Foliations by spheres with constant expansion for isolated systems without asymptotic symmetry

0.69 MB·English

by  Christopher Nerz

FOLIATIONS BY SPHERES WITH
CONSTANT EXPANSION FOR ISOLATED
SYSTEMS WITHOUT ASYMPTOTIC SYMMETRY
5 CHRISTOPHERNERZ
1
0
2 Abstract. Motivated by the foliation by stable spheres with constant mean
curvatureconstructedbyHuisken-Yau,Metzgerprovedthateveryinitialdata
n
set can be foliated by spheres with constant expansion (CE) if the manifold
a
is asymptotically equal to the standard [t=0]-timeslice of the Schwarzschild
J
solution. In this paper, we generalize his result to asymptotically flat ini-

9 tialdatasetsandweakenadditionalsmallnessassumptionsmadebyMetzger.
Furthermore,weprovethattheCE-surfacesareinawell-definedsense(asymp-

] totically)independentoftimeifthelinearmomentumvanishes.
P
A
.
h
t Introduction
a
m
MotivatedbyanideaofChristodoulouandYau[CY88],Huisken-Yauprovedthat
[ every Riemannian manifold is (near infinity) uniquely foliated by stable surfaces

with constant mean curvature (CMC) if it is asymptotically equal to the (spatial)
1
v Schwarzschild solution and has positive mass [HY96]. Their decay assumptions
7 weresubsequentlyweakenedbyMetzger,Huang,Eichmair-Metzger,andtheauthor
9 [Met07, Hua10, EM12, Ner14a]. Furthermore, the author proved that asymptotic
1
flatnessischaracterizedbytheexistenceofsuchaCMC-foliation[Ner14b]. Huisken-
2
Yau’s idea to use foliations by ‘good’ hypersurfaces was picked up by Metzger who
0
. proved that every initial data set, which is asymptotically to the standard [t=0]-
1
timeslice in the Schwarzschild solution, can (near infinity) be foliated by spheres
0
of constant expansion (CE) and that these CE-surfaces are unique within a well-
5
1 definedclassofsurfaces[Met07]. HemotivatedtheCE-foliationamongotherthings
: as foliation adapted to the apparent horizons which have zero expansion and that
v
CE-surfaces are the non-time symmetric analog of CMC-surfaces.
i
X
Note that the CMC- and the CE-foliation are not the only foliations used in
r themathematicalgeneralrelativity: Forexample,Lamm-Metzger-Schulzeachieved
a
correspondingexistenceanduniquenessresultsforafoliationbyspheresofWillmore
type [LMS11] and (in the static case) Cederbaum proved that the level-sets of the
static lapse function foliate the timeslices (near infinity) [Ced12]. However, we will
only use the CMC- and the CE-foliations in this paper.
ToexplainMetzger’sassumptionsforhisexistencetheoremfortheCE-foliation,
let us first recall that an initial data set is a tuple (M,g,k,J,%) satisfying the
Einstein constraint equations1
(1) 2%=S −(cid:12)(cid:12)k(cid:12)(cid:12)2+H2, J =div(cid:0)H g −k(cid:1).
g
Date:January12,2015.
1Wedroppedthephysicalfactor8π fornotationalconvenience.
1
2 CHRISTOPHERNERZ
Here, (M,g) is a Riemannian manifold, k is a symmetric (0,2)-tensor, J a (0,1)-
tensor, and% a function on M. This is motivated by a three-dimensional spacelike
hypersurface(M,g)withinaLorenzianmanifold(Mb,bg)withEinsteintensorG,the
secondfundamentalformkof(M,g),→(Mb,bg),itsenergydensity%..=G(ϑ,ϑ),and
the momentum density J ..= G(ϑ,·), where ϑ is the future pointing unit normal
of (M,g) ,→ (Mb,bg). If the surrounding Lorentzian manifold satisfies the Einstein
equations G = Rdic− 21Sbbg, then the Gauß-Codazzi equations of M ,→ (Mb,bg) are
equivalent to the constraint equations (1).
In this notation, Metzger assumed asymptotic to the standard [t=0]-timeslice
of the Schwarzschild solution, i.e. the existence a coordinate system x : M\L →
R3\B (0)mappingthemanifold(outsideofsomecompactsetL)totheEuclidean
1
space (outside of a closed unit ball), such that the push forward of the metric g is
asymptotically equal to the Schwarzschild metric Sg as |x| → ∞. More precisely,
heassumedthatthek-thderivativesofthedifference g −Sg ofthemetric g and
ij ij
the Schwarzschild metric Sg decays in these coordinates like |x|−1−ε−k for k ≤ 2.2
This is abbreviated with g −Sg = O2(|x|−1−ε). He furthermore assumed that the
second fundamental form k decays sufficiently fast, i.e. k = O1(|x|−2), and that
the corresponding constant is sufficiently small, i.e. |kij| ≤ η/|x|2 for some small
constant η (cid:28) 1 and correspondingly for the first derivative. He motivated this
point-wise assumption by the fact that at least in a specific example this foliation
only exists for sufficiently small second fundamental form. However, this example
of a second fundamental form is solely controlled by an integral quantity: the
ADM-linear momentum defined by Arnowitt-Deser-Misner [ADM61]. In the last
paragraph of [Met07], Metzger clarifies that (in the general setting) this foliation
is nevertheless not characterized by the ADM-linear momentum (or the ADM-
mass),i.e.smallnessofthelinearmomentumis(ingeneral)notsufficienttoensure
existence of the CE-foliation.
The first main result of this paper is the existence of the CE-foliations under
weaker decay assumptions on the metric. Furthermore, we only assume that the
secondfundamentalformisoforder|x|−2,hasasymptoticallyvanishingdivergence
J = div(Hg −k) = O0(|x|−3−ε), and need only additionally ‘smallness’ for some
integral-quantities of k.3 Here, we only state a simpler, less general version – see
Theorem 3.1 for the more general version.
Corollary 1 (Existence of CE-foliation – special case of Theorem 3.1)
Let(M,g,k,J,%)bea C2 -asymptoticallyflat,asymptoticallymaximalinitialdata
1+ε
2
set with non-vanishing mass m 6= 0. Assume that k is C0 -asymptotically anti-
2+ε
symmetric and vanishes C1-asymptotically. If the (ADM-)linear momentum is suf-
2
ficiently small, then there exist closed CE-surfaces Σ smoothly foliating M outside
σ
a compact set.
The definition of a ‘C2 -asymptotically flat initial data set’ is explained in
1+ε
2
Definition 1.3, while the other assumptions are explained in Theorem 3.1. We note
2Infact,heassumedthisdecayinamoregeometricway: g−Sg =O0(|x|−1−ε),Γijk−SΓijk=
O0(|x|−2−ε), and Ricij −SRicij = O0(|x|−3−ε) for some ε > 0, where we used the notation
explainedinSection1. Actually,healsoallowedε=0ifthecorrespondingconstantissufficiently
small.
3Notethatwecanaltertheassumptionsonk,seeRemark3.2.
FOLIATIONS BY SPHERES WITH CONSTANT EXPANSION 3
that the corresponding theorem is true for a temporal foliation (see Definition 1.4)
instead of an initial data set, i.e. every asymptotically flat temporal foliation of
a four-dimensional Lorentzian-manifold can be foliated (near infinity) by surfaces
withconstantexpansionwithrespecttothecorrespondingtimeslice(Theorem3.3).
As Metzger, we also get a uniqueness result for the CE-spheres (Theorem 3.4).
Again, we give a simple version – see Theorem 3.4 for the general version.
Corollary 2 (Uniqueness of CE-surfaces – special case of Theorem 3.4)
Let (M,g,x,k,J,%) satisfy the assumptions of Corollary 1. Let Σ1,Σ2 ,→ M be
CE-surfaces satisfying (specific) estimates. IfΣ andΣ have the same, sufficiently
1 2
small expansion, then they coincide.
Thepreciseformulationofthe‘(specific)estimates’canbefoundinTheorem3.4.
Furthermore, we can again reduce the assumptions on the initial data set – see
Theorem 3.4.
Finally, westudyhowtheseCE-surfacesevolveintimeundertheEinsteinequa-
tions (Theorem 4.1): We prove that the CE-spheres are in a well-defined sense
(asymptotically)independentoftimeiftheADM-linearmomentumvanishes. This
istobeexpectedastheauthorprovedthattheCMC-leaves(asymptotically)evolve
in time by translating in direction of the fraction of the (ADM) linear momentum
and the (ADM) mass [Ner13, Theorem 4.1] and the CE-spheres are asymptotically
just shifts of the CMC spheres (due to the results in Section 3) – and it seems
appropriate to assume that this shift is (asymptotically) independent of time. To
the best knowledge of the author, this is the first time that any evolution result
is proven for the CE-leaves. Theorem 4.1 implies the following (more descriptive)
corollary.
Corollary 3 (Time invariance of CE-surfaces – special case of Theorem 4.1)
Let (tM,tg,tk,t%,tJ)t be a (orthogonal) C21+ε-asymptotically flat temporal foliation
2
solving the Einstein equations in asymptotic vacuum. Assume that each of these
initial data sets satisfies the assumptions of Corollary 1. If the time-lapse function
is C2 -asymptotic to 1 and (asymptotically) symmetric, then the leaves of the CE-
1+ε
2
foliation evolve (asymptotically) in time by a shift in time direction (orthogonal to
each timeslice tM).
Acknowledgment. TheauthorwishestoexpressgratitudetoGerhardHuisken
for suggesting this topic and many inspiring discussions. Further thanks is owed
to Lan-Hsuan Huang for suggesting the use of the Bochner-Lichnerowicz formula
in this setting (see Proposition 3.9). Finally, thanks goes to Carla Cederbaum
for exchanging interesting thoughts about CMC- and CE-foliations and about the
interpretation of the integral quantities (see (16), (17), and Proposition 3.15).
Structure of the paper and main proof structure
In Section 1, we fix the notations and basic assumptions made in this paper.
Note that our assumptions on the decay of the second fundamental form is more
restrictive than the one used in parts of the literature, e.g. [Ner13, Hua10], but
lessrestrictivethanotherothers, e.g.[CK93,Met07]. InSection2, wecharacterize
the linearization of the map mapping a function to the expansion of its graph.
Furthermore,weexplainoneofthemainideasofthefollowingproofs. Theexistence
4 CHRISTOPHERNERZ
and uniqueness theorems are stated in full detail and proven in Section 3. In the
last main section (Section 4), we state and prove the evolution theorem.
As our main proof structure for the existence and uniqueness theorems is the
sameastheoneusedbyMetzgerin[Met07],webrieflyexplainhisproof. Hedefines
an interval I ⊆ [0;1] to be the set of all artificial times s ∈ [0;1] such that there
exists a foliation by CE-surfaces for (M,sg,sk,sJ,s%) instead of (M,g,k,J,%) and
proves by an open-closed argument that it is equal to [0;1]. Here, sg denotes the
artificial metric defined by
sg ..=Sg +s(cid:0)g −Sg (cid:1), Sg ..=(cid:18)1+ m (cid:19)4eg
ij ij ij ij ij 2|x| ij
and2s%..=sS−|sk|2 +s2H2 isanartificialenergy-density. Asitiswell-knownthat
sg
there is a CMC-foliation of the Schwarzschild standard-slice, he gets 0∈I. Hence,
I is non-empty. He proves with a convergence argument that I is also closed. In
order to prove that I is open, he uses the implicit function theorem: Let therefore
be {sr0Σ}r>r0 be the CE-foliation of (M,s0g,s0k,s0J,s0%) with s0 ∈ I and define
the map
(·H± ·P)ν :[0;1]×W2,p(sr0Σ)→Lp(sr0Σ):(s,f)7→(cid:0)sH ±sstrk(cid:1)(graphνf)
foreveryr >r0 andanyp>2,where(sH±sstrk)(graphνf)denotestheexpansion
ofgraphνf withrespecttosg andsk(seeSection2formoredetailedinformation).
Assume for a moment that the Fr´echet derivative of this map with respect to
the second component at (s ,0) is invertible. The implicit function theorem then
0
impliesthats0ΣcanbedeformedtoasurfacesΣwhichisaCE-surfacewithrespect
r r
to(M,sg,sk,sJ,s%)if|s−s0|issmallenough. ThisprovestheopennessofI under
theassumptionofinvertibilityoftheFr´echetderivativeoftheabovemap. Todeduce
thisinvertibility, heprovesmultipleestimatesforthedistanceofsuchaCE-surface
to the origin and the trace free part sk◦ of the second fundamental form sk of the
r r
surface rsΣ,→(M,sg). Here, he uses the concrete form of the Ricci-curvature of the
Schwarzschild metric and the assumed smallness of k.
We use the same approach, but replace three main arguments:
• as we know that the CMC-foliation of (M,g,x) exists [Ner14a, Thm 3.1],
we can fix the metric g instead of using the above family of metric {τg}τ;4
• we get the cruical estimate for the distance of τΣ to the coordinate origin
σ
by estimating its τ-derivative (see the Lemmas 3.11 and 3.12);
• to conclude the invertibility of the Fr´echet derivative explained above, we
use the Bochner-Lichnerowics formula and smallness of specific integral
quantities of k instead of the concrete form of the Ricci-curvature of the
Schwarzschildmetricandthepointwisesmallnessofk(seeProposition3.9).
1. Assumptions and notation
In this section, we describe the notations and decay assumptions used in this
paper. The notations used are the same as used by the author in [Ner13, Ner14a].
TheassumptionsontheRiemannianmanifoldareidenticaltotheonee.g.described
in[Ner14a,Sec.1]. Theassumptionsmadeontheotherquantitiesoftheinitialdata
4Notethattheproofof[Ner14a,Thm3.1]usestheexplainedmethodeincludingthefamilyof
metric{sg}s.
FOLIATIONS BY SPHERES WITH CONSTANT EXPANSION 5
set (respectively temporal foliation) are described in Definition 1.3 (respectively
Definition 1.4).
In order to study temporal foliations of four-dimensional spacetimes by three-
dimensional spacelike slices and foliations (near infinity) of those slices by two-
dimensional spheres, we will have to deal with different manifolds (of different
or the same dimension) and different metrics on these manifolds, simultaneously.
To distinguish between them, all four-dimensional quantities like the Lorentzian
spacetime (Mb,bg), its Ricci and scalar curvatures Rdic and Sb, and all other derived
quantities will carry a hat. In contrast, all three-dimensional quantities like the
spacelike slices (M,g), its second fundamental form k, its Ricci, scalar, and mean
curvature Ric, S, andH ..=trk, its future-pointing unit normalϑ, and all other de-
rivedquantitiescarryabar, whilealltwo-dimensionalquantitiesliketheCMCleaf
(Σ,g),itssecondfundamentalformk,thetrace-freepartofitssecondfundamental
formk◦ =k−1(trk)g, its Ricci, scalar, and mean curvature Ric, S, andH =trk, its
2
outer unit normalν, and all other derived quantities carry neither.
InSections2and4,theupperleftindexdenotesthetime-indextofthe‘current’
timeslice. In Section 3, it denotes the weight b. The only exceptions are the
upper left indices e and S which refer to Euclidean and Schwarzschild quantities,
respectively.
If different two-dimensional manifolds in one three-dimensional initial data set
(M,g,k,J,%) are involved, then the lower left index always denotes the radius R or
curvature index σ of the current leaf σΣ, i.e. the leaf with expansion σH ±σtrk =
−2/σ, where ± ∈ {−1,+1} always denotes a fixed sign. Furthermore, the two-
dimensional manifolds and metrics (and other quantities) ‘inherit’ the upper left
index of the corresponding three-dimensional manifold. We abuse notation and
suppress these indices, whenever it is clear from the context which metric we refer
to.
Here, we interpret the second fundamental form and the normal vector of a
hypersurface as quantities of the surface (and thus as ‘lower’-dimensional). For
example, if tM is a hypersurface inMb, then tϑ denotes its unit normal (and not tϑb).
The same is true for the ‘lapse function’ and the ‘shift vector’ of a hypersurfaces
arising as a leaf of a given deformation or foliation.
Finally, we use upper case Latin indices I, J, K, and L for the two-dimensional
range {2,3} and lower case Latin indices i, j, k, and l for the three-dimensional
range {1,2,3}. The Einstein summation convention is used accordingly.
As mentioned, we frequently use foliations and evolutions. These are infinitesi-
mally characterized by their lapse functions and their shift vectors.
Definition 1.1 (Lapse functions, shift vectors)
Let θ >0, σ0 ∈R be constants, I ⊇(σ0−θσ;σ0+θσ) be an interval, and (M,g)
be a Riemannian manifold. A smooth map Φ : I ×Σ → M is called deformation
of the closed hypersurface Σ = Σ = Φ(σ ,Σ) ⊆ M, if Φ(·) ..= Φ(σ,·) is a
σ0 0 σ
diffeomorphism onto its image Σ ..= Φ(Σ) and Φ ≡ id . The decomposition of
σ σ σ0 Σ
∂ Φ into its normal and tangential parts can be written as
σ
∂Φ
= u ν+ β,
∂σ σ σ σ
where ν is the outer unit normal to Σ, and β ∈ X( Σ) is a vector field. The
σ σ σ σ
function u : Σ → R is called the lapse function and β is called the shift of Φ.
σ σ σ
6 CHRISTOPHERNERZ
If Φ is a diffeomorphism (resp. diffeomorphism onto its image), then it is called a
foliation (resp. a local foliation).
In the setting of a Lorentzian manifold (Mb,bg) and a non-compact, spacelike
hypersurfaceM⊆Mb, the notions of deformation, foliation, lapseα, and shiftβ are
defined correspondingly.
As there are different definitions of ‘asymptotically flat’ in the literature, we
now give the decay assumptions used in this paper. To rigorously define these
and to shorten the statements in the following, we distinguish between the case of
a Riemannian manifold, the one of a initial data set, and the one of a temporal
foliation.
Definition 1.2 (C2 -asymptotically flat Riemannian manifolds)
1+ε
2
Let ε ∈ (0;1/2] be a constant and let (M,g) be a smooth Riemannian manifold.
The tuple (M,g,x) is called C2 -asymptotically flat Riemannian manifold if x :
1+ε
2
M\L→R3\B (0) is a smooth chart ofM outside a compact set L⊆M such that
1
(2) (cid:12)(cid:12)gij −egij(cid:12)(cid:12)+|x|(cid:12)(cid:12)(cid:12)Γijk(cid:12)(cid:12)(cid:12)+|x|2(cid:12)(cid:12)Ricij(cid:12)(cid:12)+|x|52 (cid:12)(cid:12)S(cid:12)(cid:12)≤ |x|c21+ε ∀i,j,k ∈{1,2,3}
holds for some constant c ≥ 0, where eg denotes the Euclidean metric. Arnowitt-
Deser-Misner defined the (ADM-)mass of such a manifold (M,g,x) by
ˆ
m ..= lim 1 X3 (cid:18)∂gij − ∂gjj(cid:19) νid µ,
ADM R→∞16π j=1 S2R(0) ∂xj ∂xi R R
where ν and µ denote the outer unit normal and the area measure of S2(0) ,→
R R R
(M,g) [ADM61].
In the literature, the ADM-mass is characterized using the curvature of g:
ˆ
−R S
(3) m..= lim Ric( ν, ν)− d µ,
R→∞ 8π S2(0) R R 2 R
R
see the articles of Ashtekar-Hansen, Chru´sciel, and Schoen [AH78, Sch88, Chr86].
Miao-Tamrecentlygaveaproofofthischaracterization,m =m,inthesetting
ADM
of a C2 -asymptotically flat manifold [MT14].5 We recall that this mass is also
1+ε
2
characterized by
(3’) m= lim m (cid:0)S2(0)(cid:1).
H R
R→∞
ThiscanbeseenbyadirectcalculationusingtheGaußequation,theGauß-Codazzi
equation, and the decay assumptions on metric and curvatures. Here, m (S2(0))
H R
denotestheHawking-masswhichisforanyclosedhypersurfaceΣ,→(M,g)defined
by
ˆ
r (cid:18) (cid:19)
|Σ| 1
mH(Σ)..= 16π 1− 16π H2dµ ,
Σ
whereH andµ denote the mean curvature and measure induced onΣ, respectively
[Haw03].
5TheauthorthankCarlaCederbaumforbringinghisattentiontoMiao-Tam’sarticle[MT14].
FOLIATIONS BY SPHERES WITH CONSTANT EXPANSION 7
Definition 1.3 (C2 -asymptotically flat initial data sets)
1+ε
2
Let ε ∈ (0;1/2] be a constant and let (M,g,k,J,%) be an initial data set, i.e.
(M,g) is a Riemannian manifold, k a symmetric (0,2)-tensor, % a function, and
J a one-form on M, respectively, satisfying the Einstein constraint equations (1).
The tuple (M,g,x,k,J,%) is called C2 -asymptotically flat if (M,g,x) is a C2 -
1+ε 1+ε
2 2
asymptotically flat Riemannian manifold and
(cid:12) (cid:12)
(4) |x| (cid:12)(cid:12)kij(cid:12)(cid:12)+|x|2 (cid:12)(cid:12)(cid:12)∂∂kxikj(cid:12)(cid:12)(cid:12)+|x|25 (cid:12)(cid:12)Ji(cid:12)(cid:12)≤ |x|c12+ε ∀i,j,k ∈{1,2,3},
holds in the coordinate system x. In this setting, the second fundamental form k
vanishes C1-asymptotically and C0 -asymptotically anti-symmetric if additionally
2 2+ε
(cid:12) (cid:12)
(cid:12)(cid:12)kij(cid:12)(cid:12)+|x| (cid:12)(cid:12)(cid:12)∂∂kxikj(cid:12)(cid:12)(cid:12)≤ |xc|2 and (cid:12)(cid:12)kij(x)+kij(−x)(cid:12)(cid:12)≤ |x|c2+ε,
respectively.
Definition 1.4 (C2-asymptotically flat temporal foliations)
Letε>0. ThelevelsetstM..=t−1(t)ofasmoothfunctiontonafour-dimensional
smooth Lorentzian manifold (Mb,bg) are called a (orthogonal) C21+ε-asymptotically
2
flattemporalfoliation,ifthegradientoftiseverywheretime-like,i.e.bg(∇bt,∇bt)<0,
and there is a chart (t,xb) : Ub ⊆ Mb → R × R3 of Mb, such that (tM,tg,tx ..=
xb|tM,tk,t%,tJ) is a C21+ε-asymptotically flat initial data set for all t (with not nec-
2
essarily uniform constants tc), ∂ | is orthogonal to tM for every t,6 and the time-
t tM
lapse function tα..=|bg(∂t,∂t)|1/2 ∈C1(tM) satisfies
(5) (cid:12)(cid:12)tα−1(cid:12)(cid:12)+|x| (cid:12)(cid:12)(cid:12)∂tα(cid:12)(cid:12)(cid:12)≤ tc , ∀i∈{1,2,3}.
(cid:12)∂xi(cid:12) |x|21+ε
Here, the corresponding second fundamental form tk, the energy-density t%, and
the momentum density tJ are defined by
tkij ..= 2−tα1 ∂∂tgtij, t%..=Rdic(cid:0)tϑ,tϑ(cid:1)+ Sb2, tJ ..=Rdic(cid:0)tϑ,·(cid:1), ∂t|tM =tαtϑ,
respectively,wheretϑisthefuture-pointingunitnormaltotM. Iftheconstantstcof
the above decay assumptions can be chosen independently of t, then the temporal
foliation is called uniformly C2 -asymptotically flat.
1+ε
2
Remark 1.5 (Weaker decay assumptions). We note that all the following results
remain true in the case that the above decay assumptions are only satisfied for
|x|f(|x|) instead of |x|−ε, i.e. if we replace the right hand side of (2), (4), and (5)
by |x|12 f(|x|), where f ∈ L1((1;∞)) is some smooth function with |x|f(|x|) → 0
for |x| → ∞.7 Furthermore, we can replace our pointwise assumptions by Sobolev
6Here, the orthogonality of ∂t|tM to tM is in fact not an additional assumption, as any coor-
dinatesystem(t,x)canbedeformed(usingflowsindirectionof∇bt)suchthatthisorthogonality
holdsforthenewcoordinatesystem.
7Furthermore, we have to assume 0 ≥ f0(|x|) ≥ −1/|x|, but if the above assumptions are
satisfied for some f, then there exists a f˜satisfying the above assumptions and this additional
assumptions.
8 CHRISTOPHERNERZ
assumptions, namely g −eg ∈ W31,p(R3 \B1(0)), S ∈ L1(M), and k ∈ W2,p(R3 \
2
B (0)), where p > 2 and where we used Bartnik’s definition of weighted Sobolev
1
spaces [Bar86] – compare with [Ner14a, Rem. 1.2].
Using one of DeLellis-Mu¨ller’s results [DLM05, Thm 1.1], the author proved in
[Ner14a, Prop. 2.4] (see Proposition 3.7) that every closed hypersurface which is
‘almost’ concentric and has ‘almost’ constant mean curvature is ‘almost’ umbilic,
see Proposition 3.7. Here, we call a surface ‘almost concentric’ if it is an element
of the following class of surfaces.
Definition 1.6 (Cη(c~z)-asymptotically concentric surfaces (Arε,η(c~z,c1)))
Let(M,g,x)beaC2 -asymptoticallyflatthree-dimensionalRiemannianmanifold
1+ε
2
andη ∈(0;1],c ∈[0;1),andc ≥0beconstants. Aclosed,orientedhypersurfaces
~z 1
(Σ,g),→(M,g)ofgenusgiscalledCη(c~z)-asymptoticallyconcentricwitharearadius
p
r = |Σ|/4π (and constant c1), in symbolsΣ∈Arε,η(c~z,c1) if
ˆ
c
|~z|≤c~zr+c1r1−η, r4+η ≤mΣin|x|5+2ε, ΣH2dµ−16π(1−g)≤ r1η,
where~z =(z )3 ∈R3 denotes the Euclidean coordinate center defined by
i i=1
ˆ
1
z ..= x deµ..= x deµ,
i i |Σ| i
Σ Σ
whereeµdenotesthemeasureinducedonΣbytheEuclideanmetriceg (withrespect
to x).
As it results in additional technical difficulties, we note that we cannot restrict
ourselves to ‘really’ asymptotically concentric surfaces, i.e. C1(0)-asymptotically
concentric surfaces, as the CMC-surfaces used in this work are not necessarily
within this class [CN14].
Finally, we specify the definitions of Lebesgue and Sobolev norms, we will use
throughout this article.
Definition 1.7 (Lebesgue and Sobolev norms)
Foreverycompacttwo-dimensionalRiemannianmanifold(Σ,g)withoutboundary,
the Lebesgue norms are defined by
ˆ
(cid:18) (cid:19)1
kTk ..= |T|pdµ p ∀p∈[1;∞), kTk ..=esssup|T| ,
Lp(Σ) g L∞(Σ) g
Σ Σ
whereT isanymeasurablefunction(ortensorfield)onΣandµdenotesthemeasure
induced by g. Correspondingly, Lp(Σ) is defined to be the set of all measurable
functions (or tensor fields) on Σ for which the Lp-norm is finite. If r ..= p|Σ|/4π
denotes the area radius ofΣ, then the Sobolev norms are defined by
kTk ..=kTk +rk∇Tk , kTk ..=kTk ,
Wk+1,p(Σ) Lp(Σ) Wk,p(Σ) W0,p(Σ) Lp(Σ)
wherek ∈N , p∈[1;∞], andT isanymeasurablefunction(ortensorfield)onΣ
≥0
for which the k-th (weak) derivative exists. Correspondingly, Wk,p(Σ) is the set of
all functions (or tensors fields) for which the Wk,p(Σ)-norm is finite. Furthermore,
Hk(Σ) denotes Wk,2(Σ) for any k ≥1 and H(Σ)..=H1(Σ).
FOLIATIONS BY SPHERES WITH CONSTANT EXPANSION 9
2. Pseudo stability operators of the expansion
In this section, we assume that (Mb,bg) is a Lorentzian manifold and that t is a
smooth regular function onMb such that its level sets tM..=t−1(t) form a C2 -as-
1+ε
2
ymptoticallyflattemporalfoliation(forsomeε>0)–withrespecttoafixedchart
xb.8 In particular, every level set (tM,tg,tx,tk,t%,tJ) is a three-dimensional initial
data set, where we used the same notation as in Definition 1.4. Furthermore, let
Σ,→M be a smooth, closed hypersurface in one of the timeslices M..=t0M.
The (conventional) stability operatorL ofΣ,→M is well-understood. It can be
defined as the linearization of the mean curvature map
(6) H:C2(Σ)→C0(Σ):f 7→H(graphνf)
at f ≡0, whereH(graphνf) is the mean curvature of
(7) graphνf ..=(cid:8)exp (f(p)ν) (cid:12)(cid:12) p∈Σ(cid:9)
p
with respect to the surrounding metric g. Here, exp denotes the exponential map
p
ofMatapointp∈Σ. Thisgraphisawell-definedclosedhypersurfaceifΣissmooth
and f lays in some well-defined L2(Σ)-neighborhood of zero (depending on Σ and
M). In particular, the (conventional) stability operator is well-defined for every
smooth, closed hypersurface Σ ,→ M and it is well-known that it is characterized
by
(cid:16) (cid:17)
Lf =∆f + Ric(ν,ν)+|k|2 f ∀f ∈C2(Σ).
g
AswewanttoconstructsurfaceswithconstantexpansionH ±trk(andnotwith
constant mean curvature), it is intuitive to replace the mean curvature map by the
‘expansion map’
(8) H±P:C2(Σ)→C0(Σ):f 7→(cid:0)H ±trk(cid:1)(cid:16)graph?f(cid:17)
as it was already done by Metzger [Met07]9, where ± denotes a fixed sign and
(H ±trk)(graph?f) denotes the expansion of the graph of f in ‘some direction’ –
in the mean curvature case (6), this direction was the spatial direction ν. We recall
that the expansion is the mean curvature of this graph within its future (or past)
expanding light-cone and that it is given by the mean curvature H of this graph
within (the corresponding) timeslice M plus (or minus) the two-dimensional trace
of the second fundamental form k of (the corresponding) timesliceM inMb. In this
case,therearemultiple‘intuitivedirections’inwhichthegraphcanbeconstructed.
Inthissection, wecharacterizetwoofthese(bylinearcombinationthisissufficient
for any direction):
First, we linearize this map ‘within M’, i.e. linearize the spatial expansion map
(H±P)ν :C2(Σ)→C0(Σ):f 7→(cid:0)H ±trk(cid:1)(graphνf).
where graphνf is as defined in (7), i.e. the same graph as used in the above case
of the (conventional) stability operator. To the best knowledge of the author, this
wasfirstdonebyMetzger[Met07]. Wedenotethispseudo stability operator byL .
±
ItshouldbenotedthatL doesnotariseasasecondvariationoftheareaoperator
±
8Infact,wedonotneedasymptoticallyflatnessinthissection,butonlythattMarespacelike
hypersurfacesfoliatingMb smoothly.
9NotethatMetzgerconsideredgraphνf,i.e.thegraphinspatialdirection.
10 CHRISTOPHERNERZ
f 7→ |graphνf| as the (conventional) stability operator does. The reason for this
is that the first the variation is done in null-direction (resulting in H±P) and
the second one in spatial-direction. This operator has been studied in more detail
by Andersson-Mars-Simon, Andersson-Metzger, Andersson-Eichmair-Metzger, and
others, see [AMS05, AM09, AEM11] and the citations therein.
Second, we linearize the corresponding map in time direction, i.e. linearize the
temporal expansion map
(9) (H±P)ϑ :C2(cid:0)M(cid:1)→C0(Σ):f¯7→(cid:0)H ±trk(cid:1)(cid:16)graphϑf¯(cid:17),
where graphϑf ..= {exp (f(p)ϑ) : p ∈Σ} ,→ f¯M ..= {exp (f¯(p¯)ϑ) : p¯∈M} is the
dp dp¯
graph in ‘future direction’. Here, edxpp and edxpp¯ denote the exponential map of Mb
at a point p∈Σ and at a point p¯∈M, respectively. This means in particular that
themeancurvatureandsecondfundamentalformkof(H±P)(f)atapointp∈Σ
are calculated with respect to the metric and second fundamental form of f¯M.10
The linearization of (H±P)ϑ is denoted by Lt . For notation convenience, we
±
only calculate Lt α, where α denotes the temporal lapse function of the temporal
±
foliation {tM} , i.e. we only look at functions f¯with graphϑf¯=tM for some t.
t
As first step, we calculate some identities for the Ricci curvature ofMb on a two-
dimensional deformation Φ : (t −η;t +η)×(−η;η)×Σ → M of a closed, two-
0 0
dimensionalsurfaceΣ=Φ(t ,0,Σ),→t0M. Again, werestrictourselvestothecase
0
of deformations compatible with the temporal foliation {tM} , i.e. Φ(t,σ,p) ∈ tM
t
for every t ∈ (t −η;t +η), σ ∈ (−η;η), and p ∈ Σ. Furthermore, we assume
0 0
that it is orthogonal, i.e. ∂ Φ =.. tαtϑ and ∂ Φ =.. tu tν for some smooth function
t σ σ σ
tu on tΣ ..= Φ(t,σ,Σ) and the temporal lapse function tα on tM, where tϑ again
σ σ
denotes the future pointing unit normal of tM ,→ Mb and σtν is the outer unit
normalof tΣ,→tM. Finally,wedenotethetimederivative∂ (Φ∗T)andthespatial
σ t
derivative ∂ (Φ∗T) of any quantity T by T˙ and T0, respectively.
σ
Proposition 2.1 (Curvature identities)
Let (Mb,bg) be a smooth Lorentzian manifold, {tM}t be a smooth temporal foliation
with respect to a smooth time function t on Mb. Additionally, let
Φ:(−θ+t0;θ+t0)×(−θ;θ)×Σ→Mb :(t,σ,p)7→Φ(t,σ,p)
be a smooth, orthogonal foliation-compatible deformation of a closed hypersurface
Σ ,→ t0M =.. M (see above). Suppressing the indices t and σ = 0, the tensor
0
identities
(10) Rdic(∂t,ν)=H˙ +α(cid:0)divkν−tr(cid:0)k(cid:12)k(cid:1)(cid:1)−α0trk+2kν(∇α)
(11) Rdic(cid:0)∂t,ϑ(cid:1)=(cid:0)trk+kνν(cid:1)˙−α(cid:16)(cid:12)(cid:12)k(cid:12)(cid:12)2+4(cid:12)(cid:12)kν(cid:12)(cid:12)2+kνν2(cid:17)+∆α+(cid:0)Hessα(cid:1)(ν,ν)
g g
(12) Rdic(cid:0)ϑ,∂σ(cid:1)=(cid:0)trk(cid:1)0−2kν(∇u)+u(cid:0)H kνν −divkν−tr(cid:0)k(cid:12)k(cid:1)(cid:1)
(cid:0) (cid:1)˙
(13) Rdic(ν,ν)=Ric(ν,ν)−2(cid:12)(cid:12)kν(cid:12)(cid:12)2g +kνν2+trkkνν − kανν −(cid:0)Hessα(cid:1)(ν,ν)
hold on Σ, where (k(cid:12)k)IJ ..=kIK gKLkJL, kν ..=k(ν,·), and kνν ..=k(ν,ν).
10Notethat(H±P)ϑ(f¯)dependsnotonlyonthevaluesoff¯onΣ,i.e.onf ..=f¯|Σ,butalso
onitsM-gradient∇f¯|Σ anditsM-hessianHessf¯|Σ.