Anti-de Sitter Expansion

One class of such metrics arises if one considers conformally completable Einstein metrics ˜gin dimensionn>2 with Ric[g˜] + (n−1)g˜=0. They appear as solutions to the following problem [FG85, FG12].

Definition2.2.1. Let(Σ,[γ])be a conformal structure of signature(p,q−1)of dimensionn−1.

Let M := Σ×[0, 1) be a thickening with boundary∂M = Σ× {0} and ˚M := Σ×(0, 1) and σ the parameter on[_{0, 1})_{. A}anti-de Sitter like Poincaré-Einstein metricof index(p,q)will denote a solution ˜gto the following problem.

(i) g˜has[γ]as conformal infinity.

(ii) Ric[g˜] + (n−1)g˜vanishes to infinite order inσifnis even or 3 and to orderσⁿ⁻³ ifn>3 is odd (see [FG12] for details of the order).

(iii) When written as ˜g =σ⁻² dσ²+gσ

, withgσ being a curve of metrics onΣ, then dσ²+g_σis the restriction of a smooth metrichon an open set inΣ×(−1, 1)_such that the open set andhare invariant underσ→ −σ.

The notation g^AdS = dσ²+g_σ^AdS will be used for the conformal background of the Poincaré-Einstein metric in the previous problem. Construction of a formal expansion basically works as follows [Gra00, And05b, And04]. Consider a manifold of type M = _Σ×[0, 1) and a metric of type g = dσ²+g_σ, where σ parametrises the interval [0, 1) and g_σ is a smooth family of metrics onΣ. Hence σ is a geodesic defining function forΣ× {0}, i.e. gradσ is orthonormal

52 Chapter 2: examples of almost einstein structures

The trace in the right-hand side can also be taken with respect to g, but since gradσ is in the kernel of ˙g, one is left with theg_σ-trace.

On the one hand, using Equation (1.104) for the Ricci tensor of the conformally changed metric

˜

g=σ⁻²gand respectingkdσk²_g=1 gives

σRic[g] =σ(Ric[g˜] + (n−1)g˜)−(n−2)Hess^gσ+_∆^gσg. (2.20) On the other hand, one has for vector fields tangent to the hypersurfacesΣt = _Σ× {t} that Hess^gσ(X,Y) = g(∇_Xgradσ,Y) = −g(gradσ,∇_XY) = −g(gradσ, II(X,Y)) =: −K(X,Y), where K is the scalar valued second fundamental form. Hence ∆σ = tr^gK = H is the mean curvature²ofΣσ, where the last notation is for level sets ofσ. By contracting the Gauß equation for the splitting in a local orthonormal frame{e0=gradσ,ei}one obtains

Ric[g](X,Y) =e₀g(R^g(X,e0)e0,Y) +Ric[gσ](X,Y)

(X,Y). Also for tangent vector fields, the Koszul formula pro-vides K(X,Y) = g(∇_XY, gradσ) = −¹₂L_gradσgσ

(X,Y). By assuming Ric[g˜] + (n−1)g˜ = 0 the last two formulas for the Ricci tensor then give a differential equation to the familygσ

0=σg¨_σ−(n−2)g˙_σ+2Hg_σ−σ

2 Ric[g_σ] +Hg˙+g˙²_σ

. (2.21)

A second equation is gained by exploiting the Codazzi equation for this foliation. Consider {e_i}to be a local orthonormal frame withe0=gradσandXa vector field that is tangent to the slices of M. Without loss of generality it is assumed that e_i>0 andX are p-synchronous. Then with the definitionK(X,Y)gradσ =II(X,Y)one observes³ at p thatg((∇_eiII) (X,ei), gradσ) = ∇^T_e

iK

(e_i,X)andg((∇_XII) (e_i,e_i), gradσ) =∇_X(K(e_i,e_i))fori>0. Now the Codazzi equation can be applied to the terms of the Ricci tensor and one gets

Ric[g](X, gradσ) =

Then by Equation (2.20), the left-hand side vanishes and so

0=dH(X) + (div^gσK) (X). (2.22)

2 In contrast to the second fundamental form II and to the mean curvature vector tr^gσII, the scalar valued second funda-mental form and the mean curvature depend on the choice of the normal vector.

3 The term(∇_ZII) (X,Y)is defined as ∇^⊥_Z(II(X,Y))−II ∇^T_ZX,Y

−II X,∇^T_ZY

, where∇^⊥_XYand∇^T_XYare the projections of∇XYto the normal and tangent component ofTMat the slices. By assumingXandYto bep-synchronous, the last two terms may be neglected atpif necessary.

2.2 poincaré-einstein metrics 53

Finally one can deriveL_gradσKfor tangent vector fields, which gives a Riccati equation forK L_gradσK

(X,Y) =g

∇_gradσ∇_XY, gradσ −g

∇_[gradσ,X]Y, gradσ

−g(∇_X[_gradσ,Y]_{, grad}σ)

=g(R(gradσ,X)Y, gradσ)−g(∇_Ygradσ,∇_Xgradσ).

The last term equals∑iK(X,ei)K(ei,Y). But then since[gradσ,X] = [gradσ,Y] =∇_gradσgradσ= 0, the last two terms in the first line vanish and one has

L_gradσK

(X,Y) = ∇_gradσ(K(X,Y)). Taking the trace with respect togσand adding the zero termg(R(gradσ, gradσ)gradσ, gradσ) = 0 then yield ˙H=Ric[g](gradσ, gradσ)− kKk²_gσ. Finally using Ric[g˜] + (n−1)g˜=0 gives

0=σH˙ −H+σkKk²_g

σ. (2.23)

The last result in particular implies H = 0 for the boundary surface for such metrics. Equa-tions (2.21), (2.22) and (2.23) then provide the equations for formally calculating the Fefferman-Graham expansion forgσ

gσ =g₍₀₎+σg₍₁₎+σ²g₍₂₎+. . . , (2.24) where the coefficients are g_(k) = _k!¹g^(k)_σ

_σ=0. In case where n is even, the expansion may also include logarithmic terms of typeσ^klogσh_(k)for σⁿ⁻¹and higher orders. Requiring H = 0 at σ=0, Equation (2.21) givesg₍₁₎=0. By differentiating (2.21)(j−1)times with respect toσ, i.e.

taking itsL_gradσ-Lie derivatives, the coefficients of the expansion have to satisfy

(j+₁−n)g^(j)_σ −tr^gσg^(j)_σ g_σ =|_σ=0 terms involving derivativesg⁽ⁱ⁾_σ withi<j. (2.25) Hence forj<n−1, derivativesg^(j−1)_σ can in principle be calculated from lower order terms. The formal expansion that one gets by inductively differentiating ofg_σwith respect to Equation (2.25) is referred to asFefferman-Graham expansion [FG85]. Prescribing g₍₀₎ = g0 = γ and following [Gra00, And04], the derivatives g^(j)_σ

_σ=0and hence the coefficientsg_(j)are locally determined by γand its derivatives in the induction up to order(n−2).

In particular do they vanish for odd j. This can be seen as follows. Thek-th transversal Lie derivativeL_gradσof a tensorT is denoted byT^(k). Differentiating (2.21)ktimes atσ=_{0 gives}

−(n−2−k)g^(k+1)_σ +2(Hg_σ)^(k)−k

2 Ric[g_σ] +Hg˙_σ+g˙²_σ(k−1)

=|_σ=00. (2.26) By requiring H =0 atσ =0 one already has g⁽¹⁾ = 0. An induction process then is provided by considering evenkwithk<n−2 and by assumingg^(j)=0 for all odd j∈ {1, 3, . . . ,k−1}. One now uses 2H= −tr^gσg˙ and only takes care about terms on the left-hand side that do not contain odd derivatives ofg_σ of order less thenk+1. This reduces Equation (2.26) to

−(n−2−k)g_σ^(k+1)−tr^gσg_σ^(k+1)

gσ−2kRic[gσ]^(k−1)=|_σ=00. (2.27) If Ric[g_σ]^(k−1)=0 atσ =0, then by taking the trace one obtains tr^γg^(k+1)₀ =0 and substituting the trace to Equation (2.27) would giveg^(k+1)₀ =0.

So if theg^(j)are supposed to vanish for oddj<k+1 it remains to show Ric[gσ]^(k−1)=0 along Σ0. A preliminary observation is that vanishing of odd derivativesg^(j)up to orderk−1 implies vanishing of odd derivatives g^∗(j) of the dual metric g^∗ up to order k−1. This can be seen as follows. Firstg^∗(1) is a contraction of g^∗⊗g⁽¹⁾⊗g^∗ and hence vanishes provided g⁽¹⁾ = 0.

All contractions will be summarised with a calligraphic C, i.e. g^∗(1) = C(g^∗⊗g⁽¹⁾⊗g^∗). As contractions commute with Lie derivatives this implies

g^∗(j)=

∑

|J|=j

aJCg^∗(j1)⊗g^(j2)⊗g^∗(j3) ,

54 Chapter 2: examples of almost einstein structures

with multinomial coefficients aJ. Hence forj = k−1 odd at least one of the j_i in each term of the sum has to be odd and hence all terms have to vanish.

Now derivatives of Ric[gσ] can be calculated. One starts by observing that Ric[gσ]and hence Ric[g_σ]^(j) are horizontal, i.e. vanish if evaluated with ∂_σ in one of their arguments. So it remains to calculate Ric[g_σ]^(k−1)_p (Xp,Yp) for horizontal vectors Xp and Yp. Without loss of generality assume Xp and Yp to be the values of horizontal lifts X,Y ∈ _Γ(ThM) of vector fields onΣ. Then by Remark 1.1.3 the hypersurface Ricci tensor Ric[gσ](X,Y) can be written as Ric[gσ](X,Y) = L[g^∗_σ, . . . ,g_σ^∗,gσ, . . . ,L_Xgσ, . . . ,], where L[A1, . . . ,Am] is a linear map with values in symmetric horizontal tensors that depends only on contractions of tensor products of the A_i. The horizontal tensor g_σ^∗ refers to the horizontal part of g^∗ along Σσ. The Lie brackets[gradσ,X] and [gradσ,Y] vanish as Xand Y are horizontal, so that on the one hand (L_grad^j

σRic[g_σ])(X,Y) = L^j_grad

σ(Ric[g_σ](X,Y)) and on the other hand L^j_grad

σL_X = L_XL^j_grad

σ, L^j_grad

σL_XL_Y=L_XL_YL_grad^j

σ, etc. NowL_gradσ commutes with the contraction and one gets L_grad^j _σRic[g_σ](X,Y) =L_grad^j _σ(L[g_σ^∗, . . . ,g^∗_σ,g_σ, . . . ,L_Xg, . . . ,])

=

∑

|J|=j

aJLh L_grad^j1

σg^∗_σ, . . . ,L^j_grad^p

σL_XL_Yg_σi

=

∑

|J|=j

a_JLh

g^∗_σ^(j1), . . . ,L_XL_Yg^(j_σ^p)i .

Consequently Ric[g_σ]^(k−1)(X,Y)contains contractions of tensor products of tensors of typeg^∗(j), g^(j)_σ , L_Xg_σ^(j) and L_XL_Yg_σ^(j) with j ≤ k−1. The total number |J|of derivatives in each of these products is(k−1) and hence odd, such that each product must contain at least one term with oddji. The induction process already provides that the odd derivatives ofgσvanish alongΣ0and so L_XL_Yg^(j)_σ andL_Xg^(j)_σ have to vanish as well. This finally provides vanishing of Ric[gσ]^(k−1) and hence vanishing ofg^(k+1). A different approach for showing vanishing of Ric[gσ]^(k−1)along Σ0can be found in appendix B.

The coefficientsg_(j)with jeven can now directly be calculated. g₍₂₎ =g¨0for example can be calculated from taking the Lie derivativeL_gradσ of (2.22) once. Atσ=0 one then has

−(n−3)g¨0−2 Ric[g0]−Hg^˙ 0

=0.

Taking the trace with respect to g0 = γ and having in mind that by definition tr^g0g¨0 = gradσ(tr^g0g˙)− kg˙0k²_g0 =−2 ˙H+0 then gives 2(n−2)H^˙ −τ[g0] =0. Consequently one finds

g₍₂₎= ¹

2g¨0=− ¹ n−3

Ric[γ]− ¹ n−2γ

=−2P[γ].

Equations (2.22) and (2.23) then provide constraints to the choice of the coefficient g(n−1)

[And04]. In case wherenis even, it has to be transverse-traceless with respect toγ, i.e.

tr^γg(n−1)=0 div^γg(n−1)=0 (2.28)

but is undetermined else. Also there appear no logarithmic terms in the formal expansion. gσ is then formally given by

gσ ∼γ+σ²g₍₂₎+· · ·+σⁿ⁻²g_(n−2)+σⁿ⁻¹g_(n−1)+. . . , (2.29) where higher order terms depend onγ, the choice ofg_(n−1)and its derivatives.

In case wherenis odd, the choice ofg(n−1)is constrained by

tr^γg_(n−1)=ω₁ div^γg_(n−1)=ω₂, (2.30)

2.2 poincaré-einstein metrics 55

whereω₁andω2are determined byγand its derivatives. Also the expansion contains logarith-mic terms of typeσ^j(logσ)^l withj≥n. It is given by

gσ ∼γ+σ²g₍₂₎+· · ·+σⁿ⁻³g(n−3)+σⁿ⁻¹g(n−1)+σⁿ⁻¹logσhn−1+. . . , (2.31) whereh_(n−1)is a transverse-traceless term determined byγand its derivatives.

By construction the conformal boundary is identified with(Σ,[γ]). The conformal class[γ]is of index(p,q−1). Its normal vector field gradσis spacelike.

Im Dokument On the singularitys set of Lorentzian almost Einstein structures (Seite 55-59)