4.5. Preliminary estimates for the bootstrap argument. 4.6. Energy estimates for the wave equation for k. 4.7. Transport estimates. 4.8. Conclusion of the proof of Theorem 4.6. 4.9. Extracting a limit: proof of Proposition 4.8.
5. Vanishing of the Einstein tensor. 6. Uniqueness and smoothness of solutions: proofs of Theorems 1.7 and 1.10. 6.1. Uniqueness of solutions. 6.2. Regularity of solutions. References.
1. Introduction. The Kasner spacetime ((0, +∞) × T 3 , (4) g), where
(1.1) (4)
(with p i being constants such that 3 i=1 p i = 3 i=1 p 2 i = 1) is an explicit solution to the Einstein vacuum equations (1.2) Ric( (4) g) = 0.
As long as all p i ̸ = 0, the Kasner solution moreover represents a singularity as t → 0 + . This is manifested in particular by the blowup of the Kretschmann scalar R µναβ R µναβ . In an influential paper [33], Lifshitz-Khalatnikov considered the class of spacetimes solutions to (1.2) with the form
where ω i are spatial 1-forms with a "finite limit" as t → 0 + and p i = p i (x) are now spatially-dependent functions satisfying 3 i=1 p i (x) = 3 i=1 p 2 i (x) = 1. The spacetime metrics (1.3) are Kasner-like asymptotically as t → 0 + except that the Kasner exponents are now functions. They are also sometimes called asymptotically velocity term dominated (AVTD), a terminology that is used to mean that the asymptotics near the singularity is described by a simpler system of velocity term dominated equations [16,20]. Importantly, it is argued in [33] that this class of spacetime solutions to (1.2) depend only on three "functional degrees of freedom", which is one fewer than that for the Cauchy problem of (1.2), and they are therefore expected to be highly non-generic.
In this paper, we construct a large class of solutions to (1.2) with the asymptotically Kasner-like behavior of (1.3). Our construction in fact has full three functional degrees of freedom and includes all the spacetimes considered in the heuristics in [33] (see Remark 1.5). Some previous constructions are known with either analyticity or symmetry assumptions (see Section 1.2.1); our construction is the first without such assumptions.
More precisely, our goal will be to construct a metric taking the form (4) g := -dt 2 + g
a ij t 2p max{i,j} dx i dx j , (1.4) where (t, x 1 , x 2 , x 3 ) ∈ (0, T ] × T 3 for some T > 0, p i : T 3 → R are smooth, timeindependent functions, and a ij : (0, T ] × T 3 → R are smooth functions (symmetric in i and j) which extend to continuous functions : [0, T ] × T 3 → R. Moreover, a ij obey
where c ij are some prescribed smooth functions (symmetric in i and j).
Notice that in the language of (1.3), the ansatz (1.4) imposes the condition
11 dx 1 . As we will explain in Remark 1.5, this condition is what restricts the functional degrees of freedom in our construction.
We will prove existence, uniqueness and regularity of solutions of the form (1.4). The following is our main existence theorem: THEOREM 1.1 (Existence of solution). Suppose the following assumptions hold:
(1) The (time-independent) functions c ij , p i : T 3 → R are smooth for i, j = 1, 2, 3, and that c ij = c ji .
(
) It holds that p 1 (x) < p 2 (x) < p 3 (x) < 1 pointwise. (4) It holds that c 11 (x), c 22 (x), c 33 (x) > 0.
(5) The following three asymptotic differential constraint equations are satisfied:
Then there is a C 2 solution to the Einstein vacuum equations (1.2) of the form (1.4), for a T > 0 depending on c ij , p i , which satisfies (1.5).
Remark 1.2 ((1.4) is a Lorentzian metric). Notice that under condition (3), the eigenvalues of g as in (1.4) are approximately t 2p i c ii (i = 1, 2, 3) for small t. Hence, given p i as in the theorem and the condition (1.5), it follows that (1.4) is a well-defined Lorentzian metric in all of (0, T 0 ] × T 3 , for some T 0 > 0.
Remark 1.3 (Localizing the assumptions). For technical convenience, we assume that there is a global system of coordinates on T 3 so that the assumptions of Theorem 1.1 hold. One may in principle hope to use a localization argument to construct more general spacetimes for which we require only that around every point in T 3 , there is a coordinate patch (x 1 , x 2 , x 3 ) such that the assumptions of Theorem 1.1 hold. This, however, is not carried out in the present paper.
Remark 1.4 (Asymptotic CMC condition and asymptotic constraints). The conditions (2) and ( 5) in Theorem 1.1 guarantee that a metric of the form (1.4) satisfies asymptotically, along the level sets of t, (1) the constraint equations and (2) the CMC gauge to leading order, as t → 0 + . More precisely, condition (2) is equivalent to (1.7) lim t→0 + t(tr k) = -1, lim t→0 + t 2 [R(g) -|k| 2 + (tr k) 2 ] = 0, while condition ( 5) is equivalent to (1.8) lim t→0 + t(∇ j k i j -∇ i tr k) = 0, i = 1, 2, 3; see Lemma 3.2. Note that condition (2) is algebraic in the Kasner exponents p i 's, while condition (5) is differential in the c ij 's.
Remark 1.5 (Functional degrees of freedom and considerations in [33]). Note that c ij and p i consist of 9 functions. On the other hand, the assumptions (2) and (5) in Theorem 1.1 impose a total of 5 conditions, leaving 4 functional degrees of function.
There is in fact an additional residual gauge freedom, namely, we can introduce a change of coordinates (1.9)
x 1 = x 1 , x 2 = x 2 , x 3 = f (x 1 , x 2 , x 3 ), for some smooth f such that ∂f ∂x 3 ̸ = 0, then the resulting metric will have the same form as (1.4) (in the sense that the new g 11 term is O(t 2p 1 ), the new g 12 , g 22 terms are O(t 2p 2 ), and the new g 13 , g 23 , g 33 terms are O(t 2p 3 ).)
Thus, there are a total of 3 functional degrees of freedom, which is one fewer than that for the initial value problem for the Einstein vacuum equations. It is for this reason that [33] argued that metrics of the form (1.4) are non-generic.
Notice that while we only construct a non-generic class of spacetimes, we do construct a class that includes all the metrics considered in [33] (modulo the endpoint case; see Remark 1.6). Indeed, using the change of coordinates in (1.9), one can locally change coordinates to the form g = a 11 t 2p 1 (dx 1 ) 2 + a 22 t 2p 2 (dx 2 ) 2 + a 33 t 2p 3 (dx 3 ) 2 + 2a 12 t p 2 dx 1 dx 2 + 2a 13 t p 3 dx 1 dx 3 , which is exactly the local form of the metrics considered in the work of Lifshitz-Khalatnikov; see [33, equation (3.25)].
Remark 1.6 (Some limiting cases). Our analysis degenerates in any of the limits p 3 → 1 or p i+1 -p i → 0 (see (3) in Theorem 1.1). A particularly interesting limiting case that we do not cover is when
but still assuming p 3 (x) < 1, ∀x. While we do not cover this case, it is possible that [29] is relevant. Notice that to handle possible terms with p 2 (x) = p 3 (x), we need a new argument in constructing the approximate solution in Section 2, but the analysis in the subsequent sections could in principle be carried out along the same lines.
Finally, we note that allowing p 1 (x) = -1 3 , p 2 (x) = p 3 (x) = 2 3 would also be relevant to constructing Schwarzschild-like singularities since locally the Schwarzschild singularity could be modeled by the Kasner singularity with p 1 = - 1 3 , p 2 = p 3 = 2 3 (cf. [21] and discussions in Section 1.2.1).
We now turn to uniqueness. It is hard to talk about geometric uniqueness in the above singular initial value problem, since the setup itself includes the expression (1.4) of the spacetime metric. However, we can obtain uniqueness in our gauge, i.e., within the class of metrics satisfying (1.4). More precisely, we prove that given two solutions of the form (1.4) which (1) obey the estimates (1.10) and (1.11) which is proven in Theorem 1.1 and (2) converge to each other sufficiently fast as t → 0 + , then they must in fact be the same. THEOREM 1.7 (Uniqueness of solutions). Given the assumptions of Theorem 1.1, there exists M u ∈ N sufficiently large (depending on the given data p i and c ij ) such that the following holds.
Let (4) g, (4) g be two C 3 solutions to the Einstein vacuum equations (1.2) of the form (1.4) in (0, T ] × T 3 for some T > 0, such that
• the corresponding a ij and ãij converge to c ij with the following rate
• the corresponding k i j = -1 2 (g -1 ) jℓ ∂ t g jℓ and ki j = -1 2 ( g-1 ) jℓ ∂ t gjℓ obey the following estimates
= O(min{t -1+ε , t -1+ε-2p j +2p i });
(1.11) and
• the gg and ∂ t (gg) converge to 0 sufficiently fast in the following sense:
(1.12)
Here, ε = min{min x (p 3 -p 2 )(x), min x (1-p 3 )(x)} > 0, and κ i j as in Theorem 1.1. Then (4) g = (4) g on (0, T ] × T 3 . Remark 1.8 (Asymptotics determined by approximate solutions). In the proof of our existence result (Theorem 1.1), we construct a sequence of smooth approximate solutions {g [n] } +∞ n=0 , for which we get more precise asymptotic information, as t → 0 + , as n increases; see already Sections 1.1, 2 and 3. The actual solutions that we construct in Theorem 1.1 then have asymptotics determined by an approximate solution g [n] (for some large n). From this point of view, one way to interpret our uniqueness result (Theorem 1.7) is to say that for n sufficiently large, there is in fact only one solution whose asymptotics are governed by g [n] . Remark 1.9 (Regularity implies asymptotic expansion). Given any M u ∈ N, there exists A ∈ N sufficiently large such that if (1.10) and (1.11) are replaced by the stronger regularity assumptions (1.13) |α|≤A
= O(min{t -1+ε , t -1+ε-2p j +2p i }), (1.14) then in fact the convergence condition (1.12) follows as a consequence. In fact, in this case both g and g have the leading asymptotics given by an approximate solution g [n] for large n (see Remark 1.8), which then implies (1.12). This can be proven by revisiting the argument for constructing the approximate solutions in Theorem 2.1. We omit the details.
Finally, we state our main regularity theorem. We remark that initially our proof of the existence theorem (Theorem 1.1) only constructs a solution with finite regularity. In order to obtain smoothness, we need an additional argument which relies on the uniqueness result (Theorem 1.7); see Section 1.1.5. THEOREM 1.10 (Smoothness of solutions). Given the assumptions of Theorem 1.1, there is a smooth solution to the Einstein vacuum equations (1.2) of the form (1.4) in (0, T ] × T 3 , for a T > 0 depending on c ij , p i , which satisfies (1.5).
In the remainder of the introduction, we will briefly discuss the ideas of the proof (Section 1.1) and some related works (Section 1.2).
1.1. Ideas of the proof.
1.1.1. Fuchsian analysis of a model wave equation. As far as the singularity is concerned, our basic strategy (which is quite standard, see for instance [29]) can be most easily explained by a model semilinear equation.
Consider the following nonlinear wave equation
on a Kasner spacetime (1.1) with constants p 1 < p 2 < p 3 < 1 satisfying 3 i=1 p i = 3 i=1 p 2 i = 1. (Note that the structure of the nonlinear terms plays no role here, and the nonlinearity (∂ t ϕ) 2 is chosen here for its simplicity.)
The analogue of our main result in this setup would be to construct bounded solutions to the nonlinear model equation (1.15). However, the results of [2] imply that even for the linear wave equation, generic data on say, {t = 1}, give rise to solutions that blow up as O(log 1 t ) as t → 0 + . Thus, in order to obtain bounded solutions to (1.15), the solution that we build has to be special. This is achieved by imposing the leading order behavior of ϕ(t, x) = ϕ 0 (x) + error, where ϕ 0 (x) is a prescribed smooth function which is the limit of ϕ(t, x) as t → 0 + . In fact, we build our solution as ϕ(t, x) = n j=0 ϕ j (t, x) + ϕ (d) , where ϕ j are increasingly precise approximations of ϕ, and ϕ (d) is determined by the condition lim t→0 ϕ (d) = 0.
Our strategy contains two steps:
(1) (Approximate solution) It is easy to first build an approximate solution by stipulating an ansatz ϕ [n] (t, x) = n j=0 ϕ j (t, x), where • ϕ 0 (t, x) = ϕ 0 (x) is the prescribed leading order behavior, • ϕ j obeys the better estimates |∂ α x ϕ j (t, x)| ≲ α,j t jε , and
x {□ g ϕ [n] (t, x) -(∂ t ϕ [n] ) 2 (t, x)}| ≲ α,n t -2+(n+1)ε . This expansion can simply be obtained inductively by solving (1.15) iteratively as an ODE in t. Here, we have the flexibility to carry out the expansion to an arbitrary order n so as to achieve an arbitrarily good (in terms of the t-rates as t → 0 + ) approximation to a solution to (1.15).
Without analyticity, however, one cannot hope to show that this series converges. Instead we perform energy estimates for the error.
(2) (Energy estimates) First notice that for an energy defined by
it is easy to obtain an estimate of the form (e.g. with C 0 = 2)
The issue is with the borderline singular term C 0 t E(t), which cannot be treated by Grönwall's inequality (since lim sup t→0 + t -C 0 E(t) = +∞). Nevertheless, this is where the approximation constructed in the previous step becomes useful: instead of controlling the full solution ϕ, we bound the difference quantity ϕ (d) := ϕ -ϕ [n] , which for n sufficiently large
• can be made to approach 0 with a fast polynomial rate as t → 0 + , and • satisfies an inhomogeneous nonlinear wave equation where the inhomogeneity also → 0 with a fast polynomial rate.
Define now an energy E (d) with ϕ replaced by ϕ (d) . For any large N ∈ N, we can find n ∈ N large enough (corresponding to a good enough approximation) such that under appropriate bootstrap assumptions,
where C n may depend on n, but importantly, the constant C 0 in the borderline term is independent of n. The inhomogeneous C n t N term arises from the fact that ϕ (d) satisfies an inhomogeneous equation, and N can be arbitrarily chosen as long as n is also taken to be large. Thus, we obtain an estimate
Recall now moreover that for n sufficiently large we have lim t→0 + (t -N E (d) (t)) = 0. Moreover, first choosing N large (by taking n large) and then taking t small (depending on n), it follows that N t (t -N E (d) ) on the LHS dominates
) on the RHS. This gives an estimate for t -N E (d) (t). Once such energy estimates can be proven for the error ϕ (d) , we can in fact deduce existence of solutions as follows. Choosing a sequence t I → 0 + , we solve for a sequence of solutions {ϕ I } ∞ i=0 to (1.15) with (ϕ I , ∂ t ϕ I ) ↾ t=t I = (ϕ [n] , ∂ t ϕ [n] ) ↾ t=t I . The energy estimates above allows us to show that {ϕ I } ∞ i=0 can be solved in [t I , T ] × T 3 for uniform T > 0 and that there is a limit which solves (1.15) in (0, T ] × T 3 .
1.1.2. Construction of solutions to the Einstein vacuum equations in synchronized coordinates. While the Fuchsian analysis is quite robust, we must also address the quasilinear, tensorial nature, as well as the gauge invariance, of the Einstein equations.
If one were to prescribe a wave-coordinate-type gauge, then the construction of the approximate solution will be algebraically very complex. Instead, we consider a system of synchronized coordinates, i.e., we impose that the metric takes the form
This gauge captures important anisotropic features of Kasner-like singularities. In particular, assuming that the a ij 's are C 2 up to {t = 0}, we know that
min{i,j} ; and importantly (see Lemma 2.6) that (1.17)
In such a gauge, the construction of an approximate solution becomes more tractable. The difficulty, however, is shifted to the estimates for the error terms. Indeed, even when no singularities are present, it is a priori not clear that the Einstein vacuum equations are hyperbolic in the gauge (1.16); see discussions in Section 1.1.4.
1.1.3. Constructing approximate solutions. Following ideas laid out in Section 1.1.1, we first construct approximate solutions and then use energy estimates to obtain actual solutions to the Einstein vacuum equation. In order to construct approximate solutions, the first step is to solve a system of first order evolutionary equations. The evolutionary equations will be treated as a system of ODEs in t (compare Step 1 in Section 1.1.1). In order to close the ODE analysis, we crucially rely on the bounds (1.17), which show that the spatial Ricci curvature is slightly better than critical, but we also need to additionally make use of the structure of the full system. We outline some main points here:
• The main difficulty in solving the system of ODEs is that there are many borderline terms, i.e., linear terms with O(t -1 ) coefficients. It turns out that these terms have a reductive structure. By this we mean that we can consider different components in a sequence of steps. In each step, there are two type of terms with a borderline O(t -1 ) coefficient with the following properties.
• One type can be handled by introducing an integration factor. The integration factor gives a power of t which is consistent with the initial conditions that we impose.
• Another type of terms with borderline coefficients involve only terms which have been controlled in previous steps.
• Any other linear terms must have a coefficient that is better, at least O(t -1+ε ). Such a structure is important both in estimating the metric components (Lemmas 2.11, 2.12) and the components of the (approximate) second fundamental form (Lemmas 2.8, 2.9).
• In anticipation of the energy estimates needed to construct an actual solution, we also need to treat different components on different footing in the ODE analysis. An example of this is that while for i ≤ j, we only prove that (k [n] ) i j = O(t -1 ); for i > j, we need a better estimate and the improvement we need depends on the precise i, j under consideration; see Lemma 2.8. Such estimates can be traced back to (1.17), but also require the precise structure of the system.
• Another technical difficulty is that the variable k [n] we work with is only approximately the second fundamental form.
The evolutionary equations solved in the first step roughly asserts that the spacetime Ricci curvature components Ric( (4) g [n] ) i j vanish with a very fast rate. Our second step is then to show that
• k [n] is asymptotically (as t → 0 + ) approximately the second fundamental form of the constant-t hypersurfaces, and
• all other spacetime Ricci curvature components also vanish sufficiently fast as t → 0 + . Both of these are achieved again by ODE analysis. For the first point, we need again a reductive structure, which is similar to the type used for the evolutionary equations. For the second point, the constraints as manifested both in the conditions on the Kasner exponents and asymptotic constraint equations (1.6) play a crucial role. See already Lemmas 2.13-2.15 and Proposition 3.3.
1.1.4. Energy estimates in synchronized coordinates. It is a priori unclear that under a gauge condition as in (1.16), the metric components themselves satisfy any hyperbolic system. The main new ingredient is to consider a "wave-type equation" satisfied by the second fundamental form k i j of the spatial slice. Since this is already new for a local existence problem without singularities, we will indicate the ingredients needed only for a local existence result for regular data, i.e., for this subsubsection suppose we are given geometric data (Σ, g, k) satisfying the (usual) constraint equations and we explain how to construct a spacetime solution to the Einstein vacuum equations in the gauge (1.16).
Assuming that a metric of the form (1.16) obeys the Einstein vacuum equations, we can deduce that the second fundamental form k i j obeys the following system of second order equations:
where k ⋆k ⋆k and ∂ t k ⋆k are nonlinear terms to be specified in (4.6) in Section 4.1.
Notice that (1.18) is not actually a wave equation, due to the term ∇ i ∇ j k ℓ ℓ on the RHS. The key is that the trace of k, i.e., k ℓ ℓ in fact can be proven to have additional regularity if we further use the Einstein vacuum equations. First, the Einstein vacuum equations imply that
Now we consider h = k ℓ ℓ to be a separate variable and consider the coupled system for (g, h, k):
(This system must hold for h = k ℓ ℓ if the Einstein vacuum equations are satisfied.) We then attempt to solve (1.19) with initial data where
(which is completely determined by the geometric data).
The apparent difficulty in solving (1.19) is a potential loss of derivatives. For instance, energy estimates for the second equation requires two derivative of h and give only first-derivative estimates for k. The first equation, however, does not seem to give two derivatives for h if we only have one derivative for k. A similar issue arises for g and k when we consider commutators for the second equation.
This can nevertheless be resolved by a renormalization together with elliptic estimates. As an example, we illustrate how to obtain second derivative estimates for h when only controlling one derivative of k. Commute the first equation with ∆ g so that we have, up to error terms,
The idea now is to use the second equation in (1.19) so that we obtain
This allows us to control ∆ g h even only controlling one derivative of k. The other second derivatives of h can then be bounded by ∆ g h using elliptic estimates. This avoids the loss of derivatives.
Standard energy estimates together with this renormalization/elliptic estimates trick indeed give a solution to (1.19). Furthermore, the choice of initial data and the structure of the equations allow one to propagate the symmetry of g ij and g iℓ k j ℓ . Using moreover the Hamiltonian constraint R(g) -|k| 2 + (tr k) 2 = 0, it can be shown a posteriori that h = k ℓ ℓ . In particular, we also have that ∂ t k ℓ ℓ = |k| 2 , which implies that Ric( (4) g) tt = 0.
Finally, we need to upgrade the existence result (1.19) to a bona fide existence result of solutions to the Einstein vacuum equations in the gauge (1.16), i.e., we need to show that all the Ricci components vanish (in addition to Ric( (4) g) tt ). For this purpose, first note that (after accounting for symmetries) the second equation in (1.19) gives a system of 6 first order homogeneous equations in Ric i j ( (4) g) and Ric ti ( (4) g). At the same time, three of the (contracted) second Bianchi equations give another 3 first order homogeneous equations in Ric i j ( (4) g) and Ric ti ( (4) g). (The fourth equation is redundant, and does not give us extra information.) It turns out that these 9 equations form a coupled system of wave-transport equations (see (5.8) and (5.10)). This wave-transport equations is similar in structure to (1.19), and can also be treated using energy estimates together with renormalization/elliptic estimates. Moreover, the momentum constraint and the choice of initial data, when solving (1.19), together, guarantee that Ric i j ( (4) g) and Ric ti ( (4) g) are initially vanishing. Combining all these we obtain that Ric i j ( (4) g) = 0 and Ric ti ( (4) g) = 0 everywhere, implying that the constructed solution to (1.19) indeed obeys the Einstein vacuum equations.
Obviously, in our setting, we need to handle simultaneously the existence theory and the fact that the metric becomes singular as t → 0 + . For this we combine the ideas here and Section 1.1.1. A few technical issues arise. For instance, the Kasner-type geometry dictate that we do not have uniform control of the isoperimetric constants as t → 0 + . Some care is therefore needed in the application of Sobolev embedding; in particular we need to be careful which terms are to be put in L 2 /L ∞ type spaces. Finally, we note that the Fuchsian ideas in Section 1.1.1 are used not only in solving the system (1.19), but are also used in verifying that the solution to (1.19) is indeed a solution to the Einstein vacuum equations.
1.1.5. Uniqueness and regularity. To prove uniqueness, we again rely on the wave equation satisfied by the second fundamental form, and perform tweighted energy estimates in a similar way as proving existence. The only subtlety here is that we must impose that the metrics converge to each other sufficiently fast as t → 0 + in order to close the estimate (cf. the statement of Theorem 1.7).
Finally, we prove higher regularity relying on the uniqueness result. The issue at stake here is that for each additional derivative we try to control, the estimate in terms of t worsens by one power. Thus, the approximation we choose has to be successively better for higher and higher derivatives. We then redo the construction of solutions for better and better choices of the approximations. The uniqueness result ensures that we have in fact constructed the same solution, thus showing that the already constructed solution has arbitrarily high derivative bounds.
1.2.1. Fuchsian constructions of singular spacetimes. Many works have been carried out to construct AVTD singularities in (3 + 1)-dimensional vacuum spacetimes. All previous works assume either symmetry or analyticity (or both). The symmetry classes are typically chosen so that AVTD singularities are expected to be stable within that class. We give a sample of such results, but refer the reader also to the references therein for further details.
Gowdy symmetry. AVTD singularities in (unpolarized) Gowdy symmetry was first constructed by Kichenassamy and Rendall [30] in the analytic category, in part based on the formal expansion carried out in [25]. A similar analysis was carried out by Rendall without the analyticity assumption in [36]. See also [46] for more general topologies, and [5] for a treatment in generalized wave gauges.
Polarized T 2 symmetry. Analytic AVTD singularities under polarized T 2 symmetry were first constructed in [26]; analyticity was later removed in [4].
U(1) polarized or half-polarized symmetry. Analytic solutions with AVTD behavior in polarized or half-polarized symmetry with T 3 topology were constructed by Isenberg-Moncrief in [28]. That for more general topology was later carried out in [13].
Beyond (3 + 1)-dimensional vacuum spacetimes. The first construction of analytic solutions with AVTD behavior without symmetries was carried out in [7], albeit not for the Einstein vacuum equations. Indeed, the construction in [7] was for the Einstein-scalar field or Einstein-stiff fluid system. An important difference is that in the presence of a scalar field or stiff fluid, one expects AVTD singularities to be stable [10,9]. A similar stability phenomenon is expected to occur in vacuum for spacetime dimensions ≥ 11 [19]. Correspondingly, there is a construction of AVTD singularities for high dimensional vacuum (and more general) solutions in [18]. See also Section 1.2.2.
Analytic singular spacetimes without symmetry assumptions. All the works above concern regimes (either in symmetry classes or with matter, or in high dimensions) which at least heuristically should generically have AVTD behavior near the spacelike singularity. In a recent work of Klinger [32], analytic vacuum AVTD spacetimes with no symmetry assumptions have been constructed. The work [32] can be viewed as similar to our result except for requiring the analyticity assumption and some additional inequalities on the Kasner exponents p i 's. (These additional inequalities were used in [32] to apply a black-box Fuchsian theorem.)
Asymptotically Schwarzschild singularity up to a singular 2-sphere. Finally, we mention the work [21] of the first author, who constructed a class of spacetimes approaching the Schwarzschild black hole singularity. The construction requires no symmetry or analyticity assumptions. While it does not include a full spacelike singular hypersurface, the construction does include a spacelike singular 2-sphere.
1.2.2. Stable singularities in general relativity. By "function-counting" arguments (cf. Remark 1.5), the class of spacetimes we construct are not expected to be stable. For the vacuum equations in (3 + 1) dimensions, the only known stable singularities are in fact null; see [35,34,17]. These singularities are in stark contrast with the AVTD ones, which are spacelike.
As we already mentioned in Section 1.2.1, it has been suggested that in the presence of a scalar field or stiff fluid [10,9], or in the vacuum case in spacetime dimensions ≥ 11 [19], there is an open set of initial data which give rise to asymptotically Kasner-like singularities. It is also for this reason that in these settings, the construction of spacelike singularities with AVTD behavior is simpler. Spectacular progress has recently been made which indeed proves stability of spacelike singularities in the aforementioned settings. In the case of Einsteinscalar field or Einstein-stiff fluid, this was carried out in the breakthrough work by Rodnianski-Speck [42,43] and later generalized by Speck [45]. In the case of high dimensions, assuming spacetime dimensions ≥ 39, Rodnianski-Speck has recently also constructed a class of stable spacelike singularities in vacuum [44]. (Note that the remarkable works of Rodnianski-Speck do not cover the whole regimes in [10,9,19]. Whether all of the solutions discussed in [10,9,19] are stable remains an open problem.)
Very recently, the first author and Alexakis considered the stability problem for the Schwarzschild singularity [1]. Unlike the settings studied by Rodnianski-Speck, the Schwarzschild singularity is unstable, but nonetheless it was shown in [1] to be stable within the class of polarized axisymmetric perturbations.
The understanding of AVTD singularities played an important role in understanding the strong cosmic censorship conjecture, at least under Gowdy symmetry.
The strong cosmic censorship conjecture has first been resolved in the polarized Gowdy case in [16]. The work relies in particular on [27], in which AVTD singularities in this setting were studied.
The more general case of the strong cosmic censorship conjecture in unpolarized Gowdy symmetry turned out to be significantly more difficult in view of the so-called "spikes". This has been treated in the seminal work of Ringström [39] (see also [38]). Here, a form of asymptotic velocity term domination has been established [37] and plays an important role.
It should again be stressed that outside symmetry classes (Gowdy, polarized T 2 , polarized U(1), etc.), AVTD singularities are most likely not generic, and the role of the study of AVTD singularities in the ultimate resolution of strong cosmic censorship conjecture is quite unclear.
A discussion of the large number of related numerical works will take us too far afield. For this we will refer the reader to [11] and the many references therein. 1.2.5. Linear wave equations on singular spacetimes. A closely related thread of works concerns solving the linear wave equation on a spacetime with a spacelike singularity, including Kasner, FLRW and Schwarzschild. See for instance [31,3,41,22,2,40,8,24].
1.2.6. Einstein equations in transport coordinates. At the heart of our approach is the ability to perform energy estimates in the gauge (1.16), corresponding to a choice of coordinates such that (t, x i ) are all transported by the unit normal to the spacelike hypersurfaces {t = constant}; recall Section 1.1.4. We highlight previous works where smooth solutions to the Einstein equations are constructed in gauge where the spatial x i coordinates are transported, i.e the metric takes the form
The first is the work of Rodnianski-Speck [42,43] (in which they constructed stable spacelike singularities; see discussions in Section 1.2.2 above), where α is determined by stipulating that each constant-t hypersurface has constant mean curvature. See also [23] for a different approach in handling this gauge. (Constant mean curvature foliations, but without spatially transported coordinates, have been previously used. See for instance [6], which used spatially harmonic instead of spatially transported coordinates.)
The second is the work of Choquet-Bruhat-Ruggeri [14], in which the authors consider the spacetime metric of the form (1.20) and impose the condition α = det g det e , where e is some arbitrary but fixed (i.e., t-independent) Riemannian metric. They show that in such a gauge, the Einstein equations are hyperbolic.
1.3. Outline of the paper. We end the introduction with an outline of the remainder of the paper.
The first part of the existence proof will be to construct an approximate solution. This will be carried out in Section 2, where we give the construction and show that evolutionary equations are approximately satisfied. In Section 3 we then show that the constraint equations are also approximately satisfied.
In Section 4 and 5 we then construct an actual solution, thus completing the proof of Theorem 1.1. This will be carried out in two steps: in Section 4 we will solve an appropriate system of reduced equations, then in Section 5 we show that the solutions to the reduced equations that we have constructed in fact obey the Einstein vacuum equations.
Finally, in Section 6, we end with the proofs of uniqueness (Theorem 1.7) and smoothness (Theorem 1.10).
We work under the assumptions of Theorem 1.1. In particular, we fix p i and c ij to be as in Theorem 1.1.
Unless explicitly stated otherwise, all the implicit constants (given either in the ≲ or the big-O or the • ≤ C• notation) that we have in our arguments, from now on, may depend on p i and c ij . Many estimates in this section will involve an n ∈ N or a multi-index α. Unless otherwise stated, all constants may depend also on n and α.
Our goal in this section is to construct an approximate solution, i.e., we will construct inductively a metric (4) g [n] (n ∈ N ∪ {0}), which takes the form (1.4), but with a
[n] ij in place of a ij ; as well as an approximate second fundamental form (k [n] ) i j . These a
[n] ij are constructed so that lim t→0 + a
[n] ij (t, x) = c ij (x). We will moreover show that the pairs (g [n] , k [n] ) we construct indeed form an approximate solution to the evolution equation, i.e., as n becomes larger, ∂ t (k [n] ) i j -Ric(g [n] ) i j -(k [n] ) ℓ ℓ (k [n] ) i j tends to 0 faster as t → 0 + ; see already Theorem 2.1. Unless otherwise stated, we will also be using the Einstein summation convention for repeated indices, with lower case Latin indices running through 1, 2, 3. It should be noted that sometimes we will still write out the sum explicitly in situations that confusion might arise (e.g., when one has factors of t p max{i,j} ).
Definition of (4) g [n] and k [n] . Define (4) g [0] by setting (2.1)
Now given g [n-1] , n ∈ N (and assuming that it is a Riemannian metric on (0,
subject to the following condition at t = 0:
where κ is defined by
ℓi , subject to the following condition at t = 0:
(2.5) |a
where we recall that a [n] is related to g [n] via (1.4). It readily follows from (2.4) that the inverse metric (g [n] ) -1 satisfies the equation:
(2.6) ∂ t ((g [n] ) -1 ) ij = (k [n] ) ℓ j ((g [n] ) -1 ) iℓ + (k [n] ) ℓ i ((g [n] ) -1 ) jℓ .
Our goal in this section is to establish the properties of the above sequences {g [n] } +∞ n=0 , {k [n] } +∞ n=1 given in the following theorem: THEOREM 2.1. Let p i and c ij be as in the Theorem 1.1. Define
Then for n ∈ N, there exist t n > 0 (depending on p i , c ij and n), a smooth Lorentzian metric (4) g [n] and a (1, 1)-tensor (k [n] ) i j on (0, t n ] × T 3 such that the following holds.
(1) (4) g [n] takes the following form for some smooth functions a
(2) (Convergence to initial data) For every multi-index α, every i, j and every n ∈ N, the functions a
[n] ij and (k [n] ) i j satisfy sup
for some C α,n > 0 depending on p i , c ij , in addition to α and n. (Recall the definition of κ i j immediately after (2.3).) (3) (Estimates for spatial curvature) For every multi-index α, every i, j and every n ∈ N, the spatial Ricci curvature satisfies (2.9) sup
for some C α,n > 0 depending on p i , c ij , in addition to α and n.
(4) (k [n] is an approximate second fundamental form) For every multi-index α, every i, j and every n ∈ N, (2.10)
for some C α,n > 0 depending on p i , c ij , in addition to α and n.
(5) (Evolution equations approximately satisfied) For every multi-index α, the tensors (k [n] ) i j , g
[n]
ij also satisfy
for some C α,n > 0 depending on p i , c ij , in addition to α and n.
Remark 2.2. All the ε in the error terms in Theorem 2.1 can be improved almost to 2ε (or exactly to 2ε if we allow some powers of log t in the error terms). Some estimates can even be further sharpened. We will be content with the weaker estimates for the sake of simplicity of the exposition.
Remark 2.3. The definition of ε, together with conditions (2)-(3) in Theorem 1.1, imply that (2.12)
This can be easily checked by using the following parametric form of the Kasner exponents p 1 , p 2 , p 3 :
which is valid at each point x ∈ T 3 , u = u(x).
In the rest of the section, we will prove Theorem 2.1; see the conclusion of the proof at the end of the section. (In particular, in the course of the proof, it can be seen that g [n] and k [n] are well defined.) 2.1. Estimates for g [0] . LEMMA 2.4. There exists t 0 > 0 (depending on c ij and p i ) such that the following are true for (t, x) ∈ (0, t 0 ] × T 3 :
(1) The determinant of det g [0] satisfies, for some C > 0 (depending on c ij and p i ),
(2) The eigenvalues λ 1 ≤ λ 2 ≤ λ 3 of g [0] satisfy, for some C > 0 (depending on c ij and p i ),
In particular, choosing t 0 smaller if necessary, g [0] is a Lorentzian metric on (0, t 0 ] × T 3 .
(3) For every multi-index α, the inverse metric (g [0] ) -1 satisfies, for some C α > 0 (depending on α, c ij and p i ),
c 33 t -2p 1 c 11 c 22 c 33 (c 12 c 23 -c 13 c 22 )t -2p 1 c 11 c 22 c 33 -c 12 c 33 t -2p 1 c 11 c 22 c 33 t -2p 2 c 22 (c 12 c 13 -c 11 c 23 )t -2p 2 c 11 c 22 c 33 (c 12 c 23 -c 13 c 22 )t -2p 1 c 11 c 22 c 33 (c 12 c 13 -c 11 c 23 )t -2p 2 c 11 c 22 c 33
where |∂ α x ((g -1 )
[0]
Proof. This is a simple computation and the proof is omitted. □
It will be convenient to define also
iℓ .
The following lemma gives an estimate for (k [0] ) i j .
LEMMA 2.5. For every multi-index α, there exists C α > 0 (depending on α, in addition to c ij and p) such that the following estimate holds for all (t, x) ∈ (0, t 0 ] × T 3 :
Proof. By the definition of g [0] , it is easy to see that
error ,
where
iℓ , the conclusion of the lemma can be achieved by combining the above computation with Lemma 2.4. □
The next lemma estimates the Ricci curvature of a general metric g = 3 i,j=1 a ij t 2p max{i,j} dx i dx j when a ij satisfies some basic bounds. This in particular gives an estimate for Ric(g [0] ) i j .
LEMMA 2.6. Suppose (4) g is a metric on (0, T ] × T 3 taking the form (4) g = -dt 2 + 3 i,j=1 a ij t 2p max{i,j} dx i dx j , where a ij are smooth, symmetric and obey the estimates
Then for every multi-index α, there exists C ′ α > 0 (depending on C α , in addition to c ij and p) such that the following estimate holds for all (t, x) ∈ (0, T ] × T 3 :
(2.17)
In fact, the following slightly stronger estimate holds:
(2.18)
Proof. Clearly (2.18) implies (2.17); from now on we focus on the proof of (2.18).
For notational convenience, in this proof we write g ab = (g -1 ) ab .
Here is the basic observation. For a pairing g ab ∂ c g ae (note the one contracted index), we have
). So in order to give an estimate for the Ricci curvature, we will find pairs of g -1 and derivatives of g which share at least one index.
To make the algebraic structure clear, we will focus on proving the estimate with |α| = 0 and r = 0 in Steps 1 and 2, and then indicate the necessary changes in Steps 3 and 4.
Step 1: Proof of the upper bound t -2+ε . We recall the formula for the Ricci curvature:
and that for the Christoffel symbols
Hence, we notice that every term in (2.19) has either of the forms (2.21)
where among the ℓ i 's there is an upper j and a lower i index, while the rest are contractions among themselves and with respect to a, b.
For the first kind of terms in (2.21), using Lemma 2.4, we notice that they are of order (2.22)
where the pair {ℓ 5 , ℓ 6 } contains at least one of the indices a, b, ℓ 2 , ℓ 3 . Hence, we have either -2p min{a,b} + 2p max{ℓ 5 ,ℓ 6 } ≥ 0 or -2p min{ℓ 2 ,ℓ 3 } + 2p max{ℓ 5 ,ℓ 6 } ≥ 0, leaving
for some ℓ. On the other hand, the second term in (2.21) satisfies:
where at least three from the indices a, b, ℓ 1 , ℓ 2 , ℓ 6 , ℓ 7 are contracted against three of the indices ℓ 4 , ℓ 5 , ℓ 9 , ℓ 10 . This implies that at least two pairs of exponents having opposite signs, among {-2p min{a,b} , -2p min{ℓ 1 ,ℓ 2 } , 2p max{ℓ 4 ,ℓ 5 } , -2p min{ℓ 6 ,ℓ 7 } , 2p max{ℓ 9 ,ℓ 10 } }, yield non-negative sums, thus, leaving only
Step 2: Proof of the upper bound t -2+ε-2p j +2p i . We now move on to prove the improved estimates when i > j (when i ≤ j the desired estimate follows from that proven in Step 1). As we are now familiar with this type of argument, let us just consider the contribution from the second type of term in (2.21) (the first type of terms can be treated similarly). We now separate out the factor of g jℓ (which gives a contribution of at worst of O(t -2p j )), i.e., we write
where exactly one of the ℓ m is b and exactly one of the ℓ m is i. It is easy to check that at least one of the following must hold:
• After relabelling
, so that by our basic observation g ℓ 1 ℓ 2 ∂ ℓ 3 g ℓ 4 ℓ 5 g ℓ 6 ℓ 7 ∂ ℓ 8 g ℓ 9 ℓ 10 = O(|log t| 2 ). As a result, the whole term contributes O(t -2p j |log t| 2 ), which is better than O(t -2+ε-2p j +2p i ).
• After relabelling, we have one of the following:
For the first term, after noting
For the second term, note that g df ∂ a g db = O(|log t|), g ac = O(t -2+2ε ), g jb = O(t -2p j ) and ∂ c g bi = O(t 2p i |log t|), which then again gives the desired estimate.
Step 3: Higher derivative bounds. It is easy to see that after differentiating by ∂ α
x , we at worst pick up additional powers of |log t| |α| , we then obtain the desired estimate also for higher derivatives of Ric(g) i j .
Step 4: Time derivative. For ∂ α x ∂ t Ric(g) i j , the argument is almost identical. Indeed, exploiting the form of the metric and using the estimate for
) and similar behaviors for their spatial derivatives (up to logarithms). Hence, a power of t -1 can be factored out, leaving terms with factors that behave as in the previous steps. This completes the proof of the lemma. □
LEMMA 2.7. Consider the nonlinear transport equation
where f : (0, 1) × T 3 → R is a function such that |f |(t, x) ≲ t δ for some δ > 0.
Then there exist t * ∈ (0, 1) and a unique solution u :
Proof. This is proven by a standard Picard iteration, with some extra care tracing the t dependence; we omit the details. □ LEMMA 2.8. Suppose the following holds for some N ≥ 1: there exists t N -1 > 0 such that for every 0 ≤ n ≤ N -1 and every multi-index α, g [n] satisfies the following estimate for some C α,n >0 (depending on α, n, in addition to c ij and p i ) for all (t, x) ∈ (0, t N -1 ) × T 3 :
Then, there exists t N ∈ (0, t N -1 ) sufficiently small such that for every 1 ≤ n ≤ N and every multi-index α, the following holds for all (t, x) ∈ (0, t N -1 ) × T 3 for some C ′ α,n > 0 (depending on α, n, in addition to c ij and p i ):
Proof. The key difficulty in solving (2.2) is that there are borderline terms with O(t -1 ) coefficients so that we cannot directly apply Grönwall's lemma. One can nevertheless analyze the precise structure of the equations.
Step 1: Solving an auxiliary system. We first solve an auxiliary system (2.24)
The first equation in (2.24) can be rearranged to
Using the bound |R(g [n-1] )| ≲ t -2+ε from the assumptions on g [n-1] together with Lemma 2.6, (2.25) can be solved using Lemma 2.7 with
Now the second equation in (2.24) can be rearranged to
Using (2.26), Grönwall's inequality and the estimate in Lemma 2.6, it follows that there is a unique solution (k [n] ) i j that obeys the initial condition (2.3) and satisfies
Step 2: Finishing the argument. Now that we have solved (2.24) and obtained estimates (2.26) and (2.27), in order to conclude the argument, it suffices to show that in fact h [n] = (k [n] ) ℓ ℓ . To this end, it suffices to note that
Hence, comparing this equation with (2.25), we obtain h [n] = (k [n] ) ℓ ℓ by the uniqueness statement in Lemma 2.7. □ LEMMA 2.9. Suppose the following holds for some N ≥ 2: there exists t N -1 > 0 such that for every 1 ≤ n ≤ N -1 and every multi-index α, g [n] satisfies the following estimate for some C α,n > 0 (depending on α, n, in addition to c ij and p i ) for all (t, x) ∈ (0, t N -1 ) × T 3 :
Then, taking t N ∈ (0, t N -1 ) smaller (compared to Lemma 2.8) if necessary, for every 2 ≤ n ≤ N and every multi-index α, the following holds for all (t, x) ∈ (0, t N -1 ) × T 3 for some C ′ α,n > 0 (depending on α, n, in addition to c ij and p i ):
Proof.
Step 1: Estimates on the Ricci curvature. The estimate (2.28) implies that (2.30)
for every 0 ≤ n ≤ N -1. Indeed, arguing as in the proof of Lemma 2.6, we notice that the difference of the ∂ α x derivative of the Ricci curvatures can be bounded by the differences a
(and their spatial derivatives), multiplied by a term that is controlled by t -2+2ε |log t| 2+|α| . In particular, (2.30) implies
Step 2: Estimates on (k [n] ) i j .
The assumption (2.28) implies the assumption of Lemma 2.8 holds. Hence by Lemma 2.8,
for every 2 ≤ n ≤ N .
In particular, since (by definition)
Step 3: Estimates on the difference
(2.34)
It turns out to be useful to first control the trace of k
. Taking the trace of (2.34), we obtain
This implies
By (2.31) in Step 1, the estimate (2.33) in Step 2, the condition (2.3) and Grönwall's inequality, it easily follows that (2.35)
for every 2 ≤ n ≤ N . We now return to (2.34), which we rewrite as follows.
Step 1, the estimates (2.32) and (2.33) in Step 2, the estimate (2.35) that we just proved, the condition (2.3) and Grönwall's inequality, we obtain
for every 2 ≤ n ≤ N , which is what we want to prove. □
[n] ij .
LEMMA 2.10. For n ∈ N and g
[n] ij defined by (2.4)-(2.5), the corresponding a
where k [0] is as defined in (2.16).
Proof. By (1.4) and (2.4), with repeated indices not summed, we have on the one hand
.174.251.2] Project MUSE (2025-07-03 18:11 GMT) Stanford University Libraries
and on the other hand
ℓi .
Similarly, by (1.4), (2.1) and (2.16),
Therefore, we obtain
Canceling t 2p max{i,j} on both sides, we obtain the desired equation. □ LEMMA 2.11. Suppose the following holds for some N ≥ 1: there exists t N > 0 such that for every 1 ≤ n ≤ N and every multi-index α, k [n] satisfies the estimate for some C α,n > 0 (depending on α, n, in addition to c ij and p i ) for all (t, x) ∈ (0, t N -1 ) × T 3 :
Then, after choosing t N > 0 smaller if necessary, a
[n] ij (t, x) is well defined and symmetric for all (t, x) ∈ (0, t N ] × T 3 and for every 1 ≤ n ≤ N . In addition, by reducing t N > 0 further, g
ij is symmetric, we will only estimate the six components {a
(2.41)
(2.42)
(2.43)
Here, we have used the schematic notation that when we write (a [n] -c) or a [n] without explicit indices, it can represent any component.
The key point is a reductive structure for terms with O(t -1 ) coefficients: The diagonal (a [n] -c) ii terms do not see any terms with O(t -1 ) coefficients on the right-hand side. For the remaining terms, we make the observations that (1) the linear term has coefficients which is negative and (2) by estimating the terms in the order as listed above, the only terms with O(t -1 ) coefficients have already been estimated in the previous step.
Indeed, the first three equations ((2.38)-(2.40)) give
where we have used the initial condition (2.5).
Using the fourth and fifth equations ((2.41)-(2.42)) and plugging in (2.44), we obtain
(|a [n] -c| + |c|).
(2.46)
The estimates (2.45) and (2.46) imply
(|a [n] -c| + |c|).
Finally, we consider the last equation, argue as above and plug in (2.44) and (2.47) to obtain
(|a [n] -c| + |c|).
Combining (2.44), (2.47), (2.48), and choosing t N to be sufficiently small, we obtain sup
This proves that a
[n] ij is well defined and moreover shows the first inequality in (2.37) in the case |α| = 0.
The second inequality in (2.37) (that for ∂ t a
[n] ij ) follows by applying the already derived bounds to the RHS of the system (2.38)-(2.43).
We then obtain the desired higher order estimates by induction on |α|. For example, differentiating the equation (2.36) by ∂ α
x for |α| = 1, we may treat the zeroth order terms in the differences a [n] -c as already estimated inhomogeneous terms and repeat the above argument. The same goes for ∂ α
x with |α| = 2 etc. From this we deduce the estimate (2.50) in general. We omit the details. □ LEMMA 2.12. Suppose the following holds for some N ≥ 1: there exists t N > 0 such that for every 1 ≤ n ≤ N and for every multi-index α, k [n] satisfies the following estimate for some C α,n > 0 (depending on α, n, in addition to c ij and p i ) for all (t, x) ∈ (0, t N ] × T 3 :
for every 1 ≤ n ≤ N . Then, after choosing t N > 0 smaller if necessary, for every multi-index α and
for all (t, x) ∈ (0, t N ] × T 3 and for every 1 ≤ n ≤ N .
Proof. First, we note that by Lemma 2.5 and (2.49),
(2.51)
for every 1 ≤ n ≤ N .
Subtracting the n and n -1 versions of (2.36), for i ≤ j, we have
).
(2.52)
Using equation (2.52) and the estimates in Lemmas 2.5, 2.8 and (2.49), we deduce a system of schematic equations in a similar manner as (2.38)-(2.43), namely,
(2.56)
(2.57)
(2.58)
From this point on we can argue as in Lemma 2.11, using the reductive structure of the system. Note that the system (2.53)-(2.58) is better than the system (2.38)- (2.43) in that the inhomogeneous terms O(t -1+nε )a [n-1] = O(t -1+nε ). As a result, the argument in Lemma 2.11 gives the better estimate
Now a straightforward induction argument using Lemmas 2.8, 2.9, 2.11, 2.12 shows there exists a decreasing sequence of positive times t n , such that g [n] and k [n] are well defined and smooth in (0, t n ] × T 3 , for all n ∈ N. Moreover, all the estimates in the conclusions (and proofs) of Lemmas 2.8, 2.9, 2.11, 2.12 hold. In particular, points (1), (2) in Theorem 2.1 hold true; and after using also Lemma 2.6, it can be checked that (3) in Theorem 2.1 is also verified.
In the remaining subsections, we prove points ( 4) and ( 5) in Theorem 2.1, thus completing the proof of Theorem 2.1. [n] with the second fundamental form. In this subsection, we prove point (4) of Theorem 2.1; see the main estimate in Lemma 2. 15.
The heart of the matter is the following estimates for D
[n]
ℓi .
LEMMA 2.13. For each n ∈ N, define D
ℓi . Then if (n + 1)ε > 2, after choosing t n smaller if necessary, the following estimate holds for some C α,n > 0 (depending on α, n, in addition to c ij and p i ):
Proof.
Step 1: Derivation of an equation for D
[n] ij . By (2.4),
(2.59) Therefore, (2.59) and the equation (2.2) that define k [n] , it follows that
(2.60)
is symmetric in i and j, we have
where the final estimate follows from the form of the metric, Lemmas 2.6, 2.12, and the fact that
Therefore, combining (2.60) and (2.61), we have obtained that
(2.62)
Step 2: Estimating D
13 and D
[n] 12 . By (2.62), they satisfy the following equations:
Applying the estimates for k [n] from Lemma 2.8, we obtain
(2.64)
(2.65)
To use these equations, note that when i, j, ℓ are all distinct, (2.66)
Indeed, using the estimates in Lemmas 2.8 and 2.11, one checks that D
[n]
23 , D
[n]
We then obtain (2.66) using p 1 < p 2 < p 3 .
We now use equations (2.63)-(2.65) to estimate D
[n]
ij . The key is to notice a reductive structure similar to that in the proof of Lemma 2.11, except in this situation since the different components have different rates, we argue with a bootstrap argument.
Make the bootstrap assumptions that
2 |log t| 2 , (2.67) where A is a large constant, such that denoting the implicit constant in the big-O notation in (2.63)-(2.65) by C, we require C ≪ A. Plugging (2.67) into (2.63), integrating, and using p 2 > p 1 , we obtain (2.68)
Arguing similarly, first for D
[n]
13 and then for D
[n]
12 , we also obtain
(2.70) Choosing t n sufficiently small (so that At ε ≤ 1), it is easy to check that (2.68)-(2.70) improves the bootstrap assumptions in (2.67). This gives the stated estimates for D
[n] ij in the lemma when |α| = 0. The estimates for the spatial derivatives are similar, except that we lose a factor of |log t| for each derivative we take (cf. (2.18)).
Step 3: Estimating ∂ t D
[n] ij . Finally, we plug in the estimates for D [n] ij as in Lemma 2.13, define ( D [n] ) i j := (g [n] ) jℓ D
[n]
iℓ . Then if (n + 1)ε > 2, the following estimates hold for (t, x) ∈ (0, t n ] ×
T n for some C α,n > 0 (depending on α, n, in addition to c ij and p i ):
(2.71)
Proof.
Step 1: Estimates for D [n] (when r = 0). By Lemma 2.13, the estimate ∂ α
x (g [n] ) jℓ = O(t -2p min{j,ℓ} |log t| |α| ) and the fact t ε |log t| ℓ ≲ ℓ 1, we immediately obtain
(2.72)
Step 2: Deriving evolution equations for D [n] . Contracting (2.60) with (g [n] ) jb , using (2.6) and the anti-symmetry of
ℓi .
We notice now that since
we have
ℓi -g
where in estimating the terms we have used the form of the metric, computation of the inverse metric (see (2.15)), Lemmas 2.11 and 2.12, and (2.18). Differentiating (2.74) by ∂ α x ∂ r t , and arguing similarly, we also obtain the following higher derivative bounds for r = 0, 1:
(2.75) Plugging the estimate (2.75) into (2.73), using the estimates for k [n] (by Lemma 2.8) and ∂ t k [n] (by (2.2), Lemma 2.6, (2.37) and Lemma 2.8), and relabelling the indices, we obtain
and
(2.77)
Step 3: Estimates for ∂ t D [n] and ∂ 2 t D [n] (when r = 1, 2). Plugging (2.72) into (2.76), we obtain
(2.78)
Similarly, plugging in both (2.78) and (2.72) into (2.77), we obtain
(2.79) Combining (2.72), (2.78) and (2.79) yields (2.71). □
The next lemma shows that even though k [n] is not the second fundamental form associated to g [n] , it is close to being the second fundamental form up to an error that vanishes sufficiently fast as t → 0 + . LEMMA 2.15. When (n + 1)ε > 2, the following estimates hold for (t, x) ∈ (0, t n ] × T n for some C α,n > 0 (depending on α, n, in addition to c ij and p i ):
(2.80)
Proof. By (2.4) and the definition of D
[n] ij (in Lemma 2.13) and
The desired estimate is then an immediate consequence of Lemma 2.14. □ Lemma 2.15 gives point (4) in Theorem 2.1.
(k [n] ) i j and g
[n]
ij satisfy evolution equations approximately. In this subsection we prove point (5) in Theorem 2.1 (see Proposition 2.17), which then completes the proof of the theorem. LEMMA 2.16. For every n ∈ N, the following estimates hold for (t, x) ∈ (0, t n ] × T n , for some C α,n > 0 (depending on α, n, in addition to c ij and p i ):
Proof. Going back to the proof of Lemma 2.6 and using the form of the metrics g [n] and g [n-1] , we notice that the each term in the difference of Ric(g [n] ) i j , Ric(g [n-1] ) i j has the form:
[explicit powers of t and log t with behavior O(t -2+2ε |log t| 2+|α| )]
The fact that a [n] , a [n-1] and their spatial derivatives are bounded, while |∂ α x (a [n] a [n-1] )| ≲ t nε (see Lemma 2.12), was then used in Lemma 2.9 to infer the bound (2.30). Now we verify that a time derivative acting on any of the previous type of terms, adds at worst a power of t -1 in their behavior. For the factors which are explicit powers of t this is evident. If ∂ t hits either a [n] , a [n-1] factor or their difference a [n] -a [n-1] , we make use of (2.37), (2.50) and the conclusion follows. □ PROPOSITION 2.17. For every n ∈ N, the following estimates hold for (t, x) ∈ (0, t n ] × T n , for some C α,n > 0 (depending on α, n, in addition to c ij and p i ):
Proof. Using the equation (2.2), the estimate (2.31), and Lemma 2.16, we obtain
as desired. □ Proposition 2.17 implies point (5) of Theorem 2.1. Together with the previous subsections, this completes the proof of Theorem 2.1.
3. Approximate propagation of constraints. We continue to work under the assumptions of Theorem 1.1 and take g [n] and k [n] as constructed in the beginning of Section 2 (so that for appropriately chosen t N , the estimates in Theorem 2.1 hold).
The goal of this section is to show that metrics (4) g [n] are also approximate solutions to the constraints, as t → 0, to an order that improves with the increase of n. To achieve this we argue by propagation of constraints, i.e., we use the second Bianchi identity as propagation equations and use that the constraints are asymptotically valid in the renormalized sense (1.7)-(1.8).
It will be useful to setup some notations that we use in this section. For the remainder of this section, D will denote the Levi-Civita connection of the spacetime metric (4) g [n] and ∇ will denote the Levi-Civita connection of the metric g [n] on the (spacelike) constant-t hypersurfaces. Moreover, indices are lowered and raised with respect to the metric g [n] (in particular (g [n] ) ij = ((g [n] ) -1 ) ij in this section). PROPOSITION 3.1. Let (4) g = -dt 2 + g, where g is a Riemannian metric. Define k i j := -1 2 (g -1 ) jℓ ∂ t g iℓ (the second fundamental form). Then the following identities hold:
where G( (4) g) αβ is the Einstein tensor of (4) g.
Proof. The first three identities can be found in [12,Chapter 6,(3.20)-(3.22)] (after substituting the lapse to be identically 1). The remaining identities follow from the first three by simple algebraic manipulations. □ LEMMA 3.2. Given n ∈ N ∪ {0}, (4) g [n] and k [n] given by Theorem 2.1 satisfy the estimates:
Proof. By point (3) in Theorem 2.1, it follows that
Writing also
we conclude the first estimate using condition (2) in Theorem 1.1 and the second inequality in (3), Theorem 2.1.
For the second estimate, first note that after tracing the first inequality in (3) of Theorem 2.1, we obtain
Combining this with the first estimate in (3.8) that we have just established, we obtain the second estimate in (3.8).
We now turn to the third estimate in (3.8). For notational clarity, we focus on the case |α| = 0. All the higher derivative bounds are be derived analogously after noticing the crucial algebraic structure. We compute:
jℓ )(k [n] ) i ℓ .
(3.9)
Notice now that by (2.80), we have:
Therefore, 1 2 (g [n] ) ℓb (∂ j g
(g [n] ) ℓb ∂ j g
[n]
ib (g [n] ) jc ∂ t g
[n] cℓ
where in order to show that this = O(t -1+ε ), we look at the second term, relabel the indices b ↔ j and then swap c ↔ ℓ (using that g [n] is symmetric), which then gives the negative of the first term.
For the term -1 2 (g [n] ) ℓb ∂ i g
[n]
jb (k [n] ) ℓ j , we first note that if ℓ > j, then (k [n] ) ℓ j = O(t -1-2p j +2p ℓ ) and (g [n] ) ℓb ∂ i g
[n]
jb = O(|log t|), so altogether we get an O(t -1+ε ) contribution. If ℓ < j, then (g [n] ) ℓb ∂ i g
), we get a combined contribution of O(t -1+ε ). We therefore only get the contribution when j = ℓ, i.e.
(3.11) -1 2 (g [n] ) ℓb ∂ i g
[n]
jb (k [n] ) ℓ j = -
jb (k [n] ) j j + O(t -1+ε ).
Combining (3.10) and (3.11), we have
jb (k [n] ) j j + O(t -1+ε ).
(3.12)
Plugging (3.12) into (3.9), using the estimate (2.8), and noting that by symmetry (g [n] ) jb (∂ j g
jℓ ) = 0, we obtain
jb (k [n] ) i ℓ + O(t -1+ε ).
(3.13)
Finally, notice that the second and third terms in (3.13), when j ̸ = b, contribute only O(t -1+ε ). We have thus obtained
ℓℓ (k [n] )
where in the last equality we use condition (2) in Theorem 1.1. The desired estimate now follows by employing condition (4) in Theorem 1.1. □ Combining Proposition 3.1, Lemma 3.2 and Theorem 2.1, we deduce the following bounds for the relevant curvature components of (4) g [n] . PROPOSITION 3.3. Given n ∈ N such that (n + 1)ε > 2 and (4) g [n] as in Theorem 2.1, the following estimates hold:
Proof. In this proof, the implicit constants in ≲ depend on α, n, c ij and p i .
Step 0: Estimates for Ric i j ( (4) g [n] ). According to (3.1) in Proposition 3.1 and the estimates in (2.8), Lemma 2.15 and Proposition 2.17, it follows that (3.17)
which clearly implies in particular the needed estimate for Ric( (4) g [n] ) i j in (3.14).
Also, using (2.8), Lemma 2.15 and Proposition 2.17, we also obtain the estimate (3.15) for ∂ α x ∂ t Ric i j ( (4) g [n] ). It suffices then to show that the estimates for ∂ α
x Ric ti ( (4) g [n] ), ∂ α x Ric tt ( (4) g [n] ) hold true, since all the remaining terms in (3.14) are algebraic combinations of the previous three.
Step 1: Deriving the ODEs. By virtue of the contracted second Bianchi identity we have:
jℓ Ric ti ( (4) g [n] )
jℓ Ric tt ( (4) g [n] )
iℓ Ric j i ( (4) g [n] )
where D denotes the Levi-Civita connection of (4) g [n] , and ∇ j (Ric t ) j ( (4) g [n] ) means that we take (Ric t ) j as a tensor field tangent to the constant-t hypersurfaces and then differentiate with the connection ∇, i.e.
∇ j (Ric t ) j ( (4) g [n] ) = (g [n] ) ij ∂ i (Ric t ) j ( (4) g [n] ) -(g [n] ) ij (Γ [n] ) ℓ ij (Ric t ) ℓ ( (4) g [n] ).
We now estimate the terms in (3.18) and (3.19). For term I in (3.18), we use (2.8) and (2.10) in Theorem 2.1 to obtain (3.20) I = -1 t + O(t -1+ε
) Ric ti ( (4) g [n] ).
The first term in II can be directly estimated by (3.17). To handle the second term in II, we compute using the form of the metric and (2.7) to obtain ∇ j Ric i j ( (4) g [n] ) = ∂ j Ric i j ( (4) g [n] ) -(Γ [n] ) ℓ ij Ric ℓ j ( (4) g [n] )
).
Combining all the above and using (3.17), we obtain
Plugging (3.20) and (3.21) into (3.18) yields 4) g [n] ) + O(t -2+(n+1)ε ).
(3.22) Equation (3.19) can be treated similarly. We subtract III to the LHS, use (2.8) and (2.10) to control the coefficients in IV and VII, keep the term IV, and use (3.15) for term VI so that we obtain
In a similar way, we obtain the equations for higher derivatives analogous to (3.22) and (3.23). After putting in an integrating factor, the equations read
x Ric tt ( (4) g [n] )+O(t -1+(n+1)ε ).
(3.24)
Step 2: Solving the ODEs. We will view the two equations in (3.24) as ODEs in t. In particular we will not be concerned with the loss of derivatives since we have bounds for all order of derivatives of the approximate solutions.
Note that Lemmas 2.15, 3.2 and the identities (3.2), (3.3) imply the estimates:
In particular, this means that the initial data (at {t = 0}) for t∂ α x Ric ti ( (4) g [n] ) and t 2 ∂ α
x Ric tt ( (4) g [n] ) both vanish. Now since |∂ α x Ric ti ( (4) g [n] )| ≲ t -1+ε (for all α), it follows that |∂ α
x ∇ i (Ric t ) i ( (4) g [n] )| ≲ t -3+2ε (for this we simply use that
). Hence, integrating the second equation in (3.24) and using Grönwall's inequality, we obtain
Plugging this estimate into the first equation in (3.24), we then obtain using Grönwall's inequality
Notice that (3.26) and (3.27) improves over (3.25). We now repeat the above argument, but plugging in these improve estimates to obtain (assuming n ≥ 2)
|∂ α x Ric tt ( (4) g [n] )| ≲ t -2+3ε , |∂ α x Ric ti ( (4) g [n] )| ≲ t -1+3ε .
Iterating this argument then gives the desired estimates. (The rate for ∂ α x Ric ti is limited by the last term on the RHS of the first equation in (3.24).) This completes the proof of the proposition. □ 4. Construction of an actual solution. We continue to work under the assumptions of Theorem 1.1 and take g [n] and k [n] as constructed in the beginning of Section 2 (so that for appropriately chosen t N , the estimates in Theorem 2.1 and Proposition 3.3 hold).
The main result of this section will be to prove existence of a solution to a system of reduced equations (to be introduced below in (4.9) of Section 4.1). See Theorem 4.4 for the precise statement of the main result, and see the rest of Section 4.3 for a discussion of the proof of Theorem 4.4 and an outline of the later parts of the section.
Deriving the reduced equations. As already described in Section 1.1.4 in the introduction, we will control k i j using a second-order wave-like equation. In this subsection, we derive the equation that we will use. By (3.1) in Proposition 3.1, if a metric takes the form (1.4), and k is the second fundamental form, then
Taking a ∂ t derivative of (4.1), we obtain
To compute ∂ t Ric(g) i j , we use the variation of Ricci formula (see for example equation (2.31) in [15]) and the fact ∂ t g ij = -2k ij :
where ∆ L is the Lichnerowicz Laplacian (on symmetric 2-tensors) given by
Using again ∂ t g ij = -2k ij , it follows that
We will further analyze two groups of terms on the RHS of (4.2): (1) Denoting G i := G ti (g (4) ) and considering it as a tensor on {t = constant}, we have
(2) In three dimensions, the Riemann curvature tensor can be expressed in terms of the Ricci curvature tensor (see [15, (1.62)]):
where R(g) denotes the scalar curvature of g. Therefore, the terms
can be written as some linear combinations of contractions of Ric(g) and k. Using again (4.1), we can replace Ric(g) i j by Ric( (4) g) i j + ∂ t k i j -k ℓ ℓ k i j . It therefore follows that the second fundamental form k verifies the following equation:
where
We note that the terms k ⋆ k ⋆ k and
In particular, if (4) g solves the Einstein vacuum equations, then
The equation (4.8) is almost a wave equation for k, except that there is a top order ∇ i ∇ j k ℓ ℓ term on the RHS. To proceed we think of h = k ℓ ℓ as an independent variable. If the Einstein vacuum equations were satisfied, then (3.3) in Proposition 3.1 imposes that ∂ t h = |k| 2 . It is therefore reasonable to look for a solution to the Einstein vacuum equations by solving the following coupled system of equations:
Remark that given a solution to (4.9), it follows that g -1 satisfies (4.10)
Our strategy will be to solve the system (4.9) and then a posteriori justify that it is indeed a solution to the Einstein vacuum equations (in Section 5).
Before we proceed, we introduce some notations. In the following we will consider (at least) two spacetime metrics (4) g = -dt 2 + g ij dx i dx j and (4) g [n] = -dt 2 + g We make the following definitions assuming we are given such I t , (4) g and (4) g [n] . Definition 4.1 (Constant-t hypersurfaces). Given t ∈ I t define 1) Denote by ∇ the Levi-Civita connection of g, and by ∇ [n] the Levi-Civita connection of g [n] .
(2) Denote ∇ (d) := ∇ -∇ [n] . Remark that ∇ (d) is a (1, 2)-tensor.
(3) Let r ∈ N and T be an (m, l)-tensor. Define ∇ (r) T to be the (m, l + r)tensor given by (∇ (r) T )
Definition 4.3 (Norms). ( 1) Given two rank (m, l) tensors T (1) and T (2) , define the inner product ⟨T (1) , T (2)
(2) Given a rank (m, l) tensor T , define
(3) Given a tensor T and p ∈ [1, +∞), define
, where vol Σ t = √ det g dx is the volume form induced by the metric g. For p = +∞, define
(4) For r ∈ N ∪ {0} and p ∈ [1, +∞], define the geometric Sobolev space
(5) For r ∈ N ∪ {0} and p ∈ [1, +∞], define the homogeneous geometric Sobolev space ∥T ∥ Ẇ r,p (Σ t ,g) := ∥∇ (r) T ∥ L p (Σ t ,g) .
(6) For r ∈ N ∪ {0}, define H r (Σ t , g) := W r,2 (Σ t , g), Ḣr (Σ t , g) := Ẇ r,2 (Σ t , g).
(7) Define the norm t α L 2 (Σ t , g) (for α ∈ R \ {0}) by
(8) Given any two Banach spaces X and Y , the vector spaces X + Y = {x + y : x ∈ X, y ∈ Y } and X ∩ Y are endowed with Banach space structures with norms ∥v∥ X+Y := inf v=x+y, (x,y)∈X×Y (∥x∥ X + ∥y∥ Y ), ∥v∥ X∩Y := ∥v∥ X + ∥v∥ Y .
(9) Finally, define L p (Σ t , g [n] ), W r,p (Σ t , g [n] ) and Ẇ r,p (Σ t , g [n] ) etc. as above but with g replaced by g [n] (and ∇ replaced by ∇ [n] ).
Existence of solutions to (4.9) and the main steps of the proof. Our first step of the proof of Theorem 1.1 is to build a solution to (4.9). The following is the main existence result for (4.9), whose proof will occupy the remainder of the section. THEOREM 4.4. For every s, N 0 ∈ N obeying s ≥ 5, there exists n N 0 ,s ∈ N sufficiently large such that for any n ≥ n N 0 ,s , there exist T N 0 ,s,n > 0 sufficiently small and a solution (g, h, k) to (4.9) in (0, T N 0 ,s,n ]×T 3 which satisfy the following estimates:
where
Moreover, k ij = g ℓj k i ℓ is symmetric in i and j.
We will prove Theorem 4.4 with the following steps (see the conclusion of the proof in Section 4.3.3):
(1) For T aux > 0 (with T aux ≪ T N 0 ,s,n ), we construct local solutions to (4.9) in [T aux , T aux + δ) × T 3 (with δ potentially depending on T aux ) (Lemma 4.5).
(2) For s, N , n and T N 0 ,s,n as in Theorem 4.4, we prove uniform estimates to show that the solution can be extended to [T aux , T N 0 ,s,n ]. This is carried out in a bootstrap argument and is the main step (Theorem 4.6, Corollary 4.7).
(3) Using a compactness argument, we take a sequence of auxiliary times (T aux ) i → 0 + and extract a subsequence of solutions converging to a limiting solution to (4.9) on (0, T N 0 ,s,n ] × T 3 (Proposition 4.8).
We will further elucidate these steps in the subsubsections below. Most of the proofs will then be given in later subsections.
Step 1: A local solution. We begin with the following local existence result for (4.9): LEMMA 4.5 (Local existence). For every T aux > 0 sufficiently small and n ∈ N, there exist a δ > 0 (depending a priori both on T aux and n) and a unique smooth solution (g aux , k aux , h aux ) to (4.9) in [T aux , T aux + δ] × T 3 , such that at t = T aux , (g aux , k aux , h aux ) attains the following prescribed values:
Moreover, g aux ij = g aux ji .
Such a local existence result is almost standard. The only issue is that the second equation of the system (4.9) contains the term ∇ i ∇ j h on the RHS, which seems to "have one derivative too many". This issue can be treated by deriving elliptic estimates for h, by commuting ∂ t h = |k| 2 with ∆ g and using the wave equation for k, see discussions in Section 1.1.4 and Lemma 4.34. We will use this result but will omit its straightforward proof.
Once existence is obtained, since g aux ij is symmetric at t = T aux and ∂ t (g aux ijg aux ji ) = 0, it immediately follows that g aux ij = g aux ji .
The main bootstrap argument. Our next step is to prove a uniform time of existence independent of T aux . To state the result, let us define, for (g aux , k aux , h aux ) as in Lemma 4.5,
ij , ((g -1 ) (d) ) ij := ((g aux ) -1 ) ij -((g [n] ) -1 ) ij , (4.12)
We stipulate that the metric g aux ij takes the form (1.4) and define a aux ij according to (1.4).
Introduce the following bootstrap assumptions:
The following is the main bootstrap theorem, whose proof constitutes most of this section (in Sections 4.4-4.8): THEOREM 4.6 (Bootstrap theorem). For every s, N 0 ∈ N such that s ≥ 5, there exists n N 0 ,s ∈ N sufficiently large such that for every n ≥ n N 0 ,s , the following holds for some T N 0 ,s,n > 0 sufficiently small. Suppose (g aux , k aux , h aux ) is the solution to (4.9) on a time interval [T aux , T Boot ) (for some T Boot ∈ (T aux , T N 0 ,s,n ]), with initial data at t = T aux given as in Lemma 4.5. Assume moreover that the bootstrap assumptions (4.14)-( 4.17) all hold on [T aux , T Boot ) × T 3 .
Then in fact the following estimates hold:
, where C > 0 may depend on s, N 0 and the data, but is independent of T aux . Moreover, taking T N 0 ,s,n smaller if necessary, (4.18) improves over the bootstrap assumptions (4.14)-(4.17).
As is standard, the bootstrap theorem implies immediately, using a continuity argument, that the solution can be extended up to time T N 0 ,s,n : COROLLARY 4.7. Let s, N 0 , n and T N 0 ,s,n be as in Theorem 4.6. Then the local solution given in Lemma 4.5 can in fact be extended to all of [T aux , T N 0 ,s,n ) × T 3 . Moreover, the estimates (4.18) hold.
PROPOSITION 4.8. Let s, N 0 , n and T N 0 ,s,n be as in Theorem 4.6. Them there exists a decreasing sequence of auxiliary times {T aux,I } +∞ I=1 ⊂ (0, T N 0 ,s,n ), lim I→+∞ T aux,I = 0 such that the following holds:
(1) The corresponding solutions {(g aux I , k aux I , h aux I )} +∞ I=1 given by Lemma 4.5 converge locally in C 3 × C 2 × C 2 (as I → +∞) to a limit (g, k, h).
(2) The limit, which we denote by (g, k, h), solves (4.9) in (0, T N 0 ,s,n ] × T 3 .
(
, the estimate (4.18) holds.
(4) The limit (g, k) satisfies
The proof of Proposition 4.8 will be given in Section 4.9.
Proof of Theorem 4.4. The limiting solution given by Proposition 4.8 satisfies all the conclusions of Theorem 4.4. This thus concludes the proof of Theorem 4.4. □ 4.4. Definition of the energies and an outline of the proof of Theorem 4.6. From now on until the end of Section 4.8, we focus on the proof of Theorem 4.6. To lighten our notations, in these sections we write g = g aux , a = a aux , h = h aux and k = k aux .
The crux of our proof of Theorem 4.6 is to bound an appropriate energy, which we define now.
Define the energy
Define also the modified energy
where
ren g (d) and ∇ (s+1) ren (g -1 ) (d) are the renormalized top-order quantities defined by
We remark explicitly that the modified energy and the energy differ by the following:
• The energy controls the ∇ (r) derivative of ∂ t k (d) while the modified energy controls the ∂ t derivative of ∇ (r) k (d) .
• The modified energy only controls h (d) , g (d) and (g -1 ) (d) up to s derivatives; at the top order it only controls the renormalized top-order quantities.
Since the proof will take several subsections, we give an outline of the strategy for proving Theorem 4.6.
• In Section 4.5, we begin with some preliminary estimates.
• In Section 4.6, we carry out the energy estimate for k (d) using the wave equation it satisfies.
• In Section 4.7, we carry out the energy estimates for h (d) , g (d) and (g -1 ) (d) using the transport equations they satisfy. Combining the results in Sections 4.6 and 4.7, we will obtain an estimate of the modified energy E s by the energy E s .
• In Section 4.8, we complete the proof of Theorem 4.6. The main ingredient is to control E s and E s using energy estimates, and the close everything using the Fuchsian ideas as illustrated in Section 1.1.1.
Before we proceed, we make some important remarks regarding the dependence of constants throughout the proof of Theorem 4.6.
From now on fix s ∈ N with s ≥ 5 as in Theorem 4.6. We will use C 0 and C n as general positive constants. They may change from line to line. Both C 0 and C n may depend on the data c ij , p i and also s, but importantly C n may depend on n while C 0 is not allowed to depend on n.
We always assume without loss of generality that T N 0 ,s,n ≤ 1.
Another important remark regarding the proof of the bootstrap argument is that (despite the notation) we do not know that k is the second fundamental form of the constant-t hypersurfaces. (In particular, we do not know that g ℓ[i k j] ℓ = 0.) In fact, it is only after extracting a limit in Proposition 4.8 that we know that the limiting k is an honest second fundamental form.
In this subsection we work under the assumptions of Theorem 4.6. In particular, we assume the validity of the bootstrap assumptions (4.14)-(4.17).
LEMMA 4.9. The following pointwise estimate holds for all scalar functions f on (0, T Boot ):
To get the desired estimates, we just use a very wasteful estimate that C -1
which follows directly from (4.14) and computations as in (2.15)). □ LEMMA 4.10 (Sobolev embedding). The following holds for every (m, l) Σtangent tensor T :
In particular, these inequalities imply
Proof. We first prove the inequalities (4.24) for scalar functions f . Using the Sobolev embedding for T 3 in coordinates, it follows that
where in the penultimate inequality we used Lemma 4.9 and in the last inequality we have used C -1 0 t -1 vol Σ ≤ dx ≤ C 0 t -1 vol Σ (which follows from the bootstrap assumption (4.14)).
For the second inequality in (4.24) for a scalar function f , we proceed similarly to obtain
(4.28)
Now given a general (m, l) tensor T , using (4.27) and (4.28) with f α = |T | 2 g + α 2 (α > 0) and taking α → 0, we obtain the desired inequalities in (4.24). Next, it is easy to see that (4.24) implies (4.25). Finally, by (4.25), and the fact s -ε > 2,
which is (4.26). □
We will also need to compare norms with respect to g and with respect to the trivial metric 3 i=1 (dx i ) 2 .
LEMMA 4.11. Given any rank (l, m) Σ-tangent tensor T ,
(4.29)
Proof. Using the form of the metric (1.4), the bound (2.7) on a [n] , and the fact
(4.30)
This proves (4.29) when there are no derivatives (i.e., k = 0). Define the flat connection ∇ (flat) to be Levi-Civita connection associated to 3 i=1 (dx i ) 2 , i.e.
Then, since p j < 1 -ε < 1 -ε 2 , (4.30) gives
(∂Γ [n] )T + (Γ [n] ) • • • (Γ [n] )
where we have suppressed the indices in terms where the exact contractions do not matter.
Our goal is to show that in the
norm, each term in (4.32) can be bounded above by the RHS of (4.29). By repeated application of the Cauchy-Schwarz inequality (with respect to g [n] ), and using (4.31), it suffices to prove (4.33)
which is the goal for the remainder of the proof. We first make the easy observation that
We now compute
We can pair up g [n] and (g [n] ) -1 with a common index and conclude that this expression is ≤ C n |log t| k+1 . All other terms are similar. Hence, we obtain
which is exactly (4.33). □ LEMMA 4.12. For r ≤ s -2,
For r ≤ s,
Proof. Note that
The conclusion is then an immediate consequence of Hölder's inequality and the bootstrap assumptions (4.15) and (4.16). □ LEMMA 4.13. For r ≤ s -1,
Proof. This follows from the bootstrap assumption (4.15) and Lemma 4.12. □ LEMMA 4.14. For r ≤ s + 1,
) .
Proof. This follows from the bootstrap assumptions (4.15), (4.16) and Lemma 4.12. □ Note that Lemma 4.13 fails when r = s, s + 1 as we do not control ∥g (d) ∥ W r,∞ (Σ t ,g) . On the other hand, Lemma 4.14 by itself will not be sufficient for our purpose. Instead we need the following: LEMMA 4.15. The following holds for any α > 0:
) ),
and
) ).
Proof. The main difference with Lemma 4.14 is that we may have terms which involve s or s + 1 derivatives of g (d) .
We first consider the term ∇ (s) T . When writing ∇ (s) T in terms of (∇ [n] ) (s) T , there is the term
(meaning s -1 ∇ derivatives acting on the tensor ∇ (d) ), together with other terms which are lower order and can be handled directly using the bootstrap assumptions (4.15) and (4.16). This term cannot be bounded in L ∞ , and will instead be controlled in L 2 . For this we note that by Hölder's inequality, Lemma 4.12, and the bootstrap assumptions (4.15), (4.16),
) .
This gives the first inequality in the statement of the lemma. The term ∇ (s+1) T is similar except for an additional derivative. Indeed, we need to control the terms
Both of these can be controlled in L 2 (Σ t , g) using Hölder's inequality, Lemma 4.12, and the bootstrap assumptions (4.15), (4.16) as above. □ 4.5.2. An easy consequence of the bootstrap assumption.
LEMMA 4. 16.
Proof. This follows from Lemma 4.10 (Sobolev embedding) and the bootstrap assumption (4.17). □ 4.5.3. Estimates for background quantities.
PROPOSITION 4.17. For each n ∈ N, define
Given any N ∈ N, there exists n N,s ∈ N sufficiently large such that whenever n ≥ n N,s ,
Proof. By Lemma 4.11, it suffices to show that for any given polynomial rate, n can be chosen sufficiently large so that I h [n] , (I k [n] ) i j and their coordinate derivatives tend to 0 faster than the given polynomial rate.
Step 1: Proving the estimates for I h [n] . Recall that by definition h [n] = (k [n] ) ℓ ℓ . By Proposition 3.3, (2.10) and the expression for Ric tt ( (4) g) in (3.3), it follows that given any polynomial rate in t, we can choose n ∈ N sufficiently large so that
| 2 and its coordinate derivatives go to 0 faster than the given polynomial rate in t.
Step 2: Proving the estimates for I k [n] . By (4.5) and (2.10),
)
where (G [n] ) i = Ric( (4) g [n] ) ti and L n is linearly increasing in n. By Proposition 3.3, given any polynomial rate in t, we can choose n ∈ N sufficiently large so that the terms
)
)
and their coordinate derivatives go to 0 faster than the given polynomial rate in t. □ PROPOSITION 4.21. The following estimates hold for k:
Moreover, the above estimates hold both when k is replaced by k [n] and k (d) .
Proof. That the estimates hold for k [n] follows from Propositions 4.18, 4.19 and 4.20. That the estimates hold for k (d) follows from (4.17) and Lemma 4.16.
Finally, since k = k [n] + k (d) , the estimates also hold for k. □ PROPOSITION 4.22. For any n ∈ N,
and
Proof. Recalling that we set h [n] = (k [n] ) ℓ ℓ , this can be proven in the same way as Propositions 4.18 and 4.19. □ PROPOSITION 4.23. The following estimates hold for Ric(g):
Proof. For simplicity, we write in this proof Ric = Ric(g), Ric [n] = Ric(g [n] ) and similarly for the Riemann curvature tensor.
First, notice that by Lemma 4.11 and (2.9), it follows that
As a result, Lemmas 4.13 and 4.15 imply that all the desired estimates when Ric is replaced by Ric [n] . It thus remains to estimate the difference Ric -Ric [n] . We will bound the difference of the full Riemann curvature tensor; the bounds for the Ricci curvature tensor of course follow immediately. We compute
Combining this with (4.34) and the bootstrap assumptions (4.15) and (4.16), it is easy to see that Riem -Riem [n] can be controlled by
This concludes the proof of the proposition. □ 4.5.4. Commutator estimates. We will often use the commutator formula between the Lie derivative in ∂ t and covariant derivatives in the spatial directions: PROPOSITION 4.24. The following commutation formula holds for any (m, l) Σ-tangent tensor T :
On the other hand, we compute using (4.9) and (4.10) that
Combining these computations yields the desired formula. □ PROPOSITION 4.25. Let T be an (m, l) Σ-tangent tensor. For 0 ≤ r ≤ s -
Consequently, for 0 ≤ k ≤ s, iterated commutators can be bounded as follows:
Finally, if T is a scalar function, then in fact (4.38) holds for 0 ≤ k ≤ s + 1.
Proof.
Step 1: Proof of (4.37). Using Proposition 4.24, we have the estimate
We estimate each of the terms in (4.39). Using the estimates in Proposition 4.21, the first term in (4.39) can be bounded above as follows:
Before handling the second term in (4.39), we first note make the following observations on the numerology:
• When r 1 > s -3, since we have r 1 + r 2 = r ≤ s -1, we have either r 2 = 0 or r 2 = 1. In particular, r 2 + 2 ≤ r. We can thus bound the second term in (4.39) using Proposition 4.21, (4.26) and the above observations as follows:
where the very last estimate follows simply after relabelling. Combining (4.40) and (4.41) yields (4.37).
Step 2: Proof of (4.38). When 0 ≤ k ≤ s, we compute using the triangle inequality and (4.37) to obtain
This yields (4.38).
Finally, for a scalar function f , [∂ t , ∇ i ]f = 0. Hence, in (4.42), we sum only up to r = k -1. As a result, we can take up to k = s + 1. This gives the desired improvement for scalar functions. □ 4.5.5. Estimates for general equations.
Let T be an (m, l) Σ-tangent tensor. Then
In particular,
Proof. We first note that by (4.9)
We will apply (4.45
The pointwise inequality (4.47) implies (4.43) immediately after integrating over Σ t , using (4.45), and applying again the estimates in Proposition 4.21. Finally, to derive (4.44), we simply note that by the Cauchy-Schwarz inequality,
PROPOSITION 4.27 (Energy estimates for wave equations). Let T be an (m, l) Σ t -tangent tensor such that (-∂ 2 t +∆ g )T =F for some (m, l) Σ t -tangent tensor F .
Then
Proof. Denote by E terms bounded by
Step 1: Controlling the first order terms. Applying (4.43) in Proposition 4.26, integrating by parts and using the Cauchy-Schwarz inequality,
where we have used that by Hölder's inequality and the following commutator estimate (which uses Proposition 4.25)
Step 2: Controlling the zeroth order term. It remains to control the zeroth order term ∥T ∥
2 L 2 (Σ t ,g) . For this we simply use Proposition 4.26, and then use (4.48) to obtain
Summing (4.48) and (4.49), we obtain the desired estimate. □ 4.6. Energy estimates for the wave equation for k. In this subsection we continue to work under the assumptions of Theorem 4.6. In particular, we assume the validity of the bootstrap assumptions (4.14)-(4.17).
We insert (4.13) into (4.9) to obtain evolution equations for the difference (k (d) ) i j :
where the terms (k ⋆ k ⋆ k), (∂ t k ⋆ k) are as defined in (4.6), (I k [n] ) i j is as defined in Proposition 4.17, and B i j denotes the following terms:
The following is the main energy estimates for k (d) :
PROPOSITION 4.28. Given N ∈ N, let n ∈ N be sufficiently large so that the estimates in Proposition 4.17 hold. Then
where, as before, we have used the notation
Proof. For 0 ≤ r ≤ s-1, we differentiate (4.50) by ∇ (r) to obtain the following wave equation for ∇ (r) k (d) :
For every 0 ≤ r ≤ s -1, our goal is to show that
after which we will apply Proposition 4.27.
The proof of (4.53) will be achieved in Steps 1-5 below in which we bound each term on the RHS of (4.52).
Step 1: Bounding the inhomogeneous terms. For 0 ≤ r ≤ s -1, by Proposition 4.17, (4.54)
Step 2: Bounding the terms in B i j . Recall from (4.51) that B i j consists of h terms and k terms. We first compute the exact form of the h terms:
(4.55) From (4.55), the triangle inequality and Hölder's inequality, it follows that
Term I can be directly estimated by the definition of E s (t):
By Proposition 4.22, Lemma 4.12, and the definition of E s (t), we have
s (t).
(4.58)
For term IV, we use Proposition 4.22 and the definition of E s (t) to obtain
s (t).
(4.59)
For the k terms in B i j , we compute
The terms in (4.60) are similar to those in (4.55) (with k taking the place of h) except-importantly-that (4.60) does not contain second derivative terms of k (d) . This is important because while our energy controls up to s + 1 derivatives of h (d) , it only controls up to s derivatives of k (d) . Other than this difference, the remaining terms in (4.60) can in fact be controlled very similarly as those in (4.55). We will therefore omit the details and simply give the final estimate:
Step 3: Bounding the difference of the nonlinear terms. In this step we control the H r norm (for 0
We begin with k ⋆ k ⋆ k -k [n] ⋆ k [n] ⋆ k [n] . For 0 ≤ r < s -1, we use Hölder's inequality, Proposition 4.21, and the definition of E s (t) to obtain
where we have used the shorthand
When r = s -1, we have terms as in (4.62) which can be controlled similarly, but also the following extra term, which we in addition use Sobolev embedding in
s (t).
(4.63)
We now turn to
⋆ k [n] . For 0 ≤ r < s -2, we use Hölder's inequality, Proposition 4.21, and the definition of E s (t) to obtain
(4.64) For r = s -2, we have an additional term when all derivatives hit on ∂ t (k [n] , k (d) ) so that we cannot put it in L ∞ . For this term we use Proposition 4.21 and (4.26) to obtain
s (t).
(4.65) For r = s -1, we have additionally (compared to (4.64) and (4.65)) terms where (1) all but one derivatives hit on ∂ t (k [n] , k (d) ), (2) all derivatives hit on (k [n] , k (d) ), both of which cannot be put into L ∞ . For these terms we use Proposition 4.21 and (4.26) to get
s (t).
(4.66)
Step 4: Bounding the commutator terms [-∂ 2 t + ∆ g , ∇ (r) ]. By (repeated applications of) Proposition 4.24, [-∂ 2 t , ∇ (r) ]k (d) consists exactly of terms of the form
r 1 +r 2 +r 3 =r ∇ (r 1 ) k∇ (r 2 ) k∇ (r 3 ) ∂ t k (d) .
Thus they can be controlled in exactly the same manner as in Step 3 to obtain
s (t).
On the other hand, the commutator [∆ g , ∇ (r) ] gives rise to curvature terms. In the 3-dimensional Σ t , the Riemann curvature tensor can be expressed in terms of the Ricci curvature and thus can be controlled using Proposition 4.23 to obtain
Step 5: Putting everything together. Combining Steps 1-4, we have achieved (4.53).
Therefore, for every 0 ≤ r ≤ s -1, we apply Proposition 4.27 with
where in the last line we have controlled [t - 2N -2s+2r+2
Ḣr (Σ t ,g) )] using the energy and have used (4.53).
Summing over s-1 r=0 , we obtain the desired estimate. □ 4.7. Transport estimates. In this subsection we continue to work under the assumptions of Theorem 4.6. In particular, we assume the validity of the bootstrap assumptions (4.14)-(4.17).
We prove in this subsection estimates for h (d) , g
ij , ((g -1 ) (d) ) ij , which are all derived using the transport equations they obey.
We insert (4.12) and (4.13) into (4.9) to obtain evolution equations for the differences g d) , and ((g -1 ) (d) ) ij :
We begin with the more straightforward, less than top-order, estimates for h (d) , g
ij , ((g -1 ) (d) ) ij . Commuting the equations (4.69), (4.70) and (4.71) with ∇ r , r ≤ s, we obtain:
.74) We use (4.72)-(4.74) to obtain the following estimates. PROPOSITION 4.29. Given N ∈ N, let n ∈ N be sufficiently large so that the estimates in Proposition 4.17 hold. Then d dt t -2N -2s s r=0 t 2r ∥h (d) ∥ 2 Ḣr (Σ t ,g) + 2N t t -2N -2s s r=0 t 2r ∥h (d) ∥ 2 Ḣr (Σ t ,g) ≤ (C 0 t -1 + C n t -1+ε )t -2N -2s E s (t) + C n t, (4.75) and d dt t -2N -2s s r=0 t 2r-2 (∥g (d) ∥ 2 Ḣr (Σ t ,g) + ∥(g -1 ) (d) ∥ 2 Ḣr (Σ t ,g) ) + 2N t t -2N -2s s r=0 t 2r-2 (∥g (d) ∥ 2 Ḣr (Σ t ,g) + ∥(g -1 ) (d) ∥ 2 Ḣr (Σ t ,g) )
≤ (C 0 t -1 + C n t -1+ε )t -2N -2s E s (t).
(4.76)
Proof. We will only prove (4.75); the bound (4.76) can be derived similarly (and is slightly simpler).
Applying Proposition 4.26 for T = ∇ (r) h (d) (0 ≤ r ≤ s) and M = 2N + 2s -2r, it suffices to show that (4.77)
To prove this we consider each term on the RHS of (4.72). First, by Hölder's inequality, Proposition 4.21 and (4.26), we obtain
Next, the inhomogeneous term I h [n] can be bounded using Proposition 4.17 by
Finally, by Proposition 4.25,
s (t). We next turn to the top order derivative estimates for h (d) , g (d) and (g -1 ) (d) . For this we first control the renormalized top-order quantities introduced in (4.21)-(4.23). (Subsequently we will show using elliptic estimates that the renormalized top-order quantities indeed control all top-order derivatives; see already Lemma 4.34.) PROPOSITION 4.30. Given N ∈ N, let n ∈ N be sufficiently large so that the estimates in Proposition 4.17 hold. Then
Step 1: Proof of (4.81). The main difference with the estimates in Proposition 4.29 is that there can potentially be (s + 1) derivatives of k (d) , which is not controlled by our energy E s (t). The quantity ∇ (s+1) ren h (d) is in fact designed exactly to avoid such terms after using the bounds for the wave equation for k (d) .
We begin our computations. First, (4.83)
where the error terms have at most s derivatives hitting on k (d) and thus satisfy the estimates similar to that in the proof of Proposition 4.29 (and their proofs are therefore omitted):
, however, cannot be controlled. Nevertheless, continuing our computations, we see that
(4.85)
Note that this generates a term (k
) j i which can be used to cancel the uncontrollable term in (4.83). Hence, combining (4.83), (4.84) and (4.85), we obtain
.86) We now handle to two terms in (4.86). By Proposition 4.21, (4.53) and Hölder's inequality,
On the other hand, by Proposition 4.21 and Hölder's inequality,
s (t).
ren h (d) is defined exactly to be (recall (4.21))
we thus obtain
The desired estimate (4.81) then follows directly from Proposition 4.26 (for M = 2N + 2s -2(s + 1)).
Step 2: Proof of (4.82). The main idea is similar to Step 1, so we will be brief. The main difference is that for g (d) , not only the derivatives of the inhomogeneous terms create ∇ (s+1) k, but the commutator terms also create similar terms, which have to be taken care of by a renormalization. More precisely, by (4.70) and Proposition 4.24,
where the terms denotes by • • • have at most (s + 1) derivatives on k [n] , at most s derivatives on k and at most (s + 1) derivatives on g (d) , and therefore can be bounded as in Proposition 4.29 by
It thus remains to handle all the main terms appearing on the RHS of (4.90). Now one observes that the quantity ∇
ren g (d) is designed exactly to remove this term (in a similar way as ∇ (s+1) ren h (d) is designed in Step 1) so that the additional error terms are controllable. It thus follows that (4.91)
which implies the desired estimate for ∇ (s+1) ren g (d) in (4.82) after using Proposition 4.26.. The argument for ∇ (s+1) ren (g -1 ) (d) is similar and omitted. □
We conclude this subsection by summarizing what we have achieved so far, namely that we have obtained an estimate for the modified energy by the energy: PROPOSITION 4.31. Given N ∈ N, let n ∈ N be sufficiently large so that the estimates in Proposition 4.17 hold. Then for any t ∈ [T aux , T Boot ),
Proof. This is an immediate consequence of Propositions 4.
28, 4.29 and 4.30. □ 4.8. Conclusion of the proof of Theorem 4.6. In order to conclude the proof of Theorem 4.6, we finally need to relate E s and E s (which will be achieved in Lemmas 4.32 and 4.34), and then use the energy inequality in Proposition 4.31 to deduce our desired estimates. Recalling now the difference between E s and E s (as described immediately after their definitions in (4.19)-(4.23)), we need to • relate ∂ t ∇ (r) k (d) and ∇ (r) ∂ t k (d) (achieved using a commutator estimate; see Lemma 4.32), and • relate the renormalized top-order quantities and other top-order derivatives (achieved using elliptic estimates; see Lemma 4.34). LEMMA 4.32. The following estimate holds:
Proof. We control the commutator [∂ t , ∇ (r) ]k (d) using Proposition 4.25 to obtain
from which the desired estimate follows from the definition of E s . □ LEMMA 4.33. Given any tensor ξ tangential to Σ t ,
Proof. We compute
where terms labelled error (different in the two instances) come from commuting covariant derivatives and obey an estimate
As a consequence, since on the 3-dimensional Σ t , Riem(g) can be expressed in terms of Ric(g), we can use Hölder's inequality and Proposition 4.23 to obtain
which implies (4.92) after using Young's inequality and absorbing
to the LHS. □ LEMMA 4.34. The top order part of the energy for h (d) , g (d) , (g -1 ) (d) is bounded by:
Proof. The key is to use Lemma 4.33. Consider for instance h (d) . We first note that ∆ g ∇ (s-1) h (d) can be written as a linear combination of the renormalized toporder quantity ∇ (s+1) ren h (d) and terms which has at most s derivatives of k (d) (and k [n] ) so that it can be checked that
It then follows by the elliptic estimates in Lemma 4.33 and the lower order control for h (d) by E s (t) that
The estimates for the top-order derivatives for g (d) and (g -1 ) (d) are similar. □ Combining Lemmas 4.32 and 4.34, we obtain: PROPOSITION 4.35. Given N ∈ N, let n ∈ N be sufficiently large so that the estimates in Proposition 4.17 hold. Then for any t ∈ [T aux , T Boot ),
We are now ready to conclude the proof of the bootstrap theorem (Theorem 4.6):
Proof of Theorem 4.6. Given any N ∈ N, choose n ∈ N sufficiently large so that the estimates in Proposition 4.17 hold.
Combining Propositions 4.31 and 4.35, and integrating in t (noting that we have trivial data at T aux ), we obtain that
We now choose our constants. First choose N sufficiently large so that N ≥ max{2C 0 (C 0 + 1), 2(C 0 + 1), N 0 , 7}.
We then fix an n N 0 ,s ∈ N sufficiently large so that whenever n ≥ n N 0 ,s , (4.95) holds with the given N . After fixing n, we then choose T N 0 ,s,n so that C n T ε N 0 ,s,n ≤ 1. Plugging C 0 ≤ N 2(C 0 +1) and C n T ε N 0 ,s,n ≤ 1 into (4.95), we then obtain
Notice that we have chosen N so that
dτ from both sides of (4.96) to obtain (4.97)
, which immediately implies (4.98) E s (t) ≤ t 2N +2s , after choosing T N 0 ,s,n smaller if necessary. In particular, since we have chosen N ≥ N 0 and T N 0 ,s,n ≤ 1, we obtain (4.18). Finally, we check that we have improved the bootstrap assumption. For (4.15)-(4.17), this is immediate from (4.18). For (4.14), note that (4.98) and (4.25) imply ∥g
+2s-11 . Now since N ≥ 7 and s ≥ 4, we have |a ij -a [n] ij | ≤ C 0 t 3 2 . Combining with (2.7), we thus obtain (4.99) |a ij -c ij | ≤ C 0 t ε , which improves over (4.14) after taking T N 0 ,s,n to be sufficiently small. □ As we discussed in Section 4.3.2, once we have proven Theorem 4.6, we now also obtain Corollary 4.7. 4.9. Extracting a limit: proof of Proposition 4.8. In this final subsection, we prove Proposition 4.8, which, as indicated in Section 4.3.3, is the final step of the proof of Theorem 4.4.
We begin with some easy estimates, which will allow us to extract a limit. (Notice that these estimates are allowed to degenerate as t → 0, but importantly they do not depend on T aux .) LEMMA 4.36. Let s, N 0 , n and T N 0 ,s,n be as in Theorem 4.6. For every T ′ , T ′′ satisfying 0 < T ′ < T ′′ ≤ T N 0 ,s,n , there exists a constant C > 0 independent of T aux such that the following holds (with definitions in (4.12) and (4.13)).
Let T aux ∈ (0, T ′ ] and suppose (g aux , k aux , h aux ) is the solution to (4.9) on [T aux , T N 0 ,s,n ) × T 3 given by Corollary 4.7. Then
Proof. When there is no ∂ t derivative, this just follows from (4.18) and Sobolev embedding. To obtain the estimates with the ∂ t derivatives, we use in addition the equations (4.50), (4.69), (4.70) and (4.71). □ LEMMA 4.37. Let s, N 0 , n and T N 0 ,s,n be as in Theorem 4.6. There exists a sequence of auxiliary times {T aux,I } +∞ I=1 ⊂ (0, T N 0 ,s,n ), lim I→+∞ T aux,I = 0 such that the corresponding solutions {(g aux I , k aux I , h aux I )} +∞ I=1 given by Lemma 4.5 converge locally in C 3 × C 2 × C 2 norm (as I → +∞) to a limit (g, k, h) which solves (4.9) in (0, T N 0 ,s,n ] × T 3 . Moreover, after denoting g
, the estimate (4.18) holds.
Proof. The existence of a limit follows from Lemma 4.36, the Arzela-Ascoli theorem, and a standard argument extracting a diagonal sequence. Since the limit is achieved locally in C 3 × C 2 × C 2 , it follows that the limit satisfies the system (4.9).
Finally, we prove that the limit obeys the estimate (4.18). First, note that the estimate (4.18) implies that for every t, there is a subsequence {T aux,I ℓ } +∞ ℓ=1 for which {(g aux I ℓ , k aux I ℓ , h aux I ℓ )} +∞ ℓ=1 has a weak limit satisfying (4.18). This limit must coincide with the local C 3 × C 2 × C 2 limit, thus showing the bound (4.18). □
The very final statement we need in order to complete the proof of Proposition 4.8 is that g jj ′ k i j ′ is symmetric in i and j. The key to such a statement is the following lemma. LEMMA 4.38. Suppose (g, k, h) solves (4.9). Then the term (g jj ′ k i j ′ -g ij ′ k j j ′ ) satisfies an inhomogeneous wave equation of the following form:
where X, Y and Z are some tensor products of g, g -1 and δ.
Proof.
Step 1: Easy reductions. First, a direct computation shows that
Notice that all the terms on the first line are of the form as required by the lemma. It thus follows from (4.9) that
where • • • denotes terms which are of the form as required by the lemma. (Notice in particular that the Hessian of h term drops of because it is symmetric.)
Investigating now the terms in k ⋆ k ⋆ k and ∂ t k ⋆ k, we only need to check that
where
is of the form required by the lemma. (Note that the term [(∂ t k i a )k a j ′ -(∂ t k a j ′ )k i a ] comes from combining terms in
For Q j ′ i = k i a k a j ′ , we compute
which is of the desired form. For Q j ′ i = ∂ t k i j ′ , we compute
which is of the desired form. For Q j ′ i = [(∂ t k i a )k a j ′ -(∂ t k a j ′ )k i a ], we compute [(∂ t k i a )k a j ′ -(∂ t k a j ′ )k i a ]g jj ′ -[(∂ t k j a )k a j ′ -(∂ t k a j ′ )k j a ]g ij ′
= ∂ t (k i a g j ′ a -k j ′ a g ia )k j j ′ + ∂ t (k j ′ a g ja -k j a g j ′ a ) + ∂ t k i a (k a j ′ g jj ′ -k j j ′ g aj ′ )
which is of the desired form. This concludes the proof. □
We are now ready to show that g jj ′ k i j ′ is symmetric in i and j.
LEMMA 4.39. Given a limit (g, k, h) as in Lemma 4.37, the limiting k is in fact the second fundamental form, i.e., k ij := g jj ′ k i j ′ = -1 2 ∂ t g ij .
Proof. Denoting k ij := g jj ′ k i j ′ , the equation for g implies that ∂ t g ij = -k ijk ji . Hence, in order to prove the lemma, it suffices to show that k ij is symmetric in i and j.
To this end, we define (k aux I ) ij := (g aux I ) jj ′ (k aux I ) i j ′ , and first obtain an estimate for its anti-symmetric part. By Lemma 4.38, (k aux I ) ij -(k aux I ) ji satisfies a homogeneous wave equation. By the choice of initial data for k aux I , g aux I (recall Lemma 4.5) and Lemma 2.13, it follows that ∥((k aux I ) ij -(k aux I ) ji , t∂ t ((k aux I ) ij -(k aux I ) ji )) ↾ t=T aux,I ∥ H 1 (Σ T aux,I ,g)×L 2 (Σ T aux,I ,g)
≤ Ct -1+(n+1)ε .
(4.101)
We now perform energy estimates for (k aux I ) ij -(k aux I ) ji using the wave equation in Lemma 4.38 (in a manner similar to the k energy estimates in the proof of Theorem 4.6, only simpler). By choosing n sufficiently large, the estimate (4.101) allows one to take care the borderline terms and moreover show that for any T 0 ∈ (0, T N 0 ,s,n ), Finally, since k ij is the pointwise limit of (k aux I ) ij as I → +∞ (by Lemma 4.37), the estimate (4.102) implies that k ij is symmetric in i and j, which is what we wanted to prove. □ Proof of Proposition 4.8. Proposition 4.8 follows directly from Lemmas 4.37 and 4.39. □ 5. Vanishing of the Einstein tensor. The goal of this section is to show that the solution of (4.9), constructed in Theorem 4.4 in subsection 4.3, is in fact a solution to the Einstein vacuum equations. This then concludes the proof of Theorem 1.1; see the conclusion of the proof at the end of the section.
We begin with the following: PROPOSITION 5.1. There exists N h ∈ N sufficiently large such that the following holds.
Let s ≥ 5 and N 0 ≥ N h . Then, for n ≥ n N 0 ,s , the solution (g, h, k) to (4.9) given by Theorem 4.4 satisfies h = k ℓ ℓ .
In particular, (4) Ric( (4) g) tt = 0.
Proof. Once we establish that h = k ℓ ℓ , it follows from the first equation in (4.9) that ∂ t k ℓ ℓ = |k| 2 . According to (3.3), this in turn implies that (4) Ric( (4) g) tt = 0. Taking the trace of the second equation in (4.9) and using the identity (4.7), we obtain
Since by (4.9) ∂ t h = |k| 2 , it follows that (5.1)
Note that this is a wave equation for (k ℓ ℓ -h). We can then carry out a similar energy estimates as in the proof of Theorem 4.4 to obtain t 2 ∥∂ t (k ℓ ℓ -h)∥ 2 L 2 (Σ t ,g) + 1 r=0 t 2r ∥k ℓ ℓ -h∥ 2 H r (Σ t ,g) ≤ C 0 + C n t ε t t 2 ∥∂ t (k ℓ ℓ -h)∥ 2 L 2 (Σ t ,g) + 1 r=0 t 2r ∥k ℓ ℓ -h∥ 2 H r (Σ t ,g) , (5.2) where we have used the estimates for k given in Proposition 4.21. Here, as in the previous section, we use C 0 to denote constants depending only on s, c ij and p i , while C n can depend in addition on n and N 0 . At the same time, by Theorem 4.4 and the fact that h [n] = (k [n] ) ℓ ℓ , (5.3) ∥k ℓ ℓ -h∥ 2 H 1 (Σ t ,g) + ∥∂ t (k ℓ ℓ -h)∥ 2 L 2 (Σ t ,g) ≤ 2t 2N 0 +2s-2 .
In particular, choosing N h sufficiently large, the estimates (5.2), (5.3) and Grönwall's inequality implies that ∥k ℓ ℓ -h∥ 2 H 1 (Σ t ,g) + ∥∂ t (k ℓ ℓ -h)∥ 2 L 2 (Σ t ,g) = 0, which in turn implies the desired conclusion. □ PROPOSITION 5.2. There exists N G ≥ N h and n G sufficiently large such that the following holds.
Let s ≥ 5 and N 0 ≥ N G . For n ≥ max{n N 0 ,s , n G }, take the solution (g, h, k) to (4.9) given by Theorem 4.4. Then (4) g = -dt 2 + g is in fact a solution to the Einstein vacuum equations, i.e., Ric( (4) g) = 0, and k is the corresponding second fundamental form of the constant-t hypersurfaces.
Proof. For this proof, we denote G i = G ti ( (4) g) and G ij = G ij ( (4) g), both thought of as Σ t -tangent tensors. We also use the notation that ∇ is the Levi-Civita connection for the spatial metric g.
Step 1: Derivation of a system of equations. By (4.5) and the wave equation (4.8), we have ∂ t Ric i j ( (4) g)
= ∇ i G j + ∇ j G i -3k i m Ric m j ( (4) g) + 2δ j i k m ℓ Ric ℓ m ( (4) g) -k ℓ j Ric i ℓ ( (4) g) + 2k ℓ ℓ Ric i j ( (4) g) -(k ℓ ℓ δ j i -k i j ) Ric m m ( (4) g).
(5.4)
Taking the trace of (5.4) and using the fact that Ric tt ( (4) g) = 0, we also have:
(5.5) ∂ t R( (4) g) = 2∇ j G j + 2k m ℓ Ric ℓ m ( (4) g).
The combination of (5.4) and (5.5) implies the following equation for the Einstein tensor G i j ( (4) g):
∂ t G i j ( (4) g) := ∂ t Ric i j ( (4) g) -1 2 δ i j ∂ t R( (4) g)
= ∇ i G j + ∇ j G i -δ i j ∇ ℓ G ℓ -3k i m Ric m j ( (4) g) + δ j i k m ℓ Ric ℓ m ( (4) g) -k ℓ j Ric i ℓ ( (4) g) + 2k ℓ ℓ Ric i j ( (4) g) -(k ℓ ℓ δ j i -k i j ) Ric m m ( (4) g).
(5.6)
Note that Ric i j ( (4) g) can be written in terms of G i j ( (4) g): Ric i j ( (4) g) = G i j ( (4) g) + 1 2 δ i j R( (4) g), where R( (4) g) := -R tt ( (4) g) + R ℓ ℓ ( (4) g) = R ℓ ℓ ( (4) g) by Proposition 5.1. Taking the trace we get Ric i i = G i i + 3 2 R ℓ ℓ ( (4) g) so that R ℓ ℓ ( (4) g) = 2G i i ( (4) g). It follows that (5.7) Ric i j ( (4) g) = G i j ( (4) g) + δ i j G ℓ ℓ .
We can thus rewrite (5.6) as
(5.8)
where (k ⋆ G) i j is some quadratic contraction of k and G whose exact form is unimportant.
On the other hand, by the contracted second Bianchi equations and the fact that D t G ti ( (4) g) = ∂ t G i + k i j G j , and D j G i j ( (4) g) = ∇ j G i j + k j j G i + k i j G j , we obtain (5.9) ∂ t G i = k j j G i + ∇ j G i j ( (4) g).
Taking ∂ t of (5.9), applying (5.8), and using the commutation formula in Proposition 4.24, we obtain the wave equation (5.10) where in the last equality we have also used that the curvature tensor Riem(g) can be expressed in terms of G, k and ∂ t k using (4.4), (3.1), and (5.7), so that
Here, k ⋆ k ⋆ G, etc. are in principle explicit, but we do not carry out the computations as the exact form is unimportant.
Step 1: Estimating k and k. Using the estimates (1.10) and (1.11), and arguing as in Propositions 4.18 and 4.19, we obtain (6.1) 2 r=0 t r (∥∇ (r) k∥ L ∞ (Σ t ,g) + ∥ ∇(r) k∥ L ∞ (Σ t ,g) ) ≤ Ct -1 , and (6.2)
where ∇ denotes the Levi-Civita connection of g.
Step 2: Estimating the convergence rate as t → 0 + . Let (6.3)
Define the variables g (d) := gg, (g -1 ) (d) := g -1g, h (d) := h -h, k (d) := k -k.
Given any M ′ u ∈ N we can choose M u sufficiently large so that by (1.10) and (1.12), ∥g (d) ∥ H 2 (Σ t ,g) + ∥(g -1 ) (d) ∥ H 2 (Σ t ,g) + ∥h (d) ∥ H 2 (Σ t ,g) + ∥k (d) ∥ H 1 (Σ t ,g) ≤ C ′ t M ′ u .
(6.4)
Moreover, given any M ′′ u ∈ N we can choose M u even larger so that by (1.12), (6.5) ∥ Ric(g) -Ric( g)∥ L 2 (Σ t ,g) ≤ C ′ t M ′′ u .
Now since both (4) g and (4) g solve (1.2), the RHS of (3.1) vanishes for both metrics. Hence, using (6.1), (6.4) and (6.5), we obtain (6.6)
Step 3: Energy estimates. We now carry out energy estimates for (g (d) , h (d) , k (d) ). First, we note that they satisfy a system of equations analogous to (4.50), (4.69), (4.70), (4.71) as follows.
• By definition of k and k, we immediate obtain the transport equation
j)l -2(k (d) ) (i ℓ g j)l .
• By (6.3) and (3.3), h (d) satisfies a transport equation ∂ t h (d) = |k| 2 -| k| 2 .
• Arguing as in Section 4, it follows that both k and k satisfy the wave equation (4.8) (with the corresponding metric g and g). We take the difference to obtain a wave equation for k (d) .
• There exists N r ≥ N c (where N c is as in Lemma 6.1) sufficiently large such that the following holds. Suppose s ≥ 5, N 0 ≥ N r and n ≥ n N 0 ,s , then (6.13) 1 r=0 |α|≤3-r |∂ r t ∂ α x (g N 0 ,s,n -g [n] )| = O(t M u ).
This is a direct consequence of (4.11) and Sobolev embedding. Fix (g N 0 =N r ,s=5,n 0 , k N 0 =N r ,s=5,n 0 ) on (0, T N 0 =N r ,s=5,n 0 ] × T 3 , where n 0 ≥ max{n N 0 =N r ,s=5 , n r , n G }. We want to show that this particular solution is in fact smooth. Let s 0 ≥ 5 be arbitrary. By Theorem 4.4 we obtain a solution (g N 0 =N r ,s=s 0 ,n , k N 0 ,s=s 0 ,n ) on (0, T N 0 =N r ,s=s 0 ,n ] × T 3 for some n ≥ max{n N 0 =N r ,s=s 0 , n r , n G }. We now claim that in fact on the common domain of existence (0, min{T N 0 =N r ,s=5,n 0 , T N 0 =N r ,s=s 0 ,n }] × T 3 , we have (6.14) g N 0 =N r ,s=5,n 0 ≡ g N 0 =N r ,s=s 0 ,n .
To prove the claim, it suffices to verify the conditions of Theorem 1.7:
• Since s ≥ 5 and N 0 = N r ≥ N c , the conditions (1.10) and (1.11) hold because of Lemma 6.1.
• By (6.12), (6.13) and the triangle inequality, our choice of n 0 , n, N 0 , s implies that 1 r=0 |α|≤3-r |∂ r t ∂ α x (g N 0 =N r ,s=5,n 0 -g N 0 =N r ,s=s 0 ,n )| = O(t M u ),
i.e., (1.12) holds. This establishes (6.14). As a result of (6.14), it follows that the fixed solution (g N 0 =N r ,s=5,n 0 , k N 0 =N r ,s=5,n 0 ) is in H s 0 +1 × H s 0 for every t ∈ (0, min{T N 0 =N r ,s=5,n 0 , T N 0 =N r ,s=s 0 ,n }]. Now we use energy estimates as in the proof of Theorem 4.4 to show propagation of regularity: it then follows that the solution is in H s 0 +1 × H s 0 for every t in the original time interval, i.e., for every t ∈ (0, T N 0 =N r ,s=5,n 0 ].
Since s 0 can be arbitrarily large, it follows from Sobolev embedding and the equations (4.9) that the fixed solution (g N 0 =N r ,s=5,n 0 , k N 0 =N r ,s=5,n 0 ) is in fact smooth in (0, T N 0 =N r ,s=5,n 0 ] × T 3 . This concludes the proof of the theorem. □
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