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NON SINGULAR VACUUM STATIONARY SPACE TIMES WITH A NEGATIVE COSMOLOGICAL CONSTANT

20 pages
NON-SINGULAR, VACUUM, STATIONARY SPACE-TIMES WITH A NEGATIVE COSMOLOGICAL CONSTANT PIOTR T. CHRUSCIEL AND ERWANN DELAY Abstract. We construct infinite dimensional families of non-singular stationary space times, solutions of the vacuum Einstein equations with a negative cosmological constant. Contents 1. Introduction 1 2. Definitions, notations and conventions 3 3. Isomorphism theorems 4 3.1. An isomorphism on two-tensors 4 3.2. Two isomorphisms on one-forms 5 3.3. An isomorphism on functions in dimension n 6 3.4. An isomorphism on functions in dimension 3 6 4. The equations 8 4.1. The linearised equation 9 4.2. The modified equation 9 5. The construction 12 5.1. The n-dimensional case 12 5.2. The three-dimensional case 14 6. Uniqueness 15 7. Polyhomogeneity 16 Appendix A. “Dimensional reduction” of some operators 17 A.1. Lichnerowicz Laplacian on two-tensor for a warped product metric 17 A.2. The Laplacian on one-forms for a warped product metric 18 References 19 1. Introduction A class of space-times of interest is that of vacuum metrics with a negative cosmological constant admitting a smooth conformal completion at infinity. It is natural to seek for stationary solutions with this property. In this paper we show that a large class of such solutions can be constructed by prescribing the conformal class of a stationary Lorentzian metric on the conformal boundary ∂M , provided that the boundary data are sufficiently close to, e.

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  • metrics can

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  • vacuum ein- stein equations


  • space

  • time metrics

  • boundary

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NON-SINGULAR, VACUUM, STATIONARY SPACE-TIMES
WITH A NEGATIVE COSMOLOGICAL CONSTANT
´PIOTR T. CHRUSCIEL AND ERWANN DELAY
Abstract. We construct infinite dimensional families of non-singular
stationary space times, solutions of the vacuum Einstein equations with
a negative cosmological constant.
Contents
1. Introduction 1
2. Definitions, notations and conventions 3
3. Isomorphism theorems 4
3.1. An isomorphism on two-tensors 4
3.2. Two ishisms on one-forms 5
3.3. An isomorphism on functions in dimension n 6
3.4. An ishism on functions in dimension 3 6
4. The equations 8
4.1. The linearised equation 9
4.2. The modified equation 9
5. The construction 12
5.1. The n-dimensional case 12
5.2. The three-dimensional case 14
6. Uniqueness 15
7. Polyhomogeneity 16
Appendix A. “Dimensional reduction” of some operators 17
A.1. Lichnerowicz Laplacian on two-tensor for a warped product
metric 17
A.2. The Laplacian on one-forms for a warped product metric 18
References 19
1. Introduction
Aclassofspace-timesofinterestisthatofvacuummetricswithanegative
cosmological constant admitting a smooth conformal completion at infinity.
It is natural to seek for stationary solutions with this property. In this
paper we show that a large class of such solutions can be constructed by
prescribing the conformal class of a stationary Lorentzian metric on the
conformal boundary ∂M, provided that the boundary data are sufficiently
close to, e.g., those of anti-de Sitter space-time.
Date: January 6, 2006.
1´2 P.T. CHRUSCIEL AND E. DELAY
We mention the recent papers [4,5], where we have constructed infinite
dimensional families of static, singularity free solutions of the vacuum Ein-
stein equations with a negative cosmological constant. The main point of
the current work is to remove the staticity restriction. This leads to new,
infinite dimensional families of non-singular, stationary solutions of those
equations.
n+1WethusseektoconstructLorentzianmetrics g inanyspace-dimension
n≥ 2, with Killing vector X =∂/∂t. In adapted coordinates those metrics
can be written as
i i jn+1 2 2(1.1) g =−V (dt+θ dx ) +g dx dx ,i ij|{z} | {z }
=θ =g
(1.2) ∂ V =∂ θ =∂ g =0.t t t
Our main result reads as follows (see below for the definition of non-
degeneracy; the function ρ in (1.3) is a coordinate near ∂M that vanishes
at ∂M):
Theorem 1.1. Let n=dimM ≥2, k∈Nr{0}, α∈(0,1), and consider a
static Lorentzian Einstein metric of the form (1.1)-(1.2) with strictly positive
˚V = V, g = ˚g, and θ = 0, such that the associated Riemannian metric
2 2 1 2˚ge = V dϕ +˚g on S ×M is C compactifiable and non-degenerate, with
bsmooth conformal infinity. For every smooth θ, sufficiently close to zero in
k+2,αC (∂M,T ), there exists a unique, modulo diffeomorphisms which are1
the identity at the boundary, nearby stationary vacuum metric of the form
(1.1)-(1.2) such that, in local coordinates near the conformal boundary ∂M,
˚ b(1.3) V −V =O(ρ), θ =θ +O(ρ), g −˚g =O(1).i i ij ij
Theorem 1.1 is more or less a rewording of Theorem 5.3 below, taking
into account the discussion of uniqueness in Section 6.
The (n + 1)-dimensional anti-de Sitter metric is non-degenerate in the
sense above, so Theorem 1.1 provides in particular an infinite dimensional
family of solutions near that metric.
˚The requirement of strict positivity of V excludes black hole solutions, it
would be of interest to remove this condition.
The decay rates in (1.3) have to be compared with the leading order
−2 2˚behavior ρ both for V and ˚g . A precise version of (1.3) in terms ofij
weighted function spaces (as defined below) reads
k+2,α k+2,α1 1˚(1.4) (V −V)∈C (S ×M), (g−˚g)∈C (S ×M,S ),21 2
k+2,α 1b(1.5) θ−θ∈C (S ×M,T ),12
and the norms of the differences above are small in those spaces.
2Note that our hypothesis that the metricgeis conformallyC implies that
n−1,α 3,αgeisC ∩C –conformallycompactifiableandpolyhomogeneous[9]. We
showinSection7thatoursolutionshavecompletepolneousexpan-
sions near the conformal boundary, see Theorem 7.1 for a precise statement.
Since the Fefferman-Graham expansions are valid regardless of the signa-
ture of the boundary metric, the solutions are smooth in even space-time
dimensions. In odd space-time dimensions the obstruction to smoothnessSTATIONARY SPACE-TIMES WITH NEGATIVE Λ 3
is the non-vanishing of the Fefferman-Graham obstruction tensor [13,15] of
2 −2the (Lorentzian) metric obtained by restricting −(dt+θ) +V g to the
conformal boundary at infinity.
Theorem 1.1 is proved by an implicit-function argument. This requires
the proof of isomorphism properties of an associated linearised operator.
This operator turns out to be rather complicated, its mapping properties
being far from evident. We overcome this by reinterpreting this operator
˜as the Lichnerowicz operator Δ +2n in one-dimension higher. Our non-L
˜degeneracy condition above is then precisely the condition that Δ +2n hasL
2no L –kernel. While this is certainly a restrictive condition, large classes of
Einstein metrics satisfying this condition are known [2,3,5,18].
2Because of the V multiplicative factor in front of θ in (1.1), for distinct
bθ’s the resulting space-time metrics have distinct conformal metrics at the
conformal boundary at infinity. This makes it problematic to determine
the energy of the new solutions relative e.g. to the anti-de Sitter solution;
n+1similarly for angular momentum. Now, each of our solutions g comes
associated with a family of non-stationary solutions, which asymptote to
n+1g, and which can be constructed using e.g. a technique of Friedrich [14].
To each member of such a family one can then associate global Hamiltonian
n+1charges relative to g as in [8,11]. In this approach our solutions define
the zero point of energy for each family, and there is no natural way of
comparing relative energies, angular momenta, and so on, of members of
distinct families.
2. Definitions, notations and conventions
LetN beasmooth, compact(n+1)-dimensionalmanifoldwithboundary
∂N. Let N := N\∂N, a non-compact manifold without boundary. In our
context the boundary ∂N will play the role of a boundary at infinity of
N. Let g be a Riemannian metric on N, we say that (N,g) is conformally
compact if there exists on N a smooth defining function ρ for ∂N (that
∞is ρ ∈ C (N), ρ > 0 on N, ρ = 0 on ∂N and dρ nowhere vanishing on
2 2,α ∞∂N) such that g := ρ g is a C (N)∩C (N) Riemannian metric on N,0
we will denote by gb the metric induced on ∂N. Our definitions of function
spaces follow [18]. Now if |dρ| = 1 on ∂N, it is well known (see [19] forg
instance)thatghasasymptoticallysectionalcurvature−1nearitsboundary
at infinity, in that case we say that (N,g) is asymptotically hyperbolic. If
we assume moreover than (N,g) is Einstein, then asymptotic hyperbolicity
enforces the normalisation
(2.1) Ric(g)=−ng ,
where Ric(g) is the Ricci curvature of g.
We recall that the Lichnerowicz Laplacian acting on a symmetric two-
tensor field is defined as [7,§ 1.143]
k k k klΔ h =−∇ ∇ h +R h +R h −2R h .L ij k ij ik j jk j ikjl
The operator Δ +2n arises naturally when linearising (2.1). We will sayL
2that g is non-degenerate if Δ +2n has no L -kernel.L´4 P.T. CHRUSCIEL AND E. DELAY
While we seek to construct metrics of the form (1.1), for the purpose of
the proofs we will often work with manifolds N of the form
1N =S ×M,
equipped with a warped product, asymptotically hyperbolic metric
2 2V dϕ +g,
where V is a positive function on M and g is a Riemannian metric on M.
By an abuse of terminology, such metrics will be said static.
The basic example of a non-degenerate, asymptotically hyperbolic, static
Einstein space is the Riemannian counterpart of the AdS space-time. In
nthat case M is the unit ball ofR , with the hyperbolic metric
−2g =ρ δ ,0
1 2δ is the Euclidean metric, ρ(x)= (1−|x| ), and
2 δ
−1V =ρ −1.0
q
We denote by T the set of rank p covariant and rank q contravariantp
tensors. When p = 2 and q = 0, we denote by S the subset of symmetric2 We use the summation convention, indices are lowered and raised
ijwith g and its inverse g .ij
3. Isomorphism theorems
Some of the isomorphism theorems we will use are consequences of Lee’s
theorems [18], it is therefore convenient to follow his notation for the
k,α
weighted H¨older spaces C . As described in the second paragraph be-δ
δfore proposition B of [18], a tensor in this space corresponds to ρ times a
k,αtensor in the usual C space as defined using the norm of the conformally
compact metric. This implies that, in local coordinates near the conformal
k,α k,αδboundary, a function in C is O(ρ ), a one-form in C has componentsδ δ
k,αδ−1whichareO(ρ ),andacovarianttwo-tensorinC hascomponentswhich
δ
δ−2are O(ρ ).
k,αWe will often appeal to isomorphism theorems of [18] in weighted C
spaces, for k ∈ N. Under the regularity conditions on the metric in our
definition of asymptotically hyperbolic metric, those theorems apparenly
only apply to low values of k. However, under our hypotheses, one can use
those theorems fork =2, and use scaling estimates to obtain the conclusion
for any value of k.
3.1. An isomorphism on two-tensors. We first recall a result of Lee
2(see Theorem C(c) and proposition D of [18], there is no L -kernel here by
hypothesis):
1Theorem 3.1. LetS ×M be equipped with a non-degenerate asymptotically
ehyperbolic metric ge. For 0<k+α6∈N and δ∈(0,n) the operator Δ +2nL
k+2,α k,α1 1is an isomorphism from C (S ×M,S ) to C (S ×M,S ).2 2δ δ
2 2When the metric is static of the form ge=V dϕ +g we deduceSTATIONARY SPACE-TIMES WITH NEGATIVE Λ 5
Corollary 3.2. On (M,g) we consider the operator
(W,h)7→(l(W,h),L(W,h)),
where

∗ −1 ∗ −2 2 −1 jl(W,h) = V (∇ ∇+2n+V ∇ ∇V +V |dV| )W +V ∇ V∇ Wj
i
−1 j k−V ∇ V∇ Vh +hHess V,hi .kj g g
and
1 1 −1 kL (W,h) = Δ h +nh − V ∇ V∇ hij ij ij ijL k2 2
1 −2 k k+ V (∇ V∇ Vh +∇ V∇ Vh )i kj j ki2
1 −1 k k− V (∇∇ Vh +∇ ∇ Vh )i kj j ki
2
−2 −3+2V W(Hess V) −2V ∇ V∇ VW.g ij i j
k+2,α k+2,αThen (l,L) is an isomorphism from C (M) × C (M,S ) to2δ−1 δ
k,α k,αC (M)×C (M,S ) when δ∈(0,n).2δ−2 δ
Proof. First, it is easy to see that the Laplacian commutes with the Lie
ederivative operator in the Killing direction, so the operator Δ + 2n re-L
stricted to ϕ-independent tensor field is again an isomorphism. Now, from
Lemma A.2 below, if we define P to be the set of symmetric covariant two
tensors of the form
2 i jeh=2VWdϕ +h dx dx ,ij
and if we letT denote the collection of tensors of the form
ieh=2ξ dx dϕ,i
then the Lichnerowicz Laplacian preserves the decomposition P ⊕T. In
1eparticular the operator Δ +n restricted toP is an isomorphism, and thisL2
operator is (l,L).
3.2. Two isomorphisms on one-forms. The proof of Corollary 3.2 also
shows the following (note a shift in the rates of decay, as compared to the
m,σiprevious section, due to the fact that a tensor fieldξ dx dϕ is inC if andi ρ
m,σionly if the one-form ξ dx is in C ):i ρ−1
Corollary 3.3. The operator on one-forms defined as
k −1 k −2 kL:ξ 7→ −∇ ∇ ξ +V ∇ V∇ ξ +3V ∇ V∇ Vξi k i k i i k
l −1 j+R ξ −3V ∇∇ Vξ +2nξ ,i l i j i
k+2,α k,αis an isomorphism from C (M,T ) to C (M,T ) when δ ∈ (0,n). If1 1δ−1 δ−1
2we letξ =V θ, we therefore obtain that the operatorQ on one-forms defined
−2 2as V L(V θ )i
k −1 k −1 k −2 kQ:θ 7→ −∇ ∇ θ −3V ∇ V∇ θ −2V ∇ ∇ Vθ +3V ∇ V∇ Vθi i i i ik k k k
l −1 j+R θ −3V ∇∇ Vθ +2nθ ,i l i j i
k+2,α k,α
is an isomorphism from C (M,T ) to C (M,T ) when δ∈(0,n).1 1δ+1 δ+1´6 P.T. CHRUSCIEL AND E. DELAY
WewillappealtoyetanotherresultofLee(see[18]TheoremC(c),Propo-
2sition F and Corollary 7.4, there is again no L -kernel here because of the
Ricci curvature condition):
1Theorem 3.4. OnS ×M equipped with an asymptotically hyperbolic metric
∗e e gge with negative Ricci curvature, the operator ∇ ∇− Ric acting on one-
k+2,α k,α1 1forms is an isomorphism from C (S ×M,T ) to C (S ×M,T ) when1 1δ δq
2n n|δ− |< +1.2 4
2 2When the metric is static of the form ge=V dϕ +g we deduce:
Corollary 3.5. Under the hypotheses of the preceding theorem, on (M,g)
consider the operator
j −1 jΩ 7→B(Ω) +R Ω −V ∇∇ VΩ =:B(Ω) ,i i ij i j i
where
k −1 k −2 kB(Ω) :=∇ ∇ Ω +V ∇ V∇ Ω −V ∇ V∇ VΩ .i k i k i i k
k+2,α k,α nThenB isanisomorphismfromC (M,T )toC (M,T )when|δ− |<1 1δ δ 2q
2n +1.
4
Proof. The argument is identical to the proof of Corollary 3.2 using
Lemma A.3 and the fact that, in the notation of Lemma A.3,
c j −1 je eR Ω =R Ω −V ∇∇ VΩ .ic ij i j

3.3. An isomorphism on functions in dimension n. If we assume that
2 2 1V dϕ + g is a static asymptotically hyperbolic metric on S × M, then
−2 2 −1 iit is easy to check that at infinity V |dV| = 1 and V ∇∇ V = n.i
In dimension n, we will need an isomorphism property for the following
operator acting on functions:
−3 i 3 i −1 iσ7→Tσ :=V ∇ (V ∇ σ)=∇∇ σ+3V ∇ V∇ σ .i i i
From [6, Theorem 7.2.1 (ii) and Remark (i), p. 77] we obtain:
k+2,α k+2,αTheorem 3.6. Let (V,g) be close in C (M)× C (M,S ) to an2−1 0
asymptotically hyperbolic static metric. Then T is an isomorphism from
k+2,α k,αC (M) to C (M) when 0<δ <n+2.δ δ
2Remark 3.7. Theorem 3.6 will be used with σ = O(ρ ), note that δ = 2
verifies the inequality above since n≥2.
3.4. An isomorphism on functions in dimension 3. In dimension n=
3, we will also be interested in the following operator acting on functions:
3 i −3 i −1 iω7→Zω :=V ∇ (V ∇ ω)=∇∇ ω−3V ∇ V∇ ω .i i i
The indicial exponents for this equation are μ = −1 and μ = 0 (see [6,− +
Remark (i), p. 77]). As μ > 0 we cannot invoke [6, Theorem 7.2.1] to+
6STATIONARY SPACE-TIMES WITH NEGATIVE Λ 7
conclude. Instead we appeal to the results of Lee [18]. For this we need to
3
2have a formally self-adjoint operator, so we set ω =V f, thus

3 315 3i −2 2 −1 i
2 2(3.1) Zω =V ∇∇ f− V |dV| − V ∇∇ V f =:V Zf .i i
4 2
−2 2 −1 iAt infinity V |dV| = 1 and V ∇∇ V = 3, leading to the followingi
indicial exponents
1 3
δ = , .
2 2
We want to show that Z satisfies condition (1.4) of [18],
(3.2) kuk 2 ≤CkZuk 2 ,L L
for smooth u compactly supported in a sufficiently small open set U ⊂ M
such that U is a neighborhood of ∂M. We will need the following, well
known result; we give the proof for completeness:
Lemma 3.8. On an asympotically hyperbolic manifold (M,g) with boundary
2definining function ρ we have, for all compactly supported C functions,
Z Z2n−1∗ 2u∇ ∇u≥ (1+O(ρ))u .
2
Proof. Let f be a smooth function to be chosen later, then
Z Z
−1 2 2 −2 2 2 −1|f d(fu)| = |du| +f |df| u +2f uhdf,dui≥0
An integration by parts shows that
Z Z
−1 2 −2 2 2 −1 ∗2f uhdf,dui= u f |df| +u f ∇ ∇f.
This leads toZ Z Z
∗ 2 −1 ∗ −2 2 2u∇ ∇u= |du| ≥ (−f ∇ ∇f−2f |df| )u .
n−1− 2 2
2When f = ρ the last term equals (n− 1) ku(1 +O(ρ))k /4, which2L
concludes the proof.
−2 2Lemma 3.8 combined with the fact that V |dV| = 1+O(ρ) and that
−1 ∗V ∇ ∇V =−3+O(ρ) shows that
Z Z 2(3−1) 15 9 2kuk 2kZuk 2 ≥− uZu≥ + − (1+O(ρ))u ,L L 4 4 2
which shows that Z satisfies the condition (3.2) with
2 −1/2(3−1) 15 9
C = + − =2.
4 4 2
2 1We recall that the critical weight to be in L is O(ρ ) so the function f =
−3/2 3/2 2V = O(ρ ), corresponding to ω = 1, is in the L -kernel of Z. We
prove now that this kernel equals
−3
2kerZ =V R.
2Assume f is in the L -kernel ofZ, by elliptic regularity f is smooth on M.
1,∞Let ϕ ∈W be any function on M such that ϕ =1 on the geodesic ballk k´8 P.T. CHRUSCIEL AND E. DELAY
B (k)ofradiusk centredatp,withϕ =0onMrB (k+1),and|∇ϕ |≤Cp k p k
independently of k. Such functions can be constructed by composing the
geodesic distance fromp with a test function onR. Integrating by parts one
has Z Z
3 2 2 i −30 = − V ϕ fZf =− ϕ f∇ (V ∇ f)ik k
Z
2 −3 2 −3 i= ϕ V |∇f| +2V fϕ ∇ ϕ ∇ fk k ik
Using H¨older’s inequality, the second integral can be estimated from below
by
Z Z1/2 1/2
2 −3 2 2 −3 2−2 ϕ V |∇f| f V |∇ϕ | ,kk
leading to Z Z
2 −3 2 2 −3 2ϕ V |∇f| ≤4 f V |∇ϕ | .kk
By Lebesgue’s dominated convergence theorem, the right-hand side con-
2 −1verges to zero as k tends to infinity because f ∈L , while V is uniformly
bounded, and ∇ϕ is supported in B (k+1)rB (k). So f is a constant.k p p
Using [18], Theorem C(c), we thus obtain
k+2,α k+2,αTheorem 3.9. Let (V,g) be close in C (M)× C (M,S ) to an2−1 0
asymptotically hyperbolic static metric. Then Z is an isomorphism from
k+2,α −3/2C (M)/V R toδ
Zn o
k,α −3/2f ∈C (M): V f =0 .δ
M
k+2,α3when 1/2 < δ < . Equivalently, Z is an isomorphism from C (M)/Rδ2
to Zn o
k,α −3(3.3) f ∈C (M): V f =0 .
δ
M
when −1<δ <0.
4. The equations
Rescaling the metric to achieve a convenient normalisation of the cos-
mological constant, the vacuum Einstein equations for a metric satisfying
(1.1)-(1.2) read (see, e.g., [12])

∗ 1 2V(∇ ∇V +nV)= |λ| , g4
−1 1Ric(g)+ng−V Hess V = λ◦λ, (4.1)g 22V
div(Vλ)=0,
where
2 kλ =−V (∂ θ −∂ θ ), (λ◦λ) =λ λ .ij i j j i ij i kj
In dimension n = 3 an alternative set of equations can be obtained by
iintroducing the twist potential ω. Writing dω =ω dx one setsi
V 1jk ‘ω = ε λ ⇐⇒ λ = ε ω .i ijk jk jk‘
2 VSTATIONARY SPACE-TIMES WITH NEGATIVE Λ 9
This leads to (compare [17])

∗ 1 2V(∇ ∇V +3V)= |dω| , 22V
−1 1 2Ric(g)+3g−V Hess V = (dω⊗dω−|dω| g), (4.2)g 42V ∗ −3∇ (V ∇ω)=0.
4.1. Thelinearisedequation. Wefirstconsidertheoperatorfromtheset
of functions times symmetric two tensor fields to itself, defined as

∗V V(∇ ∇V +nV)
7→ .−1g Ric(g)+ng−V Hess Vg
The two components of its linearisation at (V,g) are

∗ −1 ∗p(W,h)=V (∇ ∇+2n+V ∇ ∇V)W +hHess V,hi −hdivgravh,dVi ,g g g
1 1 −1 k ∗P (W,h) = Δ h +nh + V ∇ V(∇ h +∇ h −∇ h )−(div divgravh)ij L ij ij i kj j kj k ij ij
2 2
−2 −1+V W(Hess V) −V (Hess W) .g ij g ij
We let Tr denote the trace and we set
1 1k ∗gravh=h− Tr hg, (divh) =−∇ h , (div w) = (∇ w +∇ w ),g i ik ij i j j i
2 2
(note the geometers’ convention to include a minus in the definition of di-
vergence). It turns out to be convenient to introduce the one-form
1−1 k k −1 −2w =V ∇ Vh +∇ h − ∇ (Trh)−V ∇ W −V ∇ VW ,j kj kj j j j
2
which allows us to rewrite P(W,h) as
∗P(W,h) = L(W,h)+div w ,
where L is as in Corollary 3.2. Similarly, p(W,h) can be rewritten as
p(W,h) = l(W,h)+Vhw,dVi .g
4.2. The modified equation. We want to use the implicit function the-
orem to construct our solutions. As is well known, the linearisation of the
Ricci tensor does not lead to well behaved equations, and one adds “gauge
fixing terms” to take care of this problem. Our choice of those terms arises
from harmonic coordinates for the vacuum Einsteinequations inone dimen-
sion higher.
In dimension 3, we start by solving the following system of equations

∗ 1 2q(V,g) := V(∇ ∇V +3V +hΩ,dVi)− |dω| =0, 2 2V −1 ∗Q(V,g) := Ric(g)+3g−V Hess V +div Ωg(4.3) 1 2 − (dωdω−|dω| g)=0,4 2V ∗ −3∇ (V ∇ω)=0,
with
−Ω ≡ −Ω(V,g,U,b)j j
μ μαβ b e:= gb gb (Γ −Γ )jμ αβ αβ
‘m k k −2 j j˚ ˚= g g (Γ −Γ )+V g (U∇ U−V∇ V)jk jk‘m ‘m
1‘m −2 j j˚ ˚ ˚(4.4) = g (∇ g − ∇ g )+V g (U∇ U−V∇ V)m jj‘ ‘m jk2´10 P.T. CHRUSCIEL AND E. DELAY
˚where∇-derivatives are relative to a fixed metric b with Christoffel symbols
α˚Γ , U is a fixed positive function, latin indices run from 0 to n, and gb:=
βγ
2 0 2 α αb eV (dx ) +g withChristoffelsymbolsΓ ,whiletheΓ ’saretheChristoffelβγ βγ
2 0 2symbols of the metric U (dx ) +b, compare (A.1) below. The co-vector
field Ω has been chosen to contain terms which cancel the “non-elliptic
terms” in the Ricci tensor, together with some further terms which will
ensure bijectivity of the operators involved. The first line of the equation
above makes clear the relation of Ω to the n+1-dimensional metric gb and
its (U,b)-equivalent.
In dimension n, as a first step we will solve the system

1∗ 2q(V,g):=V(∇ ∇V +nV +hΩ,dVi)− |λ| =0, g4
−1 ∗ 1Q(V,g):=Ric(g)+ng−V Hess V +div Ω− λ◦λ=0,(4.5)g 22V 3div(Vλ)=−V dσ ,
where Ω is as in dimension 3, while the “Lorenz-gauge fixing function” σ
equals
−3 i 3σ =V ∇ (V θ ).i
A calculation shows
3 3 −1 ∗div(Vλ)+V dσ =V [−Q+2(V ∇ ∇V +n)](θ),
where Q is as in Corollary 3.3, which makes clear the elliptic character of
the third equation in (4.5).
The derivative of Ω with respect to (V,g) at (U,b) is
D Ω(U,b)(W,h)=−w,(V,g)
where w is the one-form defined in Section 4.1 with (V,g) replaced with
(U,b). Thus, the linearisation of (q,Q) at (U,b) is
D(q,Q)(U,b)=(l,L),
where (l,L) is the operator defined in Section 4.1 with (V,g) replaced with
(U,b). We will show that, under reasonable conditions, solutions of (4.3)
(resp. (4.5))aresolutionsof (4.2)(resp.(4.1)). If(ω,V,g)solves(4.3)(resp.
if (θ,V,g) solves (4.5)), we set
∗Φ:=div Ω,

1 2|dω| in the context of (4.3),42Va:= 1 2|λ| when studying (4.5),2 g4V
1 2(dωdω−|dω| g) when studying (4.3),42VA:= 1 λ◦λ when analysing (4.5).22V
With this notation, the first two equations in both (4.3) and (4.5) take the
form

∗∇ ∇V +nV +hΩ,dVi=Va,
(4.6)−1Ric(g)+ng−V Hess V +Φ=A,g
If we take the trace of the second equation in (4.6) we obtain
2 −1 ∗0 = R(g)+n +V ∇ ∇V +TrΦ−TrA
2 −1= R(g)+n −n−V hΩ(V,g),dVi+TrΦ+a−TrA.

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