Conformal Compactifications

The definition of an almost Einstein structure (M,g,σ) does not necessarily coincide with a conformal compactification of M\_Σ,σ⁻²g

. The following section examines the properties of special almost Einstein structures that admit a conformal compactification. In particular we will require(M,g)to have a compact subset ¯M ⊂ M that is the closure of a connected component of M\Σ. We will next label the sets of interest.

Figure 3.: schematics of a compactification

Let (M,g,σ) be an almost Einstein structure. Then the set M\Σ may have several connected components. This will be labelled by i.

Then fixing one component, we do the following notations M˚ i fixed connected component of M\Σ Mi connected component of ˚Mi∪_Σc M¯i connected component ofMi∪Σd

∂M˚_i := M^¯_i∩_Σ.

With this definition we have the inclusions ˚Mi ⊂ Mi ⊂ M^¯i. Provided M˚iis a conformally Einstein manifold with boundary, then∂M˚iwill be its conformal boundary.

Definition5.1.20. LetNbe a manifold. A property that is defined for everyx ∈Nwill be called openif it holds on an open subset ofN.

96 Chapter 5: almost einstein structures with vanishing almost scalar curvature

Lemma5.1.21. Let be p∈_Σc∩∂^topM˚_i a point in the topological boundary of M˚_i. Then all points q in the same connected component ofΣcalso belong to the topological boundary ofM˚_i.

Proof: We will show that the propertiesq∈Σc∩∂^topM˚_i andq∈/Σc∩∂^topM˚_iare open properties inΣc. In particular for allq∈_Σcwithq∈_Σc∩∂^topM˚_ithere is a neighbourhoodUofqsuch that for allx∈U∩_Σcit holdsx∈_Σc∩∂^topM˚_i. The same can be said forq∈/_Σc∩∂^topM˚_i.

First consider an arbitraryq ∈ _Σc such that q ∈/ _Σc∩ ∂^{t o p}M˚ i. Since ∂^{t o p}M˚ i is a closed set in M, there is an open neighbourhood U ⊂ M of q such that U∩ ∂^{t o p}M˚ i = ∅. Hence q admits a neighbourhood U∩ Σc open in Σ such that the property x ∈/ Σc∩∂^{t o p}M˚ i holds for allx ∈ U∩Σc. Hence it is an open property inΣc.

Now let q ∈ _Σc be arbitrary such that this time q ∈ _Σc∩ ∂^{t o p}M˚ _i. There is a coordinate chart ϕ = (ϕ¹, . . . ,ϕⁿ) : U → _Rⁿ for a neighbourhood of q such that ϕⁿ(x) = 0 for all x ∈ U∩_Σc. Therefore we have dϕⁿ_x(Y) = 0 for all such x and Y ∈ TxΣc and hence d ϕ¹, . . . ,ϕⁿ⁻¹

x(Y) 6= 0 for all tangent vectors ofΣc. We define ˜ϕ := ϕ¹, . . . ,ϕⁿ⁻¹,σ : U → Rⁿ. Since dσx(Y) 6= 0 for transversal vectors Y in x ∈ Σc, we find kerdϕ˜x = {0} for all x ∈ U∩ Σc and in particular for x = q. By the inverse function theorem there is a neighbourhood ˜Uofqsuch that

˜

ϕ : ˜U →_Rⁿ (5.7)

is a coordinate chart and ˜ϕ(U^˜) = B is an open ball. Since q ∈ ∂^{t o p}M˚ i it holds that ˜U∩ M˚ _i 6= ∅. Consider x ∈ U^˜ ∩ M^˚_i and without loss of generality σ(x) > 0, then by con-nectedness of the open ball we get ˜ϕ⁻¹ B∩ _Rⁿ⁻¹×(0,∞) ⊂ M^˚ _i and hence ˜U∩ _Σc =

˜

ϕ⁻¹ B∩ Rⁿ⁻¹× {0} = U^˜ ∩ Σc ⊂ ∂^{t o p}M˚ i. Consequently q ∈ Σc∩ ∂^{t o p}M˚ i is an open property too.

Now we have the disjoint union of setsΣc = _Σ¹_c∪_Σ²_c open inΣcwhereΣ¹_c :={p ∈ _Σc| p ∈

∂^{t o p}M˚ _i} and Σ²c := {p ∈ _Σc| p ∈/ ∂^{t o p}M˚ _i}. In particular the connected components of Σc

must completely belong to one of those sets.

Lemma5.1.22. ∂M˚ _icoincides with the topological boundary ∂^{t o p}M˚ _i ofM˚ _i in M.

Proof: First, we will show the inclusion∂M˚ i ⊂ ∂^{t o p}M˚ i. Consider p ∈ Σc∩∂M˚ i and therefore p ∈ Mi. By definition, Mi is connected such that there is a curve γ : [0, 1] → Mi with γ(0) = p and γ(1) =: x ∈ M^˚ i. We will consider γ to be a curve with values in Mi ∪Σ.

Let bet₀ := i n f {t ∈ [0, 1]|γ(t) ∈ _Σc}. Since Σis a closed set and γ(t) must not be in Σd, the limit p0 := γ(t0) must be an element of Σc such that the restriction γ : [0,t0] → _Σc is completely within the connected component ofΣc. For each of the neighbourhoodsUp₀ of p0

the intersectionUp₀ ∩ M^˚ i is non-empty, since due to the definition oft0there must be at⁰ > t0

such thatγ(t⁰) ∈ Up₀ ∩ M^˚ _i. Hence p0 is an element of the topological boundary. By Lemma 5.1.21the whole connected component of Σc and in particular p = γ(0) are in the topological boundary. Therefore we get

M_i ⊂ M^˚ _i. (5.8)

Now consider p ∈ Σd∩∂M˚ i. By Proposition5.1.1(ii), p is an isolated point in Σd. Therefore every neighbourhood Up of p has non-vanishing intersection with M_i, since ¯M_i is connected.

Hence p is an element of the topological boundary ∂^{t o p}Mi, which is also a subset of ∂^{t o p}M˚ i, since M_i ⊂ M^˚ _i by Equation (5.8). This completes the first inclusion ∂M˚ _i ⊂ ∂^{t o p}M˚ _i.

Second we show that the backward inclusion∂^{t o p}M˚ _i ⊂ ∂M˚_i holds. ˚M_i is a connected compo-nent of the open set M\Σ. Now considerp ∈ ∂^{t o p}M˚ i, thenp cannot be in the open set M\_Σ and hencep ∈ Σ. First let p ∈ Σcand consider ˜ϕto be the coordinate chart in a neighbourhood of p that has been defined in Equation (5.7). Since ˜U∩M^˚ i 6= ∅, there is a curve connecting p to a point in the interior ˜U∩ M^˚ _i such thatσ is strictly positive along the curve. Consequently p ∈ M_i and hence p ∈ ∂M˚_i. Now let p ∈ _Σd. Then in Morse coordinates M\_Σ consists of three connected open components. Since each of them is connected to the vertex by a path, p must be an element of ¯Mi and hence of∂M˚ i. This proves the backward inclusion.

The previous observations lead to the following proposition on the topology of the level sets in a neighbourhood of a compact conformal boundaryΣ.

5.1 topology of the conformal boundary Σ 97

Proposition5.1.23. If∂M˚ _i is compact and M geodesically null complete without boundary, then (i) The number of vertices in any connected component ofΣ∩∂M˚ i is finite.

(ii) There exists a pre-compact neighbourhoodU of∂M˚ iin M such that M˚ i can be written as M˚ i∩ U = ^[

s∈(−e,e)

Σ^s ∩M^˚ i∩ U

withΣ^s∩M^˚ _i∩ Ubeing(n−1)-dimensional hypersurfaces not leavingU, i.e.∂U ∩ _Σ^s∩ M^˚ _i

=

∅. In particular there are no critical points ofσwithin M˚ _i∩ U. (iii) The gradient vector fieldgradσif restricted to∂M˚ _i is complete.

Figure4.:topology near com-pact boundaries

Proof: The third claim is a consequence of Proposition5.1.18. By the requirements ∂M˚ i ⊂ Σ is compact and hence ∆σ is bounded on

∂M˚i ∩Σc. By Lemma5.1.21∂M˚ i∩Σc contains only connected parts of Σc as a whole and not just parts of it. Now applying Proposition 5.1.18gives the claim.

The first claim is a consequence of the compactness of∂M˚ _i. Let be p ∈ Σ_d∩∂M˚ i a vertex. By Proposition5.1.1(ii), p is an isolated point such that it admits a neighbourhood Up without any other vertices.

For p ∈ _Σc∩∂M˚ _i choose a neighbourhood Up without any vertices in it. With these choices

∂M˚ i ⊂ ^[

p∈∂M˚_i

Up

is a covering and by compactness admits a finite subcovering. Since the neighbourhood of every vertex must be an element of that covering, their number consequently must be finite.

The proof for the second claim uses the same idea. First we observe that for each p ∈ _Σthere is a connected neighbourhoodUp such that gradσ 6= 0 for all q ∈ Up\ {p}. Moreover, we demand each neighbourhood to be such that it includes only one connected component of Σ and its closureUp is supposed to be compact. Those neighbourhoods provide an open covering of∂M˚ _ithat must have a finite subcovering

U = ^[

j∈ {1,..,m}

Up_j.

Its closureU = ^Si∈ {1,..,m}Up_i is compact and so is its boundary∂U. By construction we have

∂U∩∂M˚ _i =_{∅. (}⁵_.⁹₎

Without loss of generality assume ˚Mi to be a subset of σ⁻¹( (0,∞) ). We will now prove by contradiction that there is an e such that for all 0 < δ < e we have Σ^δ ∩ M^¯ _i ∩∂U = _{∅. So} assume that for alle > 0 there exists aδ ∈ (0,e] such thatΣ^δ∩ M^¯ _i∩∂U 6= ∅. Hence we can choose for everye = ¹_i an element of that intersection

x_i ∈ _Σ¹ⁱ ∩ M^¯ _i∩∂U.

By the compactness of ¯U there is a subsequence {x_ij} that converges to an x ∈ ∂U. By con-struction we have σ(x_ij) = _i¹

j for that sequence. By continuity of σ we have σ(x) = 0 and hence x ∈ ∂M˚ i. This contradicts (5.9). The same argument holds for ˚Mi ⊂ σ⁻¹( (−∞, 0) ), so the first assumption really is no loss of generality.

As a consequence there exists an e >0 such thatΣ^δ∩M^¯ _i∩∂U = _∅for all 0 <|δ| ≤ e. The connected parts ofΣ^δ within ˚M_i∩Uare hypersurfaces since by construction dσ 6= 0 onU. We define

U :=U∩σ⁻¹( (−e,e) )∪M^¯ _i^c

98 Chapter 5: almost einstein structures with vanishing almost scalar curvature

where ¯M^c_i is the complement of ¯M_i in M. the setU by definition is open, contains ∂M˚ _i and the connected parts ofΣ^δwithin ˚Mi∩Uforδ ∈ (−e,e). Hence we get

U ∩ M^˚ _i = U ∩σ⁻¹( (−e,e) )∩ M^˚ _i

and the second claim follows directly by replacingσ⁻¹( (−e,e) )with the union of level sets.

Remark. By restrictingσto the open set ˚Mi∩ U, we are left with a smooth functionσ : ˚Mi∪ U → R. σ⁻¹( [α₀,α₁] ) are compact sets for 0 < α_i < e if σ is positive on ˚Mi and −e < α_i < 0 else. Hence by Theorem 1.5.1 the sets σ⁻¹(∞,α₀) and σ⁻¹(∞,α₁) are diffeomorphic and the former set is a deformation retract of the latter. Sincedσ is non-singular and closed in the constructed neighbourhood of∂M˚ _i, the level sets ofσrepresent a1-dimensional foliation of ˚M_i in a neighbourhood of its boundary.