Geometric characterizations of asymptotically hyperbolic Riemannian 3-manifolds by the existence of a suitable CMC-foliation

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by  Christopher Nerz

GEOMETRIC CHARACTERIZATIONS OF ASYMPTOTICALLY
HYPERBOLIC RIEMANNIAN 3-MANIFOLDS BY THE
EXISTENCE OF A SUITABLE CMC-FOLIATION
7
1 CHRISTOPHERNERZ
0
2
Abstract. In 1996, Huisken-Yau proved that every three-dimensional Rie-

b mannianmanifoldcanbeuniquelyfoliatednearinfinitybystableclosedsur-
e faces of constant mean curvature (CMC) if it is asymptotically equal to the
F (spatial)Schwarzschildsolution. Usingtheirmethod,Riggerprovedthesame
theoremforRiemannianmanifoldsbeingasymptoticallyequaltothe(spatial)
0
(Schwarzschild-)Anti-de Sitter solution. This was generalized to asymptoti-
2
callyhyperbolicmanifoldsbyNeves-Tian,Chodosh,andtheauthoratalater

stage. In this work, we prove the reverse implication as the author already
]
P didintheEuclideansetting,i.e.anythree-dimensionalRiemannianmanifold
A is asymptotically hyperbolic if it (and only if) possesses a CMC-cover satis-
fying certain geometric curvature estimates, a uniqueness property, and each
.
h surfacehascontrolledinstability. Astoyapplicationofthesegeometriccharac-
t terizations of asymptotically Euclidean and hyperbolic manifolds, we present
a
amethodforreplacinganasymptoticallyhyperbolicbyanasymptoticallyEu-
m
clidean end and apply this method to prove that the Hawking mass of the
[ CMC-surfacesisboundedbytheirlimitbeingthetotalmassoftheasymptot-
ically hyperbolic manifold, where equality holds only for the t=0-slice of the
2 (Schwarzschild-)Anti-deSitterspacetime.
v
1
8
1
9 1. Introduction
0
. In1996, Huisken-Yauprovedthatmanifoldswhichareasymptotictothespatial
1
Schwarzschild metric with positive mass possesses a foliation by stable constant
0
7 mean curvature (CMC) hypersurfaces, [HY96]. They used this foliation as a defi-
1 nition for the center of mass of the manifold and also gave a coordinated version
: of this center. Since then, this foliation proved to be a suitable tool for the study
v
i of asymptotically Euclidean (i.e. asymptotically flat Riemannian) manifolds and
X
several generalizations of Huisken-Yau’s result were made, e.g. by Metzger, Huang,
r Eichmair-Metzger, and the author, [Met07, Hua10, EM12, Ner15a]. In 2004, Rig-
a
ger used Huisken-Yau’s method—the mean curvature flow—to prove the existence
and uniqueness of such a foliation for manifolds asymptotic to the t=0-slice of the
(Schwarzschild-)Anti-deSitterspacetime,[Rig04]. Thisresultwasgeneralizedusing
othermethodstomoregeneralasymptoticallyhyperbolicmanifoldsbyNeves-Tian,
Chodosh, and the author, [NT09, NT10, Cho14, Ner16].
In [Ner15b], the author proved that the existence of a CMC-foliation is not only
an implication of asymptotic flatness but a characterization of it, i.e. an arbitrary
Riemannian 3-manifold possesses a ‘suitable’ CMC-foliation if and only if it is
asymptotically Euclidean. In this article, we prove the equivalent theorem for the
Date:February21,2017.
1
2 CHRISTOPHERNERZ
hyperbolic setting or more precisely the missing implication: if a Riemannian 3-
manifold possesses a ‘suitable’ CMC-foliation, then it is asymptotically hyperbolic.
As a toy application of these characterizations of asymptotically Euclidean and
hyperbolic manifolds, we show that we can replace any asymptotically hyperbolic
end by an asymptotically Euclidean one. Using this construction, [Bra97, HI01]
prove that if S ≥ −6, then the (hyperbolic) Hawking mass is monotone along
the leaves of the CMC-foliation and bounded from above by the total mass of
the surrounding (asymptotically hyperbolic) manifold, where equality holds if and
only if the surrounding manifold is a compact perturbation of the [t=0]-slice of the
(Schwarzschild-)Anti-de Sitter spacetime.
1.1. The main results.
Theorem I (CMC-characterization of asymptotically hyperbolic manifolds)
Let $ ∈(5;3) and υ ≥$+ 1 be constants and (M, g) be a Riemannian manifold.
2 2 −
(M, g) is C2 -asymptotically hyperbolic with1 timelike mass vector m~ if and only if
$,υ
there exists a family { Σ} of hypersurfaces of M such that
σ σ>σ0
(a) { Σ} is a W2,∞-asymptotically round CMC-cover;
σ σ>σ0 $,υ
(b) { Σ} covers M outside a compact set K =K(σ ) for every σ >σ ;
σ σ>σ1 1 1 0
(c) { Σ} is locally unique;
σ σ>σ0
(d) { Σ} has uniformly timelike and bounded Ricci-mass.
σ σ>σ0
Furthermore, the (coordinate-independent) hyperbolic Hawking mass of Σ con-
σ
verges to the total mass m..=|−m~|R3,1 of (M, g) as σ →∞.
The definitions used here are given as Definitions 3.4, 3.6, 3.8, and 3.11 on
pages 4–6. The existence of such a round cover for C2 -asymptotically hyperbolic
$,υ
manifold with $ ∈ (5;3) and υ > 3 was proven by the author in [Ner16]. In
2
this article, we prove the reverse implication, i.e. that the existence of a suitable
CMC-foliation implies the existence of a C2 -asymptotically hyperbolic chart.
$,υ
Furthermore, we prove that the Hawking mass is monotone increasing along the
foliationandboundedbythetotalmass,whereequalityonlyholdsforSchwarzschild-
Anti de Sitter.
Theorem II
Let $ ∈(5;3) be a constant and (M, g) be a C2-asymptotically hyperbolic manifold
2 $
with0≤e|x|(S+6)∈L1(M),wherexisanyC2-asymptoticallyhyperboliccoordinate
$
system2. The function mh : (σ ;∞) → R : σ 7→ mh( Σ) mapping the mean
H 0 H σ
curvature radius to the hyperbolic Hawking mass of the corresponding CMC-leaf
σΣ with mean curvature σH ≡ −2csoinshh((σσ)) is a monotone non-decreasing function
converging to (cid:12)(cid:12)|m−~|R3,1(cid:12)(cid:12) as σ →∞, where the later is the total mass3 of (M, g).
Here, equality holds for some large mean curvature radius if and only if M is
(outside of the corresponding CMC-leaf) isometric to the standard [t=0]-timeslice
(outside of a ball) of the Schwarzschild-Anti-de Sitter spacetime.
1Note that the mass vector −m~ itself depends on the asymptotically hyperbolic coordinates
system,butthetotalmassm..=−|−m~|R3,1 doesnot,andthereforeitisacoordinateindependent
propertywhetherthemassvectoristimelikeornot,[CH03].
2Morepreciselye|x|(S+6)hasonlytobeintegrable,where|x|iswell-defined.
3withrespecttoanyasymptoticallyhyperboliccoordinatesystem
GEOM. CHARAC. OF ASYMP. HYPERB. MFLDS BY EX. OF A SUIT. CMC-FOLIATION 3
Remark 1.1. IncontrasttotheEuclideansetting,TheoremIIisnotnecessarilytrue
ifwereplace{ Σ} withsomeotherfoliationofM,i.e.therearesmooth(arbitrarily
σ σ
round) hypersurfacesΣ within M having larger Hawking mass than the total mass
ofM. Thiscanbeseenasastraightforwardcalculationprovesthatthemassvector
−
m~(y) for any non-balanced asymptotically hyperbolic coordinate system y of M
satisfies
(cid:12)(cid:12)mhH(cid:0){|y|=r}(cid:1)(cid:12)(cid:12)2 −r−→−∞→(cid:12)(cid:12)m0(y)(cid:12)(cid:12)2 =−(cid:12)(cid:12)m−~(y)(cid:12)(cid:12)2R3,1 +(cid:12)(cid:12)(cid:12)(cid:0)mi(y)(cid:1)3i=1(cid:12)(cid:12)(cid:12)2R3 >−(cid:12)(cid:12)m−~(y)(cid:12)(cid:12)2R3,1.
Here, weusedNeves-Tian’sdefinitionofabalancedcoordinatesystem, see[NT10]:
a coordinate system x of an asymptotically hyperbolic manifold is called balanced
−
if and only if m~(x)=(m(x),0,0,0).
Theorem II is actually a direct corollary of the monotony of the Hawking mass
along the CMC-foliation, [Bra97], and under the inverse mean curvature flow,
[HI01], combined with the following corollary of Theorem I and [Ner15b]:
Corollary 1.2 (Replacing the asymptotic end)
Let $ ∈(5;3) and υ ∈[$+ 1;2$] be arbitrary constants.4 For every C2 -asymp-
2 2 $,υ
toticallyhyperbolicmanifold(M, g),thereexistsametricfg onMwiththefollowing
properties
• fg is C2 -asymptotically Euclidean;
$−2,υ
• the CMC-foliations with respect to g and fg are identical;
• g|TσΣ2 =fg|TσΣ2 for every σ >σ0 and g =fg in M\SσσΣ;
• fν =cosh(σ) gν for every σ >σ and the outer unit normals fν and ν of
σ σ 1 σ σ
σΣ with respect to fg and g, respectively;
• mh( Σ)=m ( Σ) for every σ >σ ;
H σ H σ 1
• if gS ≥−6, then fS|σΣ ≥0 for every σ >σ1.
Here, σ denotes the infimum of the mean curvature radius of the canonical CMC-
0
foliation and σ >σ is arbitrary.
1 0
Note that the behavior of fS is also in Sσσ1>σ0σΣ well-controlled—i.e. in the
region between the one where [g=fg] and the one where [fS≥0]—, see the con-
struction (14) on page 25.
Acknowledgment. The author thanks Stephen McCormick and Katharina
RadermacherforhelpfuldiscussionsonapplicationsofTheoremIIandtheAlexan-
der von Humboldt Foundation for ongoing financial support via the Feodor Lynen
scholarship.
2. Structure of the paper
In Section 3, we give the basic definitions and explain the notations used in
this article. In particular, we define what W2,p -asymptotically round spheres and
$,υ
covers are. We prove in Section 4 that W2,p -asymptotically round spheres satisfy
$,υ
strictestimatesontheirextrinsiccurvatureandotherregularitypropertiesofthese
objects. Then, we use them in Section 5 to conclude strict estimates on W2,p -as-
$,υ
ymptotically round covers and to show that such a cover always has a well-defined
mass. In Section 6, we then explain and present the proof of Theorem I. Finally,
we prove Theorem II and Corollary 1.2 in the last Section 7.
4υ maybeevenbein[$;2$]ifS ≥−6.
4 CHRISTOPHERNERZ
3. Assumptions and notation
Notation 3.1 (Notations for the most important tensors)
In order to study foliations (near infinity) of three-dimensional Riemannian man-
ifolds by two-dimensional spheres, we have to deal with different manifolds (of
different or the same dimension) and different metrics on these manifolds, simul-
taneously. To distinguish between them, all three-dimensional quantities like the
surroundingmanifold(M, g),itsRicciandscalarcurvatureRicandS andallother
derived quantities carry a bar, while all two-dimensional quantities like the CMC
leaf (Σ, g), its second fundamental form k, the trace-free part of its second funda-
mental form k◦ ..= k− 1(trk)g, its Ricci, scalar, and mean curvature Ric, S, and
2
H ..=trk, its outer unit normalν, and all other derived quantities do not.
As explained, we interpret the second fundamental form and the normal vector
of a hypersurface as quantities of the surface (and thus as two-dimensional). For
example, if Σ is a hypersurface in M, then ν denotes its normal (and not ν).
σ σ σ
The same is true for the ‘lapse function’ and the ‘shift vector’ of a hypersurfaces
arisingasaleafofagivendeformationorfoliation. Furthermore,westressthatthe
sign convention used for the second fundamental form, i.e. k(X,Y) = g(∇XY,ν)
for X,Y ∈ X(Σ), results in the negative mean curvature eH(S2) ≡ −2 for the
r r
two-dimensional Euclidean sphere of radius r.
Notation 3.2 (Left indexes and accents of tensors)
If different two-dimensional manifolds or metrics are involved, then the lower left
index denotes the mean curvature index σ of the current leaf Σ, i.e. the leaf with
σ
mean curvature σH ≡ −2csoinshh((σσ)), or the radius r of a coordinate sphere S2r(0).
Quantitiescarrytheupperleftindexh,e,andΩiftheyarecalculatedwithrespect
to the hyperbolic metric hg, the Euclidean metric eg, and the standard metric σΩ
oftheEuclideansphereS2(0),correspondingly. Furthermore,weusetheupperleft
σ
index r for quantities calculated with respect to the hyperbolic metric hg along a
specific(‘round’)embeddingoftheCMC-leafstothehyperbolicspace,seeSection6.
We abuse notation and suppress the left indexes, whenever it is clear from the
context which manifold and metric we refer to.
Notation 3.3 (Indexes)
We use upper case latin indices I and J for the two-dimensional range {2,3}, the
lowercaselatinindexiforthethree-dimensionalrange{1,2,3},andthegreekindex
α for the four-dimensional range {0,1,2,3}. The Einstein summation convention
is used accordingly.
As there are different definitions of ‘asymptotically hyperbolic’ in the literature,
we now give the one used in this paper.
Definition 3.4 (C2 -asymptotically hyperbolic Riemannian manifolds)
$,υ
Let $,υ > 0 be constants. A triple (M, g,x) is called (three-dimensional) C2 -
$,υ
asymptotically hyperbolic Riemannian manifold if (M, g) is a three-dimensional
smooth Riemannian manifold and x : M\L → R3 is a smooth chart of M out-
side a compact set L⊆M such that there exists a constant c≥0 with
(cid:12)(cid:12)g −hg(cid:12)(cid:12)hg+(cid:12)(cid:12)(cid:12)h∇(cid:0)g −hg(cid:1)(cid:12)(cid:12)(cid:12)hg+(cid:12)(cid:12)(cid:12)Ric−hRic(cid:12)(cid:12)(cid:12)hg ≤ce−$|x|, (cid:12)(cid:12)(cid:12)S −hS(cid:12)(cid:12)(cid:12)≤ce−υ|x|,
GEOM. CHARAC. OF ASYMP. HYPERB. MFLDS BY EX. OF A SUIT. CMC-FOLIATION 5
wherehg =dr2+sinh(|x|)2ΩandΩdenotethehyperbolicmetricandthestandard
metricoftheEuclideanunitsphereS2,respectively. Here,thesequantitiesareiden-
tifiedwiththeirpush-forwardalongx. Finally,(M, g,x)iscalledC2-asymptotically
$
hyperbolic if it is C2 -asymptotically hyperbolic.
$,$
We often abuse notation and suppress the chart x.
Remark 3.5 (Boundedness of the scalar curvature). For everything, we do in this
article the assumption on the scalar curvature can also be reduced to
−ce−υ|x| ≤S −hS, e|x|(cid:16)S +hS(cid:17)∈L1(cid:0)M(cid:1),
see also Remarks 3.14, 4.7, 4.9, and 5.2. However, we then have to assume that
the mass of (M, g) is future pointing timelike instead of only assuming that it is
timelike, see Theorem I.
Definition 3.6 (Controlled instability)
LetΣ,→(M, g)beahypersurfacewithinathree-dimensionalRiemannianmanifold
and let α∈R be a constant. IfΣ has constant mean curvature, then it is called of
α-controlledinstability ifthesmallesteigenvalueofthe(negative)stabilityoperator
−L is greater than (or equal to) α, i.e.
ˆ ˆ
(cid:16) (cid:17)
|∇f|2dµ≥ |k|2+Ric(ν,ν)+α (f −f\)2dµ ∀f ∈H2(Σ),
g g
ffl ´
where f\ ..= fdµ ..= |Σ|−1 fdµ denotes the mean value of any function f ∈
H2(Σ). The surface Σ is called stable and strictly stable if it has α-controlled
instability for α=0 and α>0, respectively.
Definition 3.7 (Hyperbolic hawking mass, e.g. [Wan01])
If (Σ, g) is a hypersurface within a three-dimensional Riemannian manifold, then
ˆ
(cid:18) (cid:19)1(cid:18) (cid:19)
mh(Σ)..= |Σ| 2 1− 1 (cid:0)H2−4(cid:1)dµ
H 16π 16π
Σ
is called hyperbolic Hawking mass of Σ.
Definition 3.8 (Round spheres)
Let (M, g) be a three-dimensional Riemannian manifold and let $ >0, η ∈(0;4],
p∈[1;∞], and c≥0 be arbitrary constants.
A hypersurfaceΣ,→(M, g) with constant mean curvature is called W2,p (c,η)-
$,υ
asymptotically round sphere of mean curvature radius σ if
(RS-1) Σ is diffeomorphic to the Euclidean sphere;
(RS-2) Σ has constant mean curvature with mean curvature radius σ, i.e. H ≡
−2cosh(σ);
sinh(σ)
(RS-3) (cid:13)(cid:13)|Ric+2g|g(cid:13)(cid:13)Lp(Σ) ≤ce−$σ|Σ|p1 and kS +6kL1(Σ) ≤ce−υσ|Σ|;
(RS-4) Σ has −(4−η)sinh(σ)−2 controlled instability;
(RS-5) one of the following assumptions is true
(RS-5a) Σ satisfies e−2σ | Σ|∈(c−1;c);
σ
(RS-5b) Σ has −1(4−η)sinh(σ)−2 controlled instability;
2
(RS-5c) Σ has c-bounded Hawking mass, i.e. |mh(Σ)|∈(c−1;c).
H
6 CHRISTOPHERNERZ
The Ricci-mass −m~(Σ)=(mα(Σ)) ∈R3,1 of such a surface is defined by
α
m0(Σ)..=− |Σ| G0, mi(Σ)..= √|Σ| Gi ∀i∈{1,2,3},
16π32σ 16 3π32σ
where σGnfn denotes the Fourier series of σG ..= (Ric(ν,ν)− 12S −1)|σΣ and ν
is a unit normal of Σ, i.e. Gn denotes the nth-coefficient of G with respect to
σ σ
the complete L2( Σ)-orthogonal system {f }∞ of eigenfunctions of the (negative)
σ n n=0
Laplace operator with corresponding eigenvaluesλ satisfyingλ ≥λ ≥0.
n n+1 n
Remark 3.9 (The definition of Ricci-mass). Note that coordinate spheres Σ =
S2(0) ..= {|x| = r} in an asymptotically hyperbolic manifold with mass vector
r
−
m~ =(m0,...,m3) satisfies
ˆ
m0(cid:16)S2r(0)(cid:17)= −8π1 rG sinh(r)dµ+O(cid:0)e−εσ(cid:1)
S2(0)
ˆ r
= −1 (cid:18)Ric− 1Sg − g(cid:19)(cid:16)ν,X0(cid:17)dµ+O(cid:0)e−εσ(cid:1)=m0+O(cid:0)e−εσ(cid:1),
8π 2
ˆ
mi(cid:16)S2r(0)(cid:17)= 81π rGxis|ixn|h(r)dµ+O(cid:0)e−εσ(cid:1)
S2(0)
ˆ r
= 1 (cid:18)Ric− 1Sg − g(cid:19)(cid:16)ν,Xi(cid:17)dµ+O(cid:0)e−εσ(cid:1) =mi+O(cid:0)e−εσ(cid:1),
8π 2
0 1 3
where X and X ,...X are the radial vector field and the composition of the
translation (in the Euclidean standard directions) and the inversion map, i.e. the
basic conformal vector fields of the hyperbolic space, see [Her15] for more informa-
tion. This motivates our coordinate independent definition.
Remark 3.10 (On the mass assumptions of the spheres). Note that a posteriori
any round sphere has even O(e−3σ) controlled instability and satisfies e−2σ| Σ| ∈
σ
(C−1;C), i.e. a posteriori it satisfies at least (RS-5a) and (RS-5b). Thus, (RS-5c)
is the strongest of the assumptions in (RS-5). We will see that if the round sphere
has sufficiently large mean curvature radius σ and is an element of a round cover
(see below) with bounded and uniformly timelike Ricci-mass, then all assumptions
in (RS-5) are equivalent.
Definition 3.11 (Round covers)
Let (M, g) be a three-dimensional Riemannian manifold and let $ >0, η ∈(0;4],
p∈[1;∞], and c≥0 be arbitrary constants.
A family M ..= {σΣ}σ>σ0 of hypersurfaces of (M, g) is called W2,p(c,η)-asymp-
totically round CMC-cover if
(RC-1) each surface Σ is a W2,p (c,η)-asymptotically round sphere with mean
σ $,υ
curvatureH ≡−2cosh(σ),
sinh(σ)
S
(RC-2) Σ covers M outside of a compact set K(σ ) ⊆ M for every σ ∈
σ>σ1σ 1 1
(σ ;∞).
0
A W2,p(c,η)-asymptotically round CMC-cover M is called locally unique if
(RC-3) for everyΣ∈M there exist q =q(M,Σ)∈(2;p) and δ =δ(M,Σ)>0 such
that the following holds for every function f ∈W2,q( Σ)
σ
kfk <δ, H(graphf)≡const =⇒ graphf ∈M.
W2,q(σΣ)
GEOM. CHARAC. OF ASYMP. HYPERB. MFLDS BY EX. OF A SUIT. CMC-FOLIATION 7
It has uniformly timelike Ricci-mass if
(RC-4) |−m~(σΣ)|R3,1 <−c−1 for every σ >σ0
and bounded Ricci-mass if
−
(RC-5) |m~(σΣ)|R3,1 ∈(−c;c) for every σ >σ0.
Finally, the Ricci-mass −m~ of a W2,p-asymptotically round cover is defined by −m~ ..=
m−~(M)..=limσ−m~(σΣ)∈R3,1 if this limit exist.
In the following, we abbreviate W2,p by W2,p.
$,$ $
Remark 3.12 (The assumptions on the mass). Recalling Remark 3.9 and [Ner16],
we see that the CMC-foliation of any C2 -asymptotically hyperbolic Riemannian
$,υ
three-manifold with timelike mass vector −m~ is W2,p(c,η)-asymptotically round for
some constant c and every p ∈ (1;∞) and η ≥ 0. Furthermore, the Ricci mass of
−
this foliation is (±|m~|R3,1,0,0,0) for some sign ±∈{−1,1}.
Note that the assumption that the Ricci-masses of the leaves of a W2,p-asymp-
toticallyroundcoverareboundedanduniformlytimelikeimpliesthatthe(absolute
value of the) Hawking mass is bounded from below, but it does a priori neither
imply that the Hawking masses are bounded from above nor that the Ricci-masses
converge,i.e.thattheRicci-massofthecoveriswell-defined. However,a posteriori
−
both is true and even m~ =(m0,0,0,0), see Proposition 5.1.5
Remark 3.13 (Locally unique covers are foliations). Note that a priori we do not
assume that the cover is a foliation, i.e. that the surfaces are disjoint. However, we
willlaterseethattheelementsofalocallyuniquecoverwithboundedanduniformly
timelikeRicci-massareinfactpairwisedisjointandthereforeaposteriorithecover
is a foliation.
Remark 3.14(Boundednessofthescalarcurvature). Wecanreducetheassumption
on the scalar curvature by only assuming integrability and one sided boundedness,
i.e.
(cid:13)(cid:13)(cid:13)(cid:0)S +6(cid:1)−(cid:13)(cid:13)(cid:13)L1(σΣ) ≤ce−υσ|σΣ|, eσ(cid:13)(cid:13)S +6(cid:13)(cid:13)L1(σΣ) ≤˜c(σ)∈L1((σ0;∞)),
where (·)− ..= min{0, ·}, see also Remarks 3.5, 4.7, 4.9, and 5.2. However, we
then have to also assume that the Hawking mass (or equivalent the 0th-component
of the Ricci-mass) of every Σ is non-negative.
σ
Finally, we use the following partition of L2(Σ) (for any asymptotically round
sphereΣ) which was introduced and motivated in [Ner16, Sect. 4].
Definition 3.15 (Canonical partition of L2)
Let Σ be a W2,p (c,η)-asymptotically round sphere of mean curvature radius σ.
$,υ
Let gb be the L2(Σ)-orthogonal projection of a function g ∈ L2(Σ) on the linear
spanofeigenfunctionsofthe(negative)Laplacianwitheigenvalueλsatisfying|λ−
2sinh(σ)−2|≤ 3 sinh(σ)−2, i.e.
2
ˆ
(cid:26) (cid:12) (cid:27)
gb ..=X fi gfidµ (cid:12)(cid:12)(cid:12) 12 ≤sinh(σ)2 λi ≤ 27 ∀g ∈L2(Σ),
Σ
5The convergence of the Ricci-mass is implied by Theorem I, as it implies that Ricci-masses
convergeandthenProposition5.1provesthisclaim.
8 CHRISTOPHERNERZ
where {f }∞ denotes a complete orthonormal system of L2(Σ) by eigenfunctions
i i=0
f of the (negative) Laplace operator with corresponding eigenvalue λ satisfying
i i
0≤λ ≤λ . Finally, gd ..=g−gb denotes the rest of such a function g ∈L2(Σ).
i i+1
Elements of L2(Σ)b ..= {fb : f ∈ L2(Σ)} are called linearized boosts and those of
L2(Σ)d ..={fd :f ∈L2(Σ)} are called deformations.
4. Regularity of W2,p-asymptotically round spheres
$
Lemma 4.1 (Boundedness of the Hawking mass implies boundedness of the area)
For each constant c > 0, there exist two constants σ = σ (c) and C = C(c) with
0 0
the following property:
If Σ is a closed, oriented hypersurface of a three-dimensional Riemannian man-
ifold satisfying (RS-2) and (RS-5c) for some σ >σ , then
0
(cid:12) (cid:12)
|R−σ|≤Ce−σ, i.e. (cid:12)|Σ|−4πsinh(σ)2(cid:12)≤Ceσ,
(cid:12) (cid:12)
where R denotes the hyperbolic area radius, i.e. |Σ|=4πsinh(R)2. In particular,Σ
satisfies (RS-5a).
Proof. AsH2−4≡4 sinh(σ)−2, we know
(cid:12) (cid:12)
12sinh(R)(cid:12)(cid:12)(cid:12)(cid:12)1− ssiinnhh((Rσ))22(cid:12)(cid:12)(cid:12)(cid:12)=(cid:12)(cid:12)mhH(cid:12)(cid:12)∈(cid:0)c−1;c(cid:1).
This implies
sinh(R)2 C 
σ ≤R =⇒ e2(R−σ) ≤ sinh(σ)2 ≤1+ sinh(R) 
=⇒ |R−σ|≤Ce−σ. ///
σ ≥R =⇒ e2(R−σ) ≥ ssiinnhh((Rσ))22 ≥1− sinCh(σ) 
Lemma 4.2 (Strictly controlled instability implies boundedness of the area)
For each c > 0, $ > 5, and η ∈ (0;4], there exist two constants σ = σ (c,$,η)
2 0 0
and C =C(c,$,η) with the following property:
If Σ is a W2,p-asymptotically round sphere satisfying (RS-5b) for some σ > σ ,
$ 0
then Σ satisfies (RS-5a) for C instead of c, too.
Proof. Thisproofisequivalenttothebeginof[Ner16,ProofofThm3.1]. Werecall
it nevertheless for the readers convenience. We start as in [NT09, Lemma 4.1]
and use the test functions´ϕi ..= xi ◦ψ−1, where ψ : S2 → Σ is a conformal
parametrization of Σ with ϕ dµ = 0. These were already used by Huisken-Yau
i
in [HY96, Prop. 5.3] and were based on an idea by Christodoulou-Yau, [CY88]. By
the controlled instability assumption, this implies
ˆ ˆ ˆ !
8π 4−η
− = x Ω∆x dΩµ= ϕ ∆ϕ dµ≤ −|k|2−Ric(ν,ν) ϕ2dµ
3 S2 i i Σ i i 2sinh(σ)2 g i
GEOM. CHARAC. OF ASYMP. HYPERB. MFLDS BY EX. OF A SUIT. CMC-FOLIATION 9
foreveryi∈{1,2,3},wherewehaveusedtheconformalinvarianceof∆fdµ. Now,
we recall that (P ϕ2)◦ψ =P x2 ≡1 to get
i i i i
ˆ
8π ≥ (cid:12)(cid:12)k◦(cid:12)(cid:12)2+H2−4 +2+Ric(ν,ν)− 4−η dµ
g 2 2sinh(σ)2
ˆ
≥(cid:13)(cid:13)k◦(cid:13)(cid:13)2L2(Σ)+ η+2sCineh((2σ−)$2)σ dµ,
i.e.
ηsinh(R)2 ≤16πsinh(σ)2+Ce(2−$)σ sinh(R)2
implyingR≤σ+C,i.e.|Σ|≤Ce2σ. Ontheotherhand,theGauß-Bonnettheorem
and the Gauß equation combined with the assumptions on Ric give
ˆ ˆ
(cid:18) 2 (cid:19)
8π = Sdµ= S −2Ric(ν,ν)−(cid:12)(cid:12)k◦(cid:12)(cid:12)2+H dµ
g 2
ˆ ˆ
2 (cid:16) (cid:17)sinh(R)2
≤C e−$σdµ+ dµ≤ 8π+Ce(2−$)σ
sinh(σ)2 sinh(σ)2
implying R≥σ−C, i.e. C−1e2σ ≤|Σ|. ///
Corollary 4.3 (Round spheres have bounded area)
For each c > 0, $ > 5, and η ∈ (0;4], there exist two constants σ = σ (c,$,η)
2 0 0
and C =C(c,$,η) with the following property:
If Σ is a W2,p-asymptotically round sphere for some mean curvature radius σ >
$
σ , then Σ satisfies (RS-5a) for C instead of c.
0
Now, let us cite two major regularity results—in the notation we introduced
above. We can apply these results due to the result in Corollary 4.3.
Lemma 4.4 ([Ner16, Prop. 3.5])
For all constants $ ∈ (2;3], η ∈ (0;4], c > 0, and p ∈ (2;∞), there exist two
constants σ =σ ($,η,c,p) and C =C($,η,c,p) with the following property:
0 0
If (Σ, g) is a W2$,p,$(c,η)-asymptotically round sphere with σ > σ0, then there
exists a conformal parametrization ϕ:S2 →Σ with corresponding conformal factor
v ∈H2(S2), i.e. ϕ∗g =e2v sinh(σ)2 Ω, such that
kvkW2,p(S2,Ω) ≤Ce(2−$)σ, (cid:13)(cid:13)k◦(cid:13)(cid:13)W1,p(Σ) ≤Ce(1+p2−$)σ,
where Ω denotes the standard metric of the Euclidean unit sphere.
Remark 4.5. By the Gauß equation and the Gauß-Bonnet theorem, Lemma 4.4
implies that every W2,1 -asymptotically round sphere satisfies |mh(Σ)−m0(Σ)|≤
2+ε H
Ce−εσ.
Proposition 4.6 ([Ner16, Prop. 4.3])
For all constants $ ..= 5 +ε∈(5;3], υ ≥3+ε, η ∈(0;4], c>0, p>2, q ∈(1;p]
2 2
withq <∞, thereexisttwoconstantsσ =σ (ε,η,c,p)andC =C(ε,η,c,p,q)with
0 0
the following property:
If (Σ, g) is a W2$,p,υ(c,η)-asymptotically round sphere with σ >σ0, then (RS-5b)
holds for η arbitrary close to 4 (depending on σ ). More precisely L is invertible
0
10 CHRISTOPHERNERZ
and if f ∈ H2(Σ) is an eigenfunction of −L with corresponding eigenvalue κ, then
either |κ|≥ 3sinh(σ)−2 or
2
(cid:12) (cid:12)
(1) (cid:13)(cid:13)(cid:13)fd(cid:13)(cid:13)(cid:13)H2(Σ) ≤Ce−(12+ε)σ(cid:13)(cid:13)f(cid:13)(cid:13)H2(Σ), (cid:12)(cid:12)(cid:12)(cid:12)κ− s6inmh0((σΣ))3(cid:12)(cid:12)(cid:12)(cid:12)≤Ce−(3+ε)σ
and for all functions g,h∈H2(Σ) the inequality
(cid:12)ˆ ˆ (cid:12)
(2) (cid:12)(cid:12)(cid:12)(cid:12) Σ(cid:0)Lgb(cid:1)hbdµ+ s6inmh0((σΣ))3 Σgbhbdµ(cid:12)(cid:12)(cid:12)(cid:12)≤Ce−(3+ε)σ(cid:13)(cid:13)gb(cid:13)(cid:13)L2(Σ)(cid:13)(cid:13)hb(cid:13)(cid:13)L2(Σ),
holds. Furthermore, the corresponding W2,p-inequalities
!
(cid:13)(cid:13)gb(cid:13)(cid:13)W2,q(Σ) ≤ 6si(cid:12)(cid:12)nmh0((σΣ))3(cid:12)(cid:12) +Ce(3−ε)σ kLgkLq(Σ),
(cid:13)(cid:13)gd(cid:13)(cid:13)W2,q(Σ) ≤Ce2σkLgkLq(Σ),
(cid:13)(cid:13)He◦ssg(cid:13)(cid:13)Lq(Σ) ≤Ce(12−ε)σkLgkLq(Σ)
hold for every function g ∈W2,1(Σ).
Remark 4.7 (Assuming only one-sided boundedness of the scalar curvature). If we
onlyassumingonesidedboundednessofS asitisexplainedinRemark3.14,then(1)
and (2) have to be weakened to
(1’) (cid:13)(cid:13)(cid:13)fd(cid:13)(cid:13)(cid:13)H2(Σ) ≤Ce−(12+ε)σ(cid:13)(cid:13)f(cid:13)(cid:13)H2(Σ), κ≥ s6inmh0((σΣ))3 +Ce−(3+ε)σ,
and
ˆ ˆ
(2’) (cid:12)(cid:12)(cid:12)(cid:12) (cid:0)Lgb(cid:1)hbdµ(cid:12)(cid:12)(cid:12)(cid:12)≤ s6inmh0((σΣ))3 gbhbdµ+Ce−(3+ε)σ(cid:13)(cid:13)gb(cid:13)(cid:13)L2(Σ)(cid:13)(cid:13)hb(cid:13)(cid:13)L2(Σ)
Σ Σ
respectively, see [Ner16, Remark 4.4]. As we then also assume that the Hawk-
ing mass (or equivalent the 0th-component of the Ricci-mass) is positive, this still
implies thatL is invertible.
Lemma 4.8 (Local estimates of the lapse function)
For all constants $ ..= 5 +ε∈(5;3], υ ≥3+ε, η ∈(0;4], c>0, p>2, q ∈(1;p]
2 2
withq <∞, thereexisttwoconstantsσ =σ (ε,η,c,p)andC =C(ε,η,c,p,q)with
0 0
the following property:
If (Σ, g) is a W2,p (c,η)-asymptotically round sphere with |mh(Σ)|>c and σ >
$,υ H
σ0, then there exist a constant δ0 > 0 and a C1-map Φ : (−δ0;δ0) × S2 with
Σ = Φ(0,S2) and such that Σ ..= Φ(δ,S2) is a W2,p-hypersurface with constant
δ
mean curvature δH ≡−2csoinshh((σσ++δδ)) for every δ ∈(−δ0;δ0). In this setting, the lapse
function u..= g(∂σΦ,ν)|Σ satisfies
(cid:13) r (cid:13)
(3) (cid:13)(cid:13)ud−1(cid:13)(cid:13)W2,q(Σ) ≤Ce(q2−1−ε)σ, (cid:13)(cid:13)(cid:13)ub− |Σ3|mm0ifi(cid:13)(cid:13)(cid:13) ≤Ce(q2−ε)σ,
(cid:13) (cid:13)
W3,q(Σ)
where (mα)3 ..= −m~ ..= −m~(Σ) and where fi are L2-orthonormal eigenfunctions of
α=0
the Laplace operator with eigenvalue λi ∈ (sinh(σ)−2;3sinh(σ)−2). In particular,
the corresponding W3,q(Σ)-estimates of ub hold. Furthermore, u is strictly positive