal curve in M intersects Σ .
0The resolution of the bounded L2 curvature conjecture in general relativity

3

in a certain functional space. We refer to the conserved quantity as the energy,
its associated functional space as the energy space, and its associated norm as theenergy norm. One can then classify such evolution equations into three categories5
1. One can prove a local existence result with a time of existence which depends
only on the energy norm of the initial data. This case is referred as energy
subcritical.
2. One can prove a local existence result for initial data in the energy space,but with a time of existence which does not only depend on the energy normof the initial data (i.e. there is no uniform lower bound on the size of thetime interval of existence for initial data with a given energy norm). Thiscase is referred as energy critical.
3. One can not prove a local existence result for initial data in the energy space.
This case is referred as energy supercritical.
As the energy supercritical case as it is still vastly open, we will focus on thetwo other cases. In the energy subcritical case, one can pile up time intervals of
existence provided by the local existence result which are all of the same size since
they only depend on the energy norm of the data which is conserved. One infers
global existence for any initial data. A nice illustration of this procedure is provided by [11] in the case of the classical Yang-Mills equations in dimension 1+3,
where the energy space is the Sobolev space H 1 .
In the energy critical case, the conjecture is that global existence holds for any
data below the energy of the first nontrivial stationary solution or solitary wave.
This conjecture has been proved in a large number of cases over the last thirty
years. A spectacular achievement of this method is the recent proof of the conjecture in the case of the 2+1 wave map problem in [38], [31], [32] and [21].
A key step in the large data results mentioned above, both for the energy subcritical and energy critical cases, is a local well-posedness result at the level ofthe energy space, which is typically a low regularity well-posedness result. In thenext section, we discuss another example of a nonlinear evolution partial differential equation for which making sense of rough solutions plays a fundamental role.

1.2.2. The proof of the weak cosmic censorship in spherical symmetry.
In this section, we briefly discuss the influential proof by Christodoulou [6] of the
weak cosmic censorship conjecture for the Einstein equations coupled to a scalar
field in spherical symmetry6 .
5 One usually defines the criticality relative to the behavior of the energy under some notion of
scaling. Here, we instead classify the equation with respect to an ability to prove local existence
results. These two classifications agree in most cases and our choice only aims at simplifying theexposition.
6 Due to Birkhoff’s theorem, the Einstein vacuum equations are non dynamical in sphericalsymmetry. To obtain a dynamical problem and yet retain the advantage of working in spherical4

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel

Let us first recall the weak cosmic censorship conjecture of Penrose. The starting point of this conjecture is the existence of space times containing singularities,
the most famous example being the Schwarzschild space-time which is sphericallysymmetric and contains a singularity at r = 0. Now, the existence of space-times
containing a singularity could be considered as an undesirable feature from thepoint of view of physics. To come to term with such space-times, Penrose formulated the celebrated weak cosmic censorship conjecture7 .
Conjecture 1.3 (Weak cosmic censorship). For generic asymptotically flat initial
data set, singularities are hidden by a black hole.
In view of this conjecture, singularities are acceptable as they not visible byan observer at infinity. At the moment, it is still an open problem in general, but
the conjecture has been proved in the case of spherical symmetry in [6] for the
Einstein equations coupled to a scalar field. This seminal work relies on the roughwell posedness result of [5]. This well posedness result - which involves regularity
assumptions at the level of a weighted bounded variation (BV) norm - allows inparticular to make sense of solutions with a jump along the backward null conefrom the singularity. This jump turns out to be essential in generating arbitrarily small perturbations of a given solution containing a singularity which are still
strong enough to cover the singularity with a black hole8 .
The result in [6] provides thus yet another example of the importance of making sense of rough solutions for nonlinear evolution PDEs. It motivates our main
result, which concerns well-posedness of rough solutions for the Einstein equationsin the absence of symmetry. Now, BV norms are only adapted to hyperbolic problems in 1+1 dimension (and hence to spherical symmetry). This will force us toabandon BV norms and to instead measure the regularity of our solution using L2
based norms which are the only norms which are propagated by the evolution inhigher dimensions.

1.3. The resolution of the bounded L2 curvature conjecture.
1.3.1. The bounded L2 curvature conjecture. In this section, we consider
the problem of going beyond the classical local existence result stated in Theorem
1.2. To make the discussion more tangible it is worthwhile to recall the form of
the Einstein vacuum equations in the wave gauge. Assuming given coordinates xα ,symmetry, one adds a matter field to the right-hand side of the Einstein equations, a scalar field
being the simplest possibility. For the sake of simplicity, we do not explicitly write down theequations in this case.
7 He also introduced the strong cosmic censorship conjecture which despite its name is independent of the weak cosmic censorship conjecture.
8 In the proof, the singularity is actually covered by a trapped region which, as it turns out, isenough.

The resolution of the bounded L2 curvature conjecture in general relativity

5

verifying9
g xα = 0, α = 0, . . . , 3,(3)

the metric coefficients gαβ = g(∂α , ∂β ), with respect to these coordinates, satisfy
the system of quasilinear wave equations10 ,g gαβ = Fαβ (g, ∂g)(4)where Fαβ are quadratic functions of ∂g, i.e. the first order derivatives of g withrespect to the coordinates xα . In the harmonic coordinates, the wave operator on
the curved background g is given by g = gµν ∂µ ∂ν . Equation (4) is obtained byexpressing the Ricci tensor Ricαβ in terms of the components of g and its firstand second order derivatives. Conversely, to verify that the solution of (4) yields
a solution to the Einstein vacuum equations (1), one has additionally to show thatthe coordinates conditions (3) propagate. This holds for solutions of (4), as wasobserved by Choquet-Bruhat, provided these coordinates conditions are satisfied
initially on Σ0 and (Σ0 , g, k) satisfies the constraint equations (2).
In a first approximation we may compare (4) with the semilinear wave equation,
φ = F (φ, ∂φ)(5)with F quadratic in ∂φ. Using standard energy estimates - i.e. differentiating (5)s − 1 times, multiplying it with ∂t ∂ s−1 φ, integrating by parts and using Gronwall’s
lemma - one obtains the following control for the Sobolev norm H s of φ
 Z t
kφ(t)ks . kφ(0)ks exp Csk∂φ(τ )kL∞ dτ .
(6)0

The classical exponent s > 3/2 + 1 arises simply from the Sobolev embedding of
H r , r > 3/2 into L∞ .
To go beyond the classical exponent, see [26], one has to replace Sobolev inequalities with Strichartz estimates of, roughly, the following type,
Z t
1/2


Z t
2
k∂φ(τ )kL∞ dτ. C k∂φ(0)kH 1+ +kφ(τ )kH 1+
00

where  > 0 can be chosen arbitrarily small. This leads to a gain of 1/2 derivatives,
i.e. we can prove well-posedness for equations of type (5) for any exponent s > 2.
The same type of improvement in the case of quasilinear wave equations requires
a highly non-trivial extension of such estimates for wave operators with non-smoothcoefficients. The first improved regularity results for quasilinear wave equations of
the type,g µν (φ)∂µ ∂ν φ = F (φ, ∂φ)(7)
9  is the covariant wave operator gαβ D D .
g
α β
10 Nonlinear wave equations are either semilinear or quasilinear according to whether the higher

order terms - here the terms containing second order derivatives - are linear or nonlinear.

6

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftelwith g µν (φ) a non-linear perturbation of the Minkowski metric mµν , are due to [1],[2], and [40], [41] and [13]. The best known results for equations of type (4) wereobtained in [14] and [28]. According to them one can lower the Sobolev exponents > 5/2 in Theorem 1.2 to s > 2. It turns out, see [22], that these results are
sharp in the general class of quasilinear wave equations of type (4). However, theEinstein equations enjoy a special structure, and it was conjectured in [12] that
one can obtain a well-posedness result at the level of s = 211 .
Conjecture 1.4 (Bounded L2 curvature conjecture). The Einstein- vacuum equations admit local Cauchy developments for initial data sets (Σ0 , g, k) with locallyfinite L2 curvature and locally finite L2 norm of the first covariant derivatives ofk 12 .

1.3.2. The bounded L2 curvature theorem. In this section, we state ourmain result which gives a positive answer to the above conjecture. We assume thespace-time (M, g) to be foliated by the level surfaces Σt of a time function t. Let
T denote the unit normal to Σt , and let k the the second fundamental form of Σt ,i.e. kab = −g(Da T, eb ), where ea , a = 1, 2, 3 denotes an arbitrary frame on Σt and
Da T = Dea T . We assume that the Σt foliation is maximal, i.e. we haveg ab kab = 0(8)where g is the induced metric on Σt .
We also recall below the definition of the volume radius on a general Riemannian
manifold M .
Definition 1.5. Let Br (p) denote the geodesic ball of center p and radius r. The
volume radius rvol (p, r) at a point p ∈ M and scales ≤ r is defined byrvol (p, r) = inf0r ≤r|Br0 (p)|,r3with |Br | the volume of Br relative to the metric on M . The volume radiusrvol (M, r) of M on scales ≤ r is the infimum of rvol (p, r) over all points p ∈ M .
Our main result is the following.
Theorem 1.6 (Main theorem). Let (M, g) an asymptotically flat solution to the
Einstein vacuum equations (1) together with a maximal foliation by space-like hypersurfaces Σt defined as level hypersurfaces of a time function t. Assume that theinitial slice (Σ0 , g, k) is such that the Ricci curvature Ric ∈ L2 (Σ0 ), ∇k ∈ L2 (Σ0 ),and Σ0 has a strictly positive volume radius on scales ≤ 1, i.e. rvol (Σ0 , 1) > 0.
11 The curvature tensor of g and the first order derivatives of the second fundamental form k
are both at the level of two derivatives of g. Thus, Conjecture 1.4 is at the level of two derivativesof g in L2 which indeed corresponds to the case s = 2.
12 As we shall see, from the precise theorem stated below, other weaker conditions, such as alower bound on the volume radius, are needed.

The resolution of the bounded L2 curvature conjecture in general relativity

7

1. L2 regularity. There exists a time
T = T (kRickL2 (Σ0 ) , k∇kkL2 (Σ0 ) , rvol (Σ0 , 1)) > 0
and a constant
C = C(kRickL2 (Σ0 ) , k∇kkL2 (Σ0 ) , rvol (Σ0 , 1)) > 0
such that the following control holds on 0 ≤ t ≤ T :kRkL∞
L2 (Σt ) ≤ C, k∇kkL∞
L2 (Σt ) ≤ C and[0,T ][0,T ]

inf rvol (Σt , 1) ≥

0≤t≤T

1
.
C

2. Higher regularity. Within the same time interval as in part (1) we also
have the higher derivative estimates13 ,

X
X 
(α)
(i)
(i)
kD RkL∞k∇ RickL2 (Σ0 ) + k∇ ∇kkL2 (Σ0 ) ,L2 (Σt ) ≤ Cm
[0,T ]
|α|≤m

|i|≤m

(9)
where Cm depends only on the previous C and m.
Let us comment on Theorem 1.6.
1. As mentioned in the previous section, the well posedness result of [28] in
H s with s > 2 is sharp for general quasilinear wave equations of type (4).
To do better, one needs to take into account the so called null structure,i.e. the special nonlinear structure of the Einstein equations. In particular,
Theorem 1.6 is the first well posedness result in which the full structure of
the quasilinear hyperbolic system, not just its principal part, plays a crucialrole.
2. The assumptions of Theorem 1.6 concern the L2 norm of the curvature tensor
of g and of the first covariant derivatives of the second fundamental form kwhich are all invariant in the sense that these objects can be defined without
reference to any coordinates system14 . This allows, when working in theframework of the solutions constructed in Theorem 1.6, to retain an essential
property of the Einstein equations, namely the freedom to pick a coordinates
system (see Remark 1.1).
3. The part of Theorem 1.6 dealing with the propagation of higher order regularity provides a continuation argument for the Einstein equations; that is
the space-time constructed by evolution from smooth data can be smoothly
continued, together with a time foliation, as long as the curvature of thefoliation and the first covariant derivatives of its second fundamental form
remain L2 - bounded along the leaves of the foliation. In fact, Theorem 1.6
13 Assuming that the initial has more regularity so that the right-hand side of (9) makes sense.
14 Note that this is not the case for instance of the result in [14] where one has to choose a fixed
coordinates system with respect to which the metric coefficients are in H s for s > 2.

8Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel

implies the break-down criterion previously obtained in [19] and improved in[24], [42]. Furthermore, this break-down criterion involves only invariant assumptions, and hence provides information on true singularities (as opposed
to coordinates singularities).
4. One may wonder whether the solutions constructed in Theorem 1.6 are as
rough as possible. To discuss this issue, observe that the light cones of aLorentzian space-time (M, g) can be obtained as the level hypersurfaces ofa solution u to the Eikonal equationgαβ ∂α u∂β u = 0.
Now, a byproduct of the proof of Theorem 1.6 is the fact that L2 bounds
on the curvature is the minimum requirement to control solutions u to the
Eikonal equation (see Remark 4.1), and hence to make sense of light cones.
As far as light cones are fundamental objects in Lorentzian space-times, it isreasonable to expect their control to be necessary to make sense of solutions
to the Einstein equations. For this reason, we conjecture that Theorem 1.6
is optimal.
The entire proof of Theorem 1.6 is contained in the sequence of papers [20] [33][34] [35] [36] [37]. In the rest of this paper, we discuss the general strategy of theproof as well as the main steps.

2. Sketch of the proof of the main theorem
2.1. Strategy of the proof. As mentioned earlier, the well posedness resultof [28] in H s for s > 2 is sharp for general quasilinear wave equations of type (4).
To do better one needs to take into account the special structure of the Einsteinequations and rely on a class of estimates which go beyond Strichartz estimates,
namely the so called bilinear estimates. In the case of semilinear wave equations,such as Wave Maps, Maxwell-Klein-Gordon and Yang-Mills, the first results whichmake use of bilinear estimates go back to [9], [10], [11]. In the particular case of theYang-Mills equation the main observation was that, after the choice of a special
gauge (Coulomb gauge), the most dangerous nonlinear terms exhibit a special, null
structure so that the system reduces to the following schematic form
φ = Qij (φ, ∇−1 φ) + ∇−1 (Qij (φ, φ)) + l.o.t,(10)where φ is vector valued15 and Qij is the null form given by
Qij (φ, ψ) = ∂i φ∂j ψ − ∂i ψ∂j φ, i, j = 1, 2, 3,(11)15 Note that (10) is a system. In particular, the schematic notation Q (φ, φ) should be underij
stood as being a linear combination of terms of the type Qij (φm , φl ) where φl and φm denotecomponents of the vector valued function φ.

The resolution of the bounded L2 curvature conjecture in general relativity

9

for which one can apply the bilinear estimates derived in [9]. With the help of
these estimates one was able to derive a well posedness result, in the flat 1 + 3
dimensional Minkowski space, for the exponent s = 116 .
To carry out a similar program in the case of the Einstein equations one would
need, at the very least to
1. Exhibit the null structure, i.e. provide a coordinate condition, relative towhich the Einstein vacuum equations verify an appropriate version of thenull condition.
2. Exploit the null structure, i.e. prove bilinear estimates for the null quadratic
terms appearing in the previous step.
Concerning the coordinate condition, let us first mention that it is a priori not
at all clear what it should be, or even if there is one for that matter.
Remark 2.1. The only known structural condition related to the classical nullcondition, called the weak null condition [23], tied to wave coordinates, fails thetest. Indeed, the following simple system in Minkowski space
2φ = 0,2ψ = φ · ∆φverifies the weak null condition and yet, according to [22], it is ill posed for s = 2.
Coordinate conditions, such as spatial harmonic17 , also do not seem to work.
We rely instead on a Coulomb type condition, for orthonormal frames, adapted
to a maximal foliation. Such a gauge condition appears naturally if we adopt a
Yang-Mills description of the Einstein field equations using Cartan’s formalism ofmoving frames, see [4]. It is important to note nevertheless that it is not at all apriori clear that such a choice would do the job. Indeed, the null form nature of
the Yang-Mills equations in the Coulomb gauge is only revealed once we commutethe resulting equations with the projection operator P on the divergence free vectorfields. Such an operation is natural in that case, since P commutes with the flatd’Alembertian. In the case of the Einstein equations, however, the corresponding
commutator term [g , P] generates18 a whole host of new terms and it is quite amiracle that they can all be treated by an extended version of bilinear estimates.
Concerning bilinear estimates, let us mention that these types of estimates
where only available for the wave operator on the Minkowski space-time. This
forces us to find an appropriate geometric framework to extend these estimatesto the wave operator on a curved space-time. To this end, we need to rely on aplane wave representation - a parametrix - for solutions of the wave equation on
16 This corresponds precisely to the s = 2 exponent in the case of the Einstein-vacuum equations.
17 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.
18 Note also that additional error terms are generated by projecting the equations on the components of the frame.

10

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftela curved background. Moreover, this parametrix, unlike in the flat case, is only
an approximate solution of the wave equation. In other words, when applying
the wave operator to the parametrix, we obtain an error term which needs to becontrolled.
Furthermore, there is another ingredient needed to establish bilinear estimateson a curved space-time. Numerous bilinear estimates need to be derived, and itturns out that the proof of several of these estimates reduces to sharp L4 (M)Strichartz estimates for a localized version of the parametrix.
Finally, the above discussion leads to the following four steps which constitutethe basic strategy of our main theorem
A. Exhibit the null structure by recasting the Einstein vacuum equations as aquasilinear Yang-Mills theory19 .
B. Prove appropriate bilinear estimates for solutions φ to the scalar wave equation on a curved space-time g φ = 0.
C. Construct an effective progressive wave representation ΦF (parametrix) for
solutions to the scalar linear wave equation g φ = F , derive appropriate
bounds for both the parametrix and the corresponding error term E = F −g ΦF and use them to derive the desired bilinear estimates.
D. Prove sharp L4 (M) Strichartz estimates for a localized version of the parametrixof step C.
While Step A is purely algebraic, Steps B, C and D all require to establish
estimates. The main difficulty is to implement these steps using only hypothetical
L2 bounds for the space-time curvature tensor, consistent with the statement ofour main theorem. To achieve this, we crucially need to exploit the null structureof the equations at every stage in the proof.
In the rest of the paper, we comment on each of these steps. We start with
Step A in the next section. We then show how to conclude the proof of the main
theorem when assuming Steps B, C and D. Finally, we discuss Steps B, C and D.

2.2. The Yang-Mills formalism (Step A). We cast the Einsteinvacuum equations in a Yang-Mills form which corresponds to step A in the strategy
outlined above. This relies on the Cartan formalism of moving frames. The idea
is to give up on a choice of coordinates and instead express the Einstein vacuum
equations in terms of the connection 1-forms associated to moving orthonormal
frames, i.e. vectorfields eα , which verify,g(eα , eβ ) = mαβ = diag(−1, 1, 1, 1).
19 The classical Yang-Mills equations are semilinear, i.e. they are defined on a given (Lorentzian)background. Here, we recast the Einstein vacuum equations as a Yang-Mills theory on the
background (M, g) solution to (1). As the background is not given but instead the unknown of
the problem itself, we obtain a quasilinear analog of the Yang-Mills equations.

The resolution of the bounded L2 curvature conjecture in general relativity

11

The connection 1-forms (they are to be interpreted as 1-forms with respect to theexternal index µ with values in the Lie algebra of so(3, 1)), defined by the formulas,(Aµ )αβ = g(Dµ eβ , eα )(12)Dµ Fµν + [Aµ , Fµν ] = 0(13)verify the equations,where, denoting (Fµν )αβ := Rαβµν ,(Fµν )αβ = Dµ Aν − Dν Aµ − [Aµ , Aν ] αβ .

(14)

In other words we can interpret the curvature tensor as the curvature of the so(3, 1)valued connection 1-form A. Note also that the covariant derivatives are taken onlywith respect to the external indices µ, ν and do not affect the internal indices α, β.
We can rewrite (13) in the form,g Aν − Dν (Dµ Aµ ) = Jν (A, DA)(15)

where,
Jν = Dµ ([Aµ , Aν ]) − [Aµ , Fµν ].
Observe that the equations (13)-(14) look just like the Yang-Mills equations on a
fixed Lorentzian manifold (M, g) except, of course, that in our case A and g are notindependent but rather connected by (12), reflecting the quasilinear structure of theEinstein equations. Just as in the case of [9], which establishes the well-posednessof the Yang-Mills equation in Minkowski space in the energy norm (i.e. s = 1), werely in an essential manner on a Coulomb type gauge condition. More precisely,
we take e0 to be the future unit normal to the Σt foliation and choose e1 , e2 , e3 anorthonormal basis to Σt , in such a way that we have, essentially div A = ∇i Ai = 0,
where A is the spatial component of A. It turns out that A0 satisfies an elliptic
equation while each component Ai = g(A, ei ), i = 1, 2, 3 verifies an equation of
the form,
g Ai = −∂i (∂0 A0 ) + Aj ∂j Ai + Aj ∂i Aj + l.o.t.
(16)
with l.o.t. denoting nonlinear terms which can be treated by more elementary
techniques (including non sharp Strichartz estimates).

2.3. The proof of the bounded L2 curvature Theorem. In thissection, assuming that step B holds - which corresponds to having appropriate
bilinear estimates at our disposal - we conclude the proof Theorem 1.6. Now, tobe in position to use these bilinear estimates, we first need to reduce the problem
to a wave equation. In view of (16), we thus need to eliminate ∂i (∂0 A0 ). To
this end, we need to project (16) onto divergence free vectorfields with the helpof a non-local operator which we denote by P. In the case of the flat Yang-Mills
equations, treated in [9], this leads to an equation of the form,
Ai = P(Aj ∂j Ai ) + P(Aj ∂i Aj ) + l.o.t.

12Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel

where both terms on the right exhibit the null structure20 . In our case however,
the operator P does not commute with g . It turns out, fortunately, that theterms generated by commutation can still be estimated by an extended class ofbilinear estimates which includes contractions with the curvature tensor. Thus, we
obtain in the end schematically for Ai
g Ai = null forms + l.o.t.,(17)

where up to (cubic) lower order terms, the quadratic terms exhibit the null structure.
We are now in position to conclude the proof of our main theorem. Recall that
the Ai are connection coefficients, and hence the curvature is a the level of one
derivative of the Ai (see (14)). In particular, controlling the curvature tensor in
L2 corresponds to the control of first order derivatives of Ai in L2 . In other words,
we need to run the energy estimate for the wave equation (17). In the case of
the standard wave equation on the Minkowski space-time, the energy estimate isbased on the usual timelike Killing vectorfield ∂t . In our case, the corresponding
vectorfield e0 = T ( the future unit normal to Σt ) is not Killing. This leads toanother class of trilinear error terms. That is to say, to control the energy estimates for the wave equation (17) we need trilinear estimates, while to control thenull forms in the right-hand side we need bilinear estimates. Assuming these bilinear and trilinear estimates hold, we finally control first order derivatives of Ai in
L2 and hence the curvature tensor in L2 . This concludes the proof of Theorem 1.6.
The rest of the paper is organized as follows. In section 3, we discuss thederivation of the bilinear estimates21 which corresponds to Step B. This derivation
relies on Step C and Step D which we discuss respectively in section 4 and 5.

3. Bilinear estimates (Step B)
3.1. The plane wave representation on a curved space-time.
In order to establish bilinear estimates on a curved space-time, we need to rely ona plane wave representation formula22 for solutions of scalar wave equations,g φ = 0.
To build such a plane wave representation, consider a plane waveeiλ

ωu(t,x), λ ∈ [0, +∞), ω ∈ S2 ,20 This corresponds to (10) where the null structure manifests itself in the presence of the null
form Qij in the right-hand side.
21 As we have seen, trilinear estimates have to be derived as well, but we skip this part for thesake of simplicity.
22 We follow the proof of the bilinear estimates outlined in [15] which differs substantially fromthat of [9] and is reminiscent of the null frame space strategy used by Tataru in his fundamental
paper [39].

The resolution of the bounded L2 curvature conjecture in general relativity

13with λ and ω parameters corresponding to Fourier variables in R3 in sphericalcoordinates. We computeg (eiλωu) = (−λ2 gαβ ∂α ( ω u)∂β ( ω u) + iλg ( ω u))eiλ

ω

u

.

The first term turns out to be the most dangerous one23 . This motivates to chooseωu solution to the Eikonal equationgαβ ∂α ( ω u)∂β ( ω u) = 0,in which case we obtaing (eiλωu) = iλg ( ω u)eiλ

ω

u

.

This yields in general an approximate solution to g (φ) = 0. We then superpose
these plane waves to obtain a full plane wave representation.
In the particular case of the standard wave equation on the Minkowski spacetime, we recover the well-known plane wave representation which is an exact solution24 . We haveX Z Z +∞
ωφ=eiλ u± (t,x) f± (λω)λ2 dλdω,
(18)
±

S2

0

where ω u± (t, x) = ±t + x · ω and f± can be explicitly computed in terms of theFourier transform of the initial data set (φ(0, .), ∂t φ(0, .)) of φ. In the general case,we superpose the basic plane waves as in the right-hand side of (18), and chooseωu± solution of the Eikonal equation with the following asymptotic behavior onΣ0ω
u± (0, x) ∼ x · ω when |x| → +∞.
This asymptotic behavior is necessary to be able to generate any initial data of
the wave equation.
In view of the previous paragraph, we consider the following representation
formula25
Z Z ∞
eiλ

φf (t, x) =S2ωu(t,x)f (λω)λ2 dλdω

(19)

0

where f represents schematically the initial data26 , and where ω u is a solution of
the eikonal equation27 ,gαβ ∂α ( ω u) ∂β ( ω u) = 0,(20)
23 λ should be understood as a Fourier variable corresponding to a derivative in physical space.
The λ2 term hence costs 2 derivatives while the wave equation only recovers one. Thus, this termis problematic as it induces a derivative loss.
24 This is special to the flat case. In the general case, we only obtain an approximate solution.
25 (19) actually corresponds to the representation formula for a half-wave. The full representation formula corresponds to the sum of two half-waves as in (18). Since the bilinear estimates
are identical for both half waves, we only consider one of them for simplicity.
26 Here f is in fact at the level of the Fourier transform of the initial data and the norm
kλf kL2 (R3 ) corresponds, roughly, to the H 1 norm of the data .
27 As we have seen above, we have ω u(t, x) = ±t + x · ω in the flat Minkowski space.

14

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftelwith the following asymptotic behavior on Σ0ω

u(0, x) ∼ x · ω when |x| → +∞.

Remark 3.1. (20) is a nonlinear transport equation. Hence, ω u needs to beprescribed not only at infinity on Σ0 as explained above, but everywhere on Σ0 .
This choice of ω u on Σ0 turns out to be delicate and is discussed in section 4 (seeStep C1 and related subsequent comments).
Remark 3.2. Note that (19) is a parametrix for a scalar wave equation. The lackof a good parametrix for a tensorial wave equation forces us to develop a strategy
based on writing the main equation in components relative to a frame, i.e. insteadof dealing with the tensorial wave equation (15) directly, we consider the system
of scalar wave equations (16). Unlike the flat case, this “scalarization” procedure
produces several terms which are potentially dangerous, and it is fortunate, as inyet another manifestation of a hidden null structure of the Einstein equations, thatthey can still be controlled by the use of an extended28 class of bilinear estimates.

3.2. Bilinear estimates on a curved space-time. The bilinear estimates all involve after some reductions the null form Qij introduced in (11). Let
us briefly explain how the structure of Qij is exploited to derive these estimates.
For simplicity, we focus on two specific bilinear estimates29 .
The first example of a bilinear estimate on a curved space-time aims at controlling the L2 (M) norm of the null form Qij (φf , ψ), where φf is given by (19).
We compute
Z Z +∞ω
Qij (φf , ψ) =Qijeiλ u(t,x) f (λω)λ2 dλdω, ψS20Z Z +∞
=
S2

i
S2

ω

u(t,x), ψ)f (λω)λ2 dλdω0

Z Z +∞=Qij (eiλeiλω

u(t,x)Qij ( ω u, ψ)f (λω)λ3 dλdω.

0

Now we have
Qij ( ω u, ψ) = ∂i ( ω u)∂j ψ − ∂j ( ω u)∂i ψ.

(21)

The fundamental observation which ultimately allows us to derive a bilinear estimate in this case is the fact that the structure of Qij is such that
Qij ( ω u, ψ) is tangent to the level hypersurfaces of ω u,(22)

28 involving contractions between the Riemann curvature tensor and derivatives of solutions ofscalar wave equations.
29 The two examples of bilinear estimates discussed here have both an analog in the semilinear
case. Indeed, they correspond to estimating the L2t,x norm of each of the term in the right-handside of (10).

The resolution of the bounded L2 curvature conjecture in general relativity

15

as can be seen from (21).
The second example of a bilinear estimate on a curved space-time aims at
controlling the following expression
k∇−1 (Qij (φf1 , φf2 ))kL2 (M) .
First, we decompose φf1 and φf2 in dyadic frequencies according to
Z Z +∞Xωφf =φf,p , φf,p =eiλ u(t,x) ψ(2−p λ)f (λω)λ2 dλdωS2p0where λ ∼ 2 on the support of ψ(2−pλ). We inferXk∇−1 (Qij (φf1 , φf2 ))kL2 (M) .
k∇−1 (Qij (φf1 ,p , φf2 ,q ))kL2 (M) .
p≥q

It is at this stage that we use the null structure of Qij by noticing that
Qij (φ, ψ) = ∂i (φ∂j ψ) − ∂j (φ∂i ψ) = ∂j (ψ∂i φ) − ∂i (ψ∂j φ)(23)

so that we may choose which derivative we factorize. We choose to factorize thederivative corresponding to the highest frequency which yields
Xk∇−1 (Qij (φf1 , φf2 ))kL2 (M) .
k∇−1 ∂(φf1 ,p ∂φf2 ,q )kL2 (M)p≥qXkφf1 ,p kL4 (M) k∂φf2 ,q kL4 (M) .

.

p≥q

The last ingredient is the sharp L4 (M) Strichartz of Step D (see section 5) which
finally yields
X |p−q|k∇−1 (Qij (φf1 , φf2 ))kL2 (M) .
2− 2 kλf1,p kL2 (R3 ) kλf2,q kL2 (R3 )p≥qkλf1 kL2 (R3 ) kλf2 kL2 (R3 ).

and concludes the proof of the second example of bilinear estimate.
Remark 3.3. The null structure of Qij is exploited differently in the two examplesof bilinear estimates presented above as can be seen by comparing (22) and (23).

4. Control of the parametrix (step C)
To prove the bilinear and trilinear estimates of Step B, we need in particular tocontrol the parametrix given by (19). To this end, it turns out that it suffices tocontrol the parametrix at initial time (i.e. restricted to the initial slice Σ0 )Z Z ∞
ωφf (0, x) =eiλ u(0,x) f (λω)λ2 dλdω
(24)
S2

0

16

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftelas well as the error term30 corresponding to (19)Z Z ∞ωEf (t, x) = g φf (t, x) = i
eiλ u(t,x) g ( ω u)f (λω)λ3 dλdω.
S2

(25)

0

This requires the following four sub steps
C1 Make an appropriate choice for the equation satisfied by ω u(0, x) on Σ0 , andcontrol the geometry of the foliation of Σ0 by the level surfaces of ω u(0, x).
C2 Prove that the parametrix at t = 0 given by (24)31 is bounded in L(L2 (R3 ), L2 (Σ0 ))using the estimates for ω u(0, x) obtained in C1.
C3 Control the geometry of the foliation of M given by the level hypersurfacesof ω u.
C4 Prove that the error term (25) satisfies the estimate kEf kL2 (M) . kλf kL2 (R3 )using the estimates for ω u and g ( ω u) proved in C3.
To achieve Step C3 and Step C4, we need, at the very least, to control g ( ω u)
in L∞ . This issue was first addressed in the sequence of papers [16] [17] [18] wherean L∞ bound for g ( ω u) was established, depending only on the L2 norm of thecurvature flux along null hypersurfaces. The proof required an interplay between
both geometric and analytic techniques and had all the appearances of being sharp,
i.e. we don’t expect an L∞ bound for g ( ω u) which requires bounds on less than
two derivatives in L2 for the metric32 .
Remark 4.1. It turns out, as a byproduct of the proof of Step C3, that the radiusof injectivity of the level hypersurfaces of ω u is controlled by the L∞ norm ofg ( ω u). Furthermore, this control appears to be sharp. In other words, we expect
to loose control over the radius of injectivity in the absence of this bound. Hence,in view of the discussion above, L2 bounds on the curvature tensor appear to beminimal for the control of the Eikonal equation.
To obtain the L2 bound for the Fourier integral operator E defined in (25),
we need, of course, to go beyond uniform estimates for g ( ω u). The classical L2
bounds for Fourier integral operators of the form (25) are not at all economicalin terms of the number of integration by parts which are needed. In our case thetotal number of such integration by parts is limited by the regularity properties ofthe function g ( ω u). To get an L2 bound for the parametrix at initial time (24)30 Note that φ is an exact solution of  φ = 0 only if  ( ω u) = 0. Hence, φ is an exactggff
solution only in flat space.
31 (24) only corresponds to the value at t = 0 of a half wave parametrix. The full parametrix at
initial data is the sum of two half waves as in (18). Step C2 actually corresponds to proving thatthe parametrix at t = 0 generates any initial data to the wave equation g φ = 0 with a suitable
control of the corresponding f± . We have chosen to provide a more restricted statement of Step
C2 to simplify the exposition.
32 classically, this requires, at the very least, the control of R in L∞ .

The resolution of the bounded L2 curvature conjecture in general relativity

17

and the error term (25) within such restrictive regularity properties we need, inparticular:• In Step C1 and Step C3, a precise control of derivatives of ω u and g ( ω u)with respect to both ω as well as with respect to various directional derivatives33 . To get optimal control we need, in particular, a very careful construction of the initial condition for ω u on Σ0 and then sharp space-time estimatesof Ricci coefficients, and their derivatives, associated to the foliation inducedby ω u.
• In Step C2 and Step C4, a careful decompositions of the Fourier integral
operators (24) and (25) in both λ and ω, similar to the first and second dyadic
decomposition in harmonic analysis, see [30], as well as a third decomposition,
which in the case of (25) is done with respect to the space-time variables
relying on the geometric Littlewood-Paley theory developed in [18].
Below, we make further comments on Steps C1-C4:
1. The choice of ω u(0, x) on Σ0 in Step C1. Let us note that the typical choiceωu(0, x) = x · ω in a given coordinate system would not work for us, since
we don’t have enough control on the regularity of a given coordinate system
within our framework. Instead, we need to find a geometric definition ofωu(0, x). A natural choice would be that u = ω u verifies
g u = 0 on Σ0
which by a simple computation turns out to be the following simple variantof the minimal surface equation34




∇u ∇u
∇u
,
=k
on Σ0 .
div|∇u||∇u| |∇u|
Unfortunately, this choice does not allow us to have enough control of thederivatives of u in the normal direction to the level surfaces of u. This forces
us to look for an alternate equation for u:




∇u
1
∇u ∇u
div
=1−
+k
,
on Σ0 .
|∇u||∇u||∇u| |∇u|This equation turns out to be parabolic in the normal direction to the levelsurfaces of u, and allows us to obtain the desired regularity in Step C1. A
closer inspection reveals its relation to the mean curvature flow on Σ0 .
2. How to achieve Step C3. The regularity obtained in Step C1, together withnull transport equations tied to the eikonal equation, elliptic systems of
33 Taking into account the different behavior in tangential and transversal directions with respect to the level surfaces of ω u.
34 In the time symmetric case k = 0, this is exactly the minimal surface equation.

18

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel

Hodge type, the geometric Littlewood-Paley theory of [18], sharp trace theorems, and an extensive use of the structure of the Einstein equations, allows
us to propagate the regularity on Σ0 to the space-time, thus achieving Step
C3.
3. The regularity with respect to ω in Steps C1 and C3. The regularity withrespect to x for u is clearly limited as a consequence of the fact that weonly assume L2 bounds on R. On the other hand, R is independent of
the parameter ω, and one might infer that u is smooth with respect to ω.
Surprisingly, this is not at all the case. Indeed, the regularity in x obtained foru in Steps C1 and C3 is better in directions tangent to the level hypersurfacesof u. Now, the ω derivatives of the tangential directions have non zero normalcomponents. Thus, when differentiating the structure equations with respectto ω, tangential derivatives to the level surfaces of u are transformed to
non tangential derivatives which in turn severely limits the regularity in ω
obtained in Steps C1 and C3.
4. How to achieve Steps C2 and C4. The classical arguments for proving L2
bounds for Fourier operators are based either on a T T ∗ argument, or a T ∗ T
argument, which requires several integration by parts either with respect tox for T ∗ T , of with respect to (λ, ω) for T T ∗ . Both methods would fail by far
within the regularity for u obtained in Step C1 and Step C3. This forces us
to design a method which allows to take advantage both of the regularity inx and ω. This is achieved using in particular the following ingredients• Geometric integrations by parts taking full advantage of the better regularity properties in directions tangent to the level hypersurfaces of u.
• The standard first and second dyadic decomposition in frequency space,with respect to both size and angle (see [30]), an additional decomposition in physical space relying on the geometric Littlewood-Paley projections of [18] for Step C4, as well as another decomposition involving
frequency and angle for Step C2.
Even with these precautions, at several places in the proof, one encounterslog-divergences which have to be tackled by ad-hoc techniques, taking fulladvantage of the null structure of the Einstein equations.

5. Sharp L4 (M) Strichartz estimates (Step D)
Recall that the parametrix constructed in Step C is also used to prove sharp L4 (M)Strichartz estimates. Indeed the proof of several bilinear estimates of Step B reduces to the proof of sharp L4 (M) Strichartz estimates for the parametrix (19)with λ localized in a dyadic shell (see section 3.2).

The resolution of the bounded L2 curvature conjecture in general relativity

19

More precisely, let j ≥ 0, and let ψ a smooth function on R3 supported in1≤ |ξ| ≤ 2.
2
Let φf,j the parametrix (19) with a additional frequency localization λ ∼ 2j
Z Z ∞
ωφf,j (t, x) =eiλ u(t,x) ψ(2−j λ)f (λω)λ2 dλdω.
(26)
S2
35

We will need the sharp

04

L (M) Strichartz estimatejkφf,j kL4 (M) . 2 2 kψ(2−j λ)f kL2 (R3 ) .

(27)

The standard procedure for proving36 (27) is based on a T T ∗ argument which
reduces it to an L∞ estimate for an oscillatory integral with a phase involving ω u.
This is then achieved by the method of stationary phase which requires quite a few
integrations by parts. In fact the standard argument would require, at the very
least37 , that the phase function u = ω u verifies,∂t,x u ∈ L∞ , ∂t,x ∂ω2 u ∈ L∞ .

(28)

This level of regularity is, unfortunately, incompatible with the regularity properties of solutions to our eikonal equation (20). In fact, based on the estimates forωu derived in step C3, we are only allowed to assume∂t,x u ∈ L∞ , ∂t,x ∂ω u ∈ L∞ .

(29)

We are thus forced to follow an alternative approach38 to the stationary phasemethod inspired by [27] and [28] .

References[1] H. Bahouri, J.-Y. Chemin, Équations d’ondes quasilinéaires et estimation de
Strichartz . Amer. J. Math., 121, 1337–1777, 1999.
[2] H. Bahouri, J.-Y. Chemin, Équations d’ondes quasilinéaires et effet dispersif . IMRN,
21, 1141–1178, 1999.
[3] Y. C. Bruhat, Theoreme d’Existence pour certains systemes d’equations aux derivees
partielles nonlineaires., Acta Math. 88, 141–225, 1952.
35 Note in particular that the corresponding estimate in the flat case is sharp.
36 Note that the procedure we describe would prove not only (27) but the full range of mixed
Strichartz estimates.
37 The regularity (28) is necessary to make sense of the change of variables involved in thestationary phase method.
38 We refer to the approach based on the overlap estimates for wave packets derived in [27] and[28] in the context of Strichartz estimates respectively for C 1,1 and H 2+ metrics. Note howeverthat our approach does not require a wave packet decomposition.

20

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel[4] E. Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces
torsion. C. R. Acad. Sci. (Paris), 174, 593–595, 1922.
[5] D. Christodoulou, Bounded variation solutions of the spherically symmetric einsteinscalar field equations, Comm. Pure and Appl. Math, 46, 1131–1220, 1993.
[6] D. Christodoulou, The instability of naked singularities in the gravitational collapseof a scalar field, Ann. of Math., 149, 183–217,1999.
[7] A. Fischer, J. Marsden, The Einstein evolution equations as a first-order quasi-linear
symmetric hyperbolic system. I, Comm. Math. Phys. 28, 1–38, 1972.
[8] T. J. R. Hughes, T. Kato, J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity,
Arch. Rational Mech. Anal. 63, 273–394, 1977.
[9] S. Klainerman, M. Machedon, Space-time estimates for null forms and the local
existence theorem, Communications on Pure and Applied Mathematics, 46, 1221–
1268, 1993.
[10] S. Klainerman, M. Machedon, Finite energy solutions of the Maxwell-Klein-Gordon
equations, Duke Math. J. 74, 19–44, 1994.
[11] S. Klainerman, M. Machedon, Finite Energy Solutions for the Yang-Mills Equations
in R1+3 , Annals of Math. 142, 39–119, 1995.
[12] S. Klainerman, PDE as a unified subject, Proceeding of Visions in Mathematics,
GAFA 2000(Tel Aviv 1999). Geom Funct. Anal. 2000, Special Volume , Part 1,
279–315.
[13] S. Klainerman, I. Rodnianski, Improved local well-posedness for quasi-linear wave
equations in dimension three, Duke Math. J. 117 (1), 1–124, 2003.
[14] S. Klainerman, I. Rodnianski, Rough solutions to the Einstein vacuum equations,Annals of Math. 161, 1143–1193, 2005.
[15] S. Klainerman, I. Rodnianski, Bilinear estimates on curved space-times, J. Hyperbolic Differ. Equ. 2 (2), 279–291, 2005.
[16] S. Klainerman, I. Rodnianski, Casual geometry of Einstein vacuum space-times withfinite curvature flux, Inventiones 159, 437–529, 2005.
[17] S. Klainerman, I. Rodnianski, Sharp trace theorems on null hypersurfaces, GAFA 16(1), 164–229, 2006.
[18] S. Klainerman, I. Rodnianski, A geometric version of Littlewood-Paley theory, GAFA
16 (1), 126–163, 2006.
[19] S. Klainerman, I. Rodnianski, On a break-down criterion in General Relativity, J.
Amer. Math. Soc. 23, 345–382, 2010.
[20] S. Klainerman, I. Rodnianski, J. Szeftel, The Bounded L2 Curvature Conjecture,
arXiv:1204.1767, 91 p, 2012.
[21] J. Krieger, W. Schlag, Concentration compactness for critical wave maps, Monographs of the European Mathematical Society, 2012.
[22] H. Lindblad, Counterexamples to local existence for quasilinear wave equations,Amer. J. Math. 118 (1), 1–16, 1996.
[23] H. Lindblad, I. Rodnianski, The weak null condition for the Einstein vacuum equations, C. R. Acad. Sci. 336, 901–906, 2003.

The resolution of the bounded L2 curvature conjecture in general relativity

21[24] D. Parlongue, An integral breakdown criterion for Einstein vacuum equations in thecase of asymptotically flat spacetimes, arXiv:1004.4309, 88 p, 2010.
[25] F. Planchon, I. Rodnianski, Uniqueness in general relativity, preprint.
[26] G. Ponce, T. Sideris, Local regularity of nonlinear wave equations in three spacedimensions, Comm. PDE 17, 169–177, 1993.
[27] H. F. Smith, A parametrix construction for wave equations with C 1,1 coefficients,
Ann. Inst. Fourier (Grenoble) 48, 797–835, 1998.
[28] H.F. Smith, D. Tataru, Sharp local well-posedness results for the nonlinear wave
equation, Ann. of Math. 162, 291–366, 2005.
[29] S. Sobolev, Methodes nouvelle a resoudre le probleme de Cauchy pour les equations
lineaires hyperboliques normales, Matematicheskii Sbornik, 1 (43), 31–79, 1936.
[30] E. Stein, Harmonic Analysis, Princeton University Press, 1993.
[31] J. Sterbenz, D. Tataru, Regularity of Wave-Maps in dimension 2 + 1, Comm. Math.
Phys. 298 (1), 231–264, 2010.
[32] J. Sterbenz, D. Tataru, Energy dispersed large data wave maps in 2 + 1 dimensions,Comm. Math. Phys. 298 (1), 139–230, 2010.
[33] J. Szeftel, Parametrix for wave equations on a rough background I: Regularity of thephase at initial time. arXiv:1204.1768, 145 p, 2012.
[34] J. Szeftel, Parametrix for wave equations on a rough background II: Construction of
the parametrix and control at initial time. arXiv:1204.1769, 84 p, 2012.
[35] J. Szeftel, Parametrix for wave equations on a rough background III: Space-time
regularity of the phase. arXiv:1204.1770, 276 p, 2012.
[36] J. Szeftel, Parametrix for wave equations on a rough background IV: Control of theerror term. arXiv:1204.1771, 284 p, 2012.
[37] J. Szeftel, Sharp Strichartz estimates for the wave equation on a rough background.
arXiv:1301.0112, 30 p, 2013.
[38] T. Tao, Global regularity of wave maps I–VII, preprints.
[39] D. Tataru, Local and global results for Wave Maps I, Comm. PDE 23, 1781–1793,
1998.
[40] D. Tataru. Strichartz estimates for operators with non smooth coefficients and thenonlinear wave equation. Amer. J. Math. 122, 349–376, 2000.
[41] D. Tataru, Strichartz estimates for second order hyperbolic operators with non smoothcoefficients, J.A.M.S. 15 (2), 419–442, 2002.
[42] Q. Wang, Improved breakdown criterion for Einstein vacuum equation in CMC gauge,Comm. Pure Appl. Math. 65 (1), 21–76, 2012.
Department of Mathematics, Princeton University, Princeton NJ 08544
E-mail: seri@math.princeton.edu
Department of Mathematics, Princeton University, Princeton NJ 08544 & Departmentof Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
E-mail: irod@math.pinceton.edu & irod@math.mit.edu
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75005 Paris
E-mail: jeremie.szeftel@upmc.fr