We will now prove the supplementary lemmas that were used in the proof of Theorem2 above.
Lemma 6 LetI =(M3,g,K, μ,J)be a C21/2+ε-asymptotically Euclidean initial data set with non-vanishing energy E = 0, withx:M3\B →R3\BR(0)denoting the asymp-totic coordinate chart. Assume in addition that K satisfies the potentially stronger decay assumptions|K| ≤CI |x|−δ−εfor someδ≥ 32and allx∈R3\BR(0).
Let∅ =U ⊆ [0,1]be an open subset of[0,1]and defineIτ as in the proof of Theorem 2for eachτ ∈ U . Let a ∈ [0,1), b ≥0,η ∈ (0,1]be fixed. Then there exist constants σ >0and C>0, depending only onε,δ, a, b,η, CI, and E such that the following holds for anyσ > σ: Assume there exists a C1-mapFσ:U ×S2 → M3 such that for every τ ∈U the surfaceτσ := Fσ(τ,S2)is in the a priori classA(a,b, η)and has constant spacetime mean curvatureH(τσ) ≡ 2/σ with respect to the initial data setIτ. Assume further thatFσ is anormal variation mapin the sense that there exists a continuous lapse function u=uστ:τσ →Rsuch that∂τFσ =uν, whereν=ντσ is the unit normal toτσ in(M3,g). Then we have
uW2,2(στ)≤Cσ5−2δ−2ε, udW2,2(στ)≤Cσ92−2δ−3ε, (65) andLu=O(σ1−2δ−2ε).
Proof In this proof,C >0 andσ >0 denote generic constants that may vary from line to line, but depend only onε,δ,a,b,η,CI, andE. The surfacesτσhave constant spacetime
mean curvatureH(τσ) ≡ 2/σ in the initial data set Iτ. For clarity, we will write this constant spacetime mean curvature with an explicit reference to the initial data setIτ as H(τσ,Iτ)≡2/σfor allτ ∈U. Hence
∂τH(στ,Iτ)=0,
which gives us the following linear elliptic PDE on the closed surfaceτσ for the a priori only continuous lapse functionu=uστ:στ →R:
Lu= τ (trστ K)2
H(στ) , (66)
where the elliptic operatorL is (up to a certain factor) the linearization of the spacetime mean curvature operator for the surfaceτσin the initial data setIτ, defined in (27). Then Proposition2implies thatu ∈W2,2(τσ), and that such auis unique. Together with (25) andP=O(σ−δ−ε), (66) implies that
Lu=O(σ1−2δ−2ε). (67)
As a consequence, by Corollary1and (49), we get udW2,2(τσ)≤Cσ2LudL2(στ)
Using the Cauchy–Schwarz Inequality, integration by parts, and (34), together with fi ≈ 3
Combining the last three estimates with a Triangle Inequality, it follows that utL2(τσ)≤ σ3
6|mH| ˆ
τσuLfidμ
+Cσ−εutL2(στ)+ σ3 6|mH|
ˆ
τσudL fidμ
≤Cσ5−2δ−2ε+Cσ−εutL2(τσ)+Cσ12−εudL2(τσ), so that
utL2(τσ)≤Cσ5−2δ−2ε+Cσ12−εudL2(τσ). Recalling (68), we now get aW2,2-estimate forud, namely
udW2,2(στ)≤Cσ92−2δ−3ε. From this, as a consequence of (67) and Corollary1, we also have
utW2,2(τσ)≤ udW2,2(στ)+ uW2,2(στ)
≤ udW2,2(στ)+Cσ3LuL2(τσ)≤Cσ5−2δ−2ε.
Lemma6enables us to prove the following result.
Lemma 7 LetI =(M3,g,K, μ,J)be a C21/2+ε-asymptotically Euclidean initial data set with non-vanishing energy E = 0, with x denoting the asymptotic coordinate chart. Let
∅ =U ⊆ [0,1]be an open subset of[0,1]and defineIτas in the proof of Theorem2for eachτ ∈U . Let a ∈ [0,1), b ≥0,η ∈(0,1]be fixed. Then there exist constantsσ >0 and C>0, depending only onε, a, b,η, CI, and E such that the following holds for any σ > σ: Assume there exists a C1-mapFσ:U×S2→ M3such that for everyτ ∈U the surfaceτσ := Fσ(τ,S2)is in the a priori classA(a,b, η)and has constant spacetime mean curvatureH(τσ)≡2/σwith respect to the initial data setIτ. Assume further that Fσis a normal variation map in the sense explained in Lemma6. Then
∂τ
|x| ◦Fσ≤Cσ1−ε, (69) ∂τ|τσ|≤Cσ32−ε, (70)
|∂τ(z◦Fσ)| =O(σ1−2ε). (71) Proof In this proof,C >0 andσ >0 denote generic constants that may vary from line to line, but depend only onε,a,b,η, andCI, andE. Letu:στ →Rdenote the lapse function as in Lemma6. Then Lemma6applied withδ = 32 and the Sobolev Embedding Theorem in the form of Lemma12imply that|∂τFσ| = |u| ≤Cσ1−ε. Then (69) is proved by the elementary estimate
∂τ
|x| ◦Fσ= 3
i=1(xi◦Fσ)(∂τFσ)
|x| ◦Fσ ≤Cσ1−ε.
In order to prove (70), we first recall that the mean curvature ofστ satisfies H = σ2 + O(σ−2−ε), see (25). The first variation of area formula, the fact that the eigenfunctions
used to spanL2(τσ)areL2(στ)-orthogonal so that in particular we have´
τσutdμ=0, combined with Lemma6forδ= 32 lead to
∂τ|στ|= ˆ
στ H u dμ
≤ ˆ
τσH uddμ +
ˆ
στ(H− 2 σ)utdμ
≤CudL2(στ)+Cσ−1−εutL2(στ)
≤Cσ32−ε,
where we also used the Cauchy–Schwarz Inequality.
A very similar analysis demonstrates∂τ|τσ|δ≤ Cσ32−ε. Finally, we prove (71): By definition,
zi◦Fσ = 1
|τσ|δ
ˆ
τσxidμδ.
Using the variation of area formula, the Cauchy–Schwarz Inequality, (19), (20), and Lemma 6withδ= 32we compute
∂τ(zi◦Fσ)= 1
|στ|δ
ˆ
τσuνidμδ+ ˆ
τσxiu H dμδ
− 1
|στ|2δ∂τ|τσ|δ
=O(σ1−2ε).
This proves (71).
Lemma 8 LetI =(M3,g,K, μ,J)be a C21/2+ε-asymptotically Euclidean initial data set with non-vanishing energy E=0. Let∅ =U ⊆ [0,1]be an open, connected subset of[0,1] and defineIτas in the proof of Theorem2for eachτ∈U .
Let a,a∈ [0,1), b,b∈ [0,∞),η∈(0,2ε), andη∈(0,1]. Then there exists a constant σ > 0, depending only onε, a, a, b, b, η,η, CI, and E such that the following holds for anyσ > σ: Assume there exists a C1-mapFσ:U ×S2 → M3 such that for every τ ∈U the surfaceτσ := Fσ(τ,S2)is in the a priori classA(a,b, η)and has constant spacetime mean curvatureH(τσ) ≡ 2/σ with respect to the initial data setIτ. Assume further thatFσis a normal variation map in the sense explained in Lemma6. Now suppose in addition thatστ0 ∈A(a,b, η)for someτ0 ∈U . Then in factτσ ∈A(a,bτ, ητ)with bτ =b+O(σ−min{2ε−η,ε})andητ =η+O(σ−ε)for anyτ ∈U . Here, the constants in the O-notation depend only onε, a, a, b, b,η,η, CI, and E.
Remark 10 Note that the assumption η ∈ (0,2ε) of the lemma is not restrictive as the inclusionA(a,b, η1)⊆A(a,b, η2)for 0< η2< η1≤1 implies that we may without loss of generality decreaseη∈(0,1]to achieveη∈(0,2ε).
Proof We drop the explicit reference toσfor notational convenience, asσwill not be modified in this proof. Letrτandzτdenote the area radius and the coordinate center ofτ, respectively, and let (slightly abusing notation)xτdenote the restriction of the coordinate vectorxtoτ, wherexdenotes the asymptotic coordinate chart. The mean curvature ofτ is denoted by Hτ. In this proofσ >0,C>0 and constants involved in theO-notation may vary from line to line but depend only onε,a,a,b,b,η,η,CI, andE.
We first show that there existsητ =η+O(σ−ε)such that the second inequality describing the fact thatτ ∈A(a,bτ, ητ)in (15) holds, namely
(rτ)2+ητ ≤ |xτ|52+ε. (72) Sinceτ ∈A(a,b, η)for allτ ∈U, by the Mean Value Theorem combined with (70) and (20) we have
4π(rτ)2= |τ| = |τ0| +O(σ32−ε)=4π(rτ0)2(1+O(σ−12−ε)), hence
rτ=rτ0(1+O(σ−12−ε)). (73) Similarly, combining (69) with (19) and (20) we conclude that
|xτ| = |xτ0|(1+O(σ−ε)). (74) Sinceτ0 ∈A(a,b, η)we have
(rτ0)2+η≤ |xτ0|52+ε, which in the view of (73) and (74) can be written as
(rτ(1+O(σ−1/2−ε)))2+η≤
|xτ|(1+O(σ−ε))5
2+ε. Consequently, we have
(rτ)2+η(1−Cσ−ε)≤ |xτ|52+ε. Choosing
ητ:=η+logrτ(1−Cσ−ε), (75) (72) follows. Note that by (19) we have
ητ =η+ln(1−Cσ−ε)
lnrτ =η+O(σ−ε(lnσ )−1)=η+O(σ−ε).
We will now apply a similar method and adjust the value of the constantbτ so that the first and the third inequality in (15) describing the fact thatτ ∈A(a,bτ, ητ), that is
|zτ| ≤arτ+bτ(rτ)1−ητ (76)
and ˆ
τ
Hτ2dμ−16π≤ bτ
(rτ)ητ, (77)
hold withητas defined in (75).
First, we deal with (76). Sinceτ0∈A(a,b, η)we have
|zτ0| ≤arτ0+b(rτ0)1−η. Combining this with (73) and (71) we obtain
|zτ| +O(σ1−2ε)≤arτ+O(σ12−ε)+b(rτ+O(σ12−ε))1−η,
which in the view of (20) may be further rewritten as
|zτ| ≤arτ+(b+O(ση−2ε)+O(ση−12−ε))(rτ)1−η. Sinceε≤12, we conclude that
|zτ| ≤arτ+(rτ)1−η(b+O(ση−2ε)).
Recall that by our definition (75) ofητwe haveη=ητ−logrτ(1−Cσ−ε). Hence
|zτ| ≤arτ+rτ1−ητ(1−Cσ−ε)(b+O(ση−2ε)).
Consequently, we have
|zτ| ≤arτ+(rτ)1−ητ(b+Cσ−min{2ε−η,ε}). (78) Next, we address (77). We recall thatHτ2=H2−τ2P2= σ42−τ2P2which implies that
∂τHτ2 = O(σ−3−2ε). Consequently, we may compute using the variation of area formula and (65) withδ= 32that
∂τ ˆ
τ
Hτ2dμ= ˆ
τ
∂τHτ2dμ+ ˆ
τ
u Hτ3dμ=O(σ−2ε). (79) Again, sinceτ0∈A(a,b, η)we have
ˆ
τ0
Hτ20dμ−16π≤ b (rτ0)η.
As before, we use the Mean Value Theorem, (79), and (73) to rewrite this as ˆ
τ
Hτ2dμ−16π+O(σ−2ε)≤b(rτ+O(σ12−ε))−η which in the view of (20) may be further rewritten as
ˆ
τ
Hτ2dμ−16π≤(rτ)−η(b+O(σ−12−ε)+O(ση−2ε)).
Substitutingη=ητ−logrτ(1−Cσ−ε)in the view ofε≤ 12gives ˆ
τ
Hτ2dμ−16π≤(rτ)−ητ(1−Cσ−ε)(b+O(ση−2ε)).
Consequently, we have ˆ
τ
Hτ2dμ−16π≤(rτ)−ητ(b+Cσ−min{2ε−η,ε}). (80) Together, the inequalities (78) and (80) imply the existence of the constant bτ = b+ O(σ−min{2ε−η,ε})such that (76) and (77) hold. This concludes the proof as we can choose
aτ =aas the above computations show.
Lemma 9 Under the assumptions of Theorem2, there exists a constantσ >0, depending only onε, a, b,η, CI1, and E, and a compact setK ⊂M3such that the map1:(σ ,∞)×S2→ M3\K defined by(64)is a bijective C1-map.
Proof To prove the claim, we need to show that1isC1, injective, and surjective onto a suitably chosen exterior region ofM3. We already proved in Theorem2thatFσ and thus 1isC1with respect to theS2-component. The differentiability with respect toσ can be proven following the Implicit Function Theorem argument of Lemma4, where the graphical spacetime mean curvature map is to be interpreted as a function ofσ∈(σ ,∞)instead of as a function ofτ ∈ [0,1]. This is to be viewed in light of the uniqueness results in Sect.6.3.
Injectivity.In order to show injectivity of1, we need to assert that1σ1∩σ12 = ∅for any choice ofσ1 =σ2,σ1, σ2 > σ. This can be done by analyzing the lapse function of the variation1:(σ,∞)×S2→M3with respect toσ, namelyu=uσ1 :=g(∂σ1, ν), where ν=ν1σ denotes the outward unit normal of1σ →(M3,g). Ifu>0 on(σ ,∞)×S2, we can conclude that1is injective.
We will in fact show thatu=1+O(σ−21−ε). Again,C >0 andσ >0 denote generic constants that may vary from line to line, but depend only onε,a,b,η,CI1, andE. First, we note that since the spacetime mean curvature ofσ1 in the initial data setI1is constant, H(1σ)≡2/σ, we have
Lu=
1− P
H 2
∂σH(1σ)
=(1+O(σ−1−ε)) ∂σ 2
σ
= − 2
σ2 +O(σ−3−ε),
(81)
which uniquely determinesu ∈ W2,2(1σ)by Proposition2. Furthermore, by (81), (25), (16), and the asymptotic decay assumptions onI1, we have
L(u−1)=Lu+ |A|2+Ric(ν, ν)+ P
H(∇νtrgK − ∇νK(ν, ν))
= − 2
σ2 +O(σ−3−ε)+ H2
2 + |A˚|2+Ric(ν, ν)+ P
H(∇νtrgK − ∇νK(ν, ν))
=Ric(ν, ν)+O(σ−3−ε),
(82) whereH = H(1σ),P = P(1σ). This shows thatL(u−1) = O(σ−52−ε)which is not sufficient for concluding thatu=1+O(σ−ε)and thusu >0 via Corollary1and Lemma 12: such an argument would requireL(u−1) = O(σ−3−ε)which we obviously do not have.
Instead, we argue as in the proof of Lemma6. Forv:= u−1, the above computation shows thatLv=O(σ−52−ε), which in combination with Corollary1and (49) gives
vdW2,2(σ1)≤Cσ2LvdL2(σ
1)
≤Cσ2
LvL2(σ1)+ LvtL2(1σ)
≤Cσ2
σ−32−ε+ LvtL2(σ
1)
≤Cσ12−ε+σ−12−εvtL2(σ
1).
(83)
In addition, fori =1,2,3, by adding a rich zero and using the orthogonality ofvd and fi, directly carries over to our spacetime context), and integration by parts, we obtain as in Lemma6that
Combining these estimates, we get, again grouping terms as in Lemma6, that vtL2(σ1)≤Cσ12−εvdL2(σ1)+Cσ1−ε, strictly positive for allσ > σ. This shows that1is indeed injective.
Surjectivity.By construction, the STCMC-surfacesσ = 1(σ,S2)forσ > σ are in the class of asymptotically centered surfacesA(0,b, η)for someb>0 andη∈(0,1]. In particular, recalling Proposition1, eachσcan be written as a graph over a sphere enclosing the interior region of M3. Suppose p ∈ M3 is in the exterior region of someσ with σ > σ. By comparability of the coordinate and mean curvature radii for surfaces in the class A(0,b, η)(see (19) and (20)), we can findσ > σ such thatplies in the region enclosed byσ, and hence in the annulus Aσ,σ betweenσ = 1(σ,S2)andσ = 1(σ ,S2). Since1: [σ,∞)×S2→M3is continuous it follows thatAσ,σ =1([σ,σ] ×S2)hence p = 1(σ ,ˆ q) for someσˆ ∈ [σ,σ]andq ∈ S2. Asσ > σ was arbitrary, this proves
surjectivity.