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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.