5.1 topology of the conformal boundary Σ
5.1.3. Flow of grad σ
ϕ
exp−1p (U)∩C˜
⊃ ϕ˜
exp−p1(U)∩S˜
= ϕ˜
exp−p1(U∩Σc)
= ϕ(U∩Σc)
= V× {0}.
C = CpM∩U˜ topologically is an open subset ofCpM and hence an (n−1)-dimensional sub-manifold. Consequently(exp−1p (U)∩S˜, ˜ϕ)must be a chart for a neighbourhood of exp−p1(x) in ˜C. In particular exp−1p (U)∩S˜ is a open neighbourhood of exp−1p (x)in ˜C. By arbitrariness of x ∈ Σc∩U˜ we get that ˜S is open in ˜C. Hence S\ {0} = S˜ ∩K is an open subset of C\ {0} =C˜ ∩K. Therefore we haveC\ {0} =S\ {0}and hence
C= S.
We now choose Up := expp(U). Then sinceU⊂ K, we get CpM∩U= exp−1p Σc∩Up
. (5.5)
Proposition1.2.8(Equation (1.85)) states that in a normal neighbourhood the geodesic null cone Cp(Up)in p is the image of the tangent null cone under the exponential map. Hence
Cp(Up) = expp(CpM∩U)
(5.5)
= Σc∩Up.
This proves the claim.
5.1.3. Flow of grad σ
In this section we will give a more detailed analysis of the level sets ofσ in a neighbourhood of vertices p ∈ Σd. This will help to identify some properties of the flow of gradσ on Σc. We already named the level set of level zero byΣ=σ−1(0). The remaining level sets will be denoted
Σs:=σ−1(s).
fors∈ R. In a more abstract context we will also use the notationΣσ for the level sets. In the following we will not distinguish between the future and past time cone or between the future and past null cone. Nevertheless the inside and outside of a local null cone will be specified as follows.
Definition5.1.13. Let p ∈ Σd be a vertex of Σand U a normal neighbourhood withU⊂ TpM being its preimage. Then theinside I(p,U)of the local geodesic null cone is the set generated by all timelike vectors inU, i.e. I(p,U) = expp TpM∩U
. Theoutsideis the set generated by all spacelike vectors inU, i.e.O(p,U):=expp U\KpM
.
We point out that since U is required to be a normal neighbourhood, we have the disjoint union
U =I(p,U)∪O(p,U)∪ Cp(U). (5.6) Lemma5.1.14. Let p∈Σdbe a vertex ofΣ. Then there is a normal neighbourhoodU of p such that the inside and outside of the geodesic null cone in p are characterised by the causality character ofgradσin the following sense:
I(p,U) = {x∈ U |gradσxtimelike} O(p,U) = {x∈ U |gradσxspacelike}.
5.1 topology of the conformal boundary Σ 93
Proof: Due to Corollary1.4.12, there is a neighbourhood ofpsuch thatρ(x)6=0 for allx in that neighbourhood. Now consider the Morse chart(V,ϕ) with ϕ(p) = 0, as found in Proposition 5.1.1. We will requireVto be small enough such thatρis non-vanishing within it. In those coor-dinates we haveσ =s
ϕ02
− ϕ12
− · · · − ϕn−12
, wheres=sgn(ρ(p)). FromS(g,σ) =0 we conclude
g(gradσ, gradσ) = −2ρσ
=|ρ|
−ϕ0 2
+ϕ1 2
+· · ·+ϕn−1 2
for all x ∈ V. Further we notice that for all q ∈ V we have q ∈ Σ ⇐⇒ρ6=0 ρ(q)σ(q) = 0 ⇔ gq(gradσq, gradσq) =0 and hence by Proposition5.1.12there is a normal neighbourhoodU ⊂ V such that for allq∈ U we have
q∈ Cp(U)⇐⇒gq(gradσq, gradσq) =0.
We conclude that gradσmust not change its causal character within the connected components ofI(p,U)andO(p,U). For dimensionn>2 the outsideO(p,U)is connected, whileI(p,U)has two connected components. Hence we just have to calculate the causal character of gradσ for one point in each connected component.
Lemma 5.1.3 states that there is aκ ∈ R+ such that the canonic frame {κ∂µ} with respect to the coordinates is orthonormal in p and [∂0]p is a timelike vector. Hence the curveγ(t) := ϕ−1(±t·(1, 0, . . . , 0)) is a timelike curve for t small enough and hence maps to I(p,U). The sign in the definition ofγ determines the connected component of I(p,U), to which it maps.
In addition we havegγ(t)(gradσ, gradσ) = −|ρ◦γ(t)|t2 < 0 independently of the sign in the definition ofγsuch that gradσ is timelike for allq∈ I(p,U). We conclude
q∈ I(p,U) =⇒gq(gradσq, gradσq)<0.
We now define the curve γ(t) := ϕ−1(±t·(0, 1, 0, . . . , 0)) such that gγ(t)(gradσ, gradσ) =
|ρ◦γ(t)|t2>0 and hence by the decomposition (5.6)γ(t)must be a curve outside the geodesic null cone in p. Moreover, gradσ must be spacelike for allg ∈O(p,U), which together with the
last result completes the proof.
Corollary5.1.15. Let be p∈ Σd a vertex ofΣ. Then locally the inside and outside of the geodesic null cone can be distinguished by the sign ofρσas follows.
I(p,U) = {x∈ U |ρ(x)σ(x)>0} O(p,U) = {x∈ U |ρ(x)σ(x)<0}.
The corollary is a consequence of the equalityg(gradσ, gradσ) =−2ρσand the last lemma.
Proposition5.1.16. Let p∈Σdbe a vertex. Then there is a normal neighbourhoodU of p such that (i) all level setsΣs inside the local null cone in p are spacelike hypersurfaces, i.e. all connected
compo-nents ofΣs∩I(p,U)have a timelike normal vector.
(ii) all level sets Σs outside the local null cone are Lorentzian hypersurfaces, i.e. on any connected component ofΣs∩I(p,U), the normal vector field is spacelike.
Proof: Following the proof of Proposition 5.1.1 we assert p to be isolated in the set of critical points of σ. Hence there is a neighbourhood U of p such that gradσq 6= 0 for all q ∈ U \ {p}. Due to Corollary5.1.15U can be reduced to an open normal neighbourhood such that inside and outside of the local geodesic coneCp(U)are characterised by the sign ofρσ.
Now letΣs be the level set to the levels. Then we have gradσq 6=0 for allq∈Σs∩ U. Hences is a regular value ofσ if restricted toU and henceΣs∩ U and all its connected components are n−1 dimensional submanifolds. Fors6= 0 we haveg(gradσ, gradσ) 6=0 and therefore gradσ
94 Chapter 5: almost einstein structures with vanishing almost scalar curvature
is transversal to Σs and orthogonal to its tangent space. By Lemma 5.1.14 gradσ is timelike inside the local null cone and hence the hypersurfacesΣs∩ U inside the null cone are spacelike.
Outside the null cone gradσis spacelike and therefore the hypersurfacesΣs∩ U outside the cone
are Lorentzian hypersurfaces.
Lemma5.1.17. Let(M,g)be geodesically null complete. If ∆σ is bounded onΣc, then every maximal integral curveγ : (α,β) →Σcofgradσthat starts onΣcis complete.
Proof: For every p ∈ Σd let Up be an attracting or repelling neighbourhood as constructed in Corollary5.1.9and define
U := [
p∈Σd
Up.
If γ(t) ∈ U for a t ∈ (α,β) then by definition of attractors or repellers either α = −∞ or β = ∞. Hence γis at least complete in one direction.
Now consider the case where γ(t) ∈ U/ for all t ∈ (α0,β) for some α0 > α. By Lemma 5.1.6(iv) we findγto be a null pregeodesic and hence by Lemma1.2.3there is a reparametrisation h :(h0,h1) →(α0,β)to a null geodesicη := γ◦h : (h0,h1) →Σc. Without loss of generality let beh(h1) =β. We will showηto be inextendible toh1and therefore h1 = ∞due to the null completeness of(M,g).
Assumeη to be extendible to the interval(h0,h1] andη(h1) = p ∈ M. Then p is inΣsince Σ= σ−1(0)is a closed subset of M. Moreover, as we requiredγ(t)∈ U/ the limit pmust be in Σcandγ(t)→ pand ˙γ(t) →gradσp 6= 0 fort→ β. Since gradσp6= 0 there is a coordinate neighbourhood (U,ϕ) of p such that for one component of gradσ in those coordinates we have(gradφ)k(x) > δfor all x ∈U. For any ballB(0)in these coordinates, there is at0such that ϕ◦γ(t) ∈ B for all t > t0. If we require B to be small enough such that B2e ⊂ ϕ(U), we can apply Lemma1.1.16and hence there is a t > t0 such that ϕ◦γ(t) ∈/ B, which is a contradiction.
Henceηis inextendible toh1and we findηto be defined on the interval(h0,∞), in particular η :(h0,∞) →Σc.
Now we will show that the reparametrisationh in direction of h1 = ∞must be complete, i.e.
h(s) →∞fors→ ∞.
The reparametrisation constructed in Lemma1.2.3fulfils Equation (A.4) s−h0 =
Z h(s) α0
exp[C(t) ]d t with C(t) = Rt
t0 c(x)d x. In particular for the integral curveγ we have c(x) = −ρ◦γ(x) =
−1n (∆σ) ◦γ(x). The Laplacian of σ is bounded on Σ. Let be |∆σn | ≤ B < ∞, then the right-hand side can be estimated to
s−h0 ≤ Z h(s)
α0
exp[B(t−t0) ]d t
= exp(−B t0)
B (exp(B h(s) )−exp(Bα0) ).
Henceh(s) →∞fors →∞. Therefore the integral curve is defined on the interval (α,∞). The same arguments hold for the interval(α,β0)for some β0 < β. In particular either γ is repelled by ap ∈ Σdand henceα = ∞or complete in that direction by changing the sign in the
above arguments.
A consequence of the last lemma is the following proposition.
Proposition5.1.18. Assume that(M,g)is geodesically null complete and∆σ is bounded onΣc, then the maximal flow Φ : U ⊂ R× M → M of gradσ is complete on Σc, i.e. the restriction Φ : R×Σc →Σc is well defined on the hole intervalRand
Φ(t,Σc) =Σc
5.1 topology of the conformal boundary Σ 95
for all t∈ R.
Remark. Following the last claims, the gradient vector field gradσ is complete if restricted to Σc and hence Φ : R×Σc → Σc is a one parameter family of diffeomorphisms [O’N83, Lemma1.54].
Lemma5.1.19. Let Φ : DΦ ⊂ R×Σc → Σc be the maximal flow of a vector field V ∈ X(Σc). Furthermore let be S ⊂ Σc a n−2)-dimensional submanifold ofΣcwith the following properties
(∗) ∀t6=0,(t,S) ⊂ DΦ : Φ(t,S)∩S = ∅ (∗ ∗) ∀x ∈ S : TxΣc = TxS⊕ hV(x)i
where hV(x)i is the line in TxM spanned by V(x). If f : U ⊂ Rn−2 → S is a diffeomorphism parametrising S, then the mapΦ˜ := Φ◦(id, f)defined by
Φ˜ : R×U ⊃ DΦ˜ → Φ DΦ (t,α) 7 → Φ(t,f(α) )
withDΦ˜ := (id,f)−1(R× f(U) )∩ DΦis a diffeomorphism onto its image.
Proof: Since f is a diffeomorphism, the map ˜Φis surjective by definition. For showing injectivity considerΦ(t1,f(α1) ) =Φ(t2,f(α2) ). Since Φis the maximal flow, this means that f(α1) = Φ(0, f(α1) ) = Φ(t2−t1, f(α2) ). From requirement (∗) we immediately get t2−t1 = 0.
Henceα1 = α2as f is a diffeomorphism.
Now consider the differential of ˜Φin some arbitrary point(t,α) ∈ DΦ˜ evaluated on(X0,X) ∈ Rn−1. Then we get
d[Φ◦(id, f) ](t,α)(X0,X) =dΦ(t,f(α) )(X0,d fα(x) )
= X0VΦ(t,f(α) ) + [dΦt]f(α)(d fα(X) ),
where Φt = Φ(t, ˙) is the t-th stage of Φ. As Φt is a local diffeomorphism and d fα(X) is transversal toVf(α), also[dΦt]f(α)(d fα(X) )is transversal to VΦ(t,f(α) ) providedX6=0. Con-sequentlydΦ˜(t,α)is an isomorphism for all (t,α)∈ DΦ˜.