A Note on Existence and Non-existence of Minimal Surfaces in Some Asymptotically Flat 3-manifolds

0.14 MB·English

by  Pengzi Miao

6
A Note on Existence and Non-existence of Minimal
0
0
Surfaces in Some Asymptotically Flat 3-manifolds
2

n
a
Pengzi Miao∗†
J

9
1

]
G Abstract
D
Motivated by problems on apparent horizons in general relativity,
. weprovethe followingtheoremonminimalsurfaces: Letg be ametric
h
3 3
t on the three-sphere S satisfying Ric(g)≥2g. If the volume of (S ,g)
a
isnolessthanonehalfofthe volumeofthe standardunitsphere,then
m
there are no closed minimal surfaces in the asymptotically flat mani-
[ fold (S3\{P},G4g). Here G is the Green’s function of the conformal
1 Laplacian of (S3,g) at an arbitrary point P. We also give an exam-
v ple of (S3,g) with Ric(g)> 0 where (S3\{P},G4g) does have closed
0
minimal surfaces.
8
4
1
1 Introduction
0
6
0 Let(N3,g,p)beaninitialdatasetsatisfyingthedominantenergyconstraint
/
h condition in general relativity. It is a fascinating question to ask underwhat
at conditions an apparent horizon (of a back hole) exists in (N3,g,p). Here an
m apparent horizon is a 2-surface Σ2 ⊂ N3 satisfying
:
v
H = Tr p, (1)
i Σ Σ
X
r where H is the mean curvature of Σ in N and Tr p is the trace of the
a Σ Σ
restriction of p to Σ.
Afundamentalresultof Schoen andYau states thatmatter condensation
causes apparent horizons to be formed [11]. Their result is remarkable not
only because it provides a general criteria to the existence question, butalso
∗Current address: Department of Mathematics, University of California, Santa Bar-
bara, CA 93106, USA.E-mail: [email protected]
†Address after April 3 06: School of Mathematical Sciences, Monash University, Vic-
toria 3800, Australia.
1
because it leads to a refined problem – besides matter fields, what is the
pure effect of gravity on the formation of apparent horizons?
To analyze this refined problem, one considers an asymptotically flat ini-
tial data set (N3,g,p) in a vacuum spacetime. As thefirststep, one assumes
(N3,g,p) is time-symmetric (i.e. p ≡ 0). In this context, an apparent hori-
zon is simply a minimal surface, and the relevant topological assumption
is that N3 is diffeomorphic to R3. (If N3 has nontrivial topology, a closed
minimal surface always exists by [8].)
There is a geometric construction of such an initial data set. Let [g]
be a conformal class of metrics on the three-sphere S3. Recall the Yamabe
constant of (S3,[g]) is defined by
[8|∇v|2 +R(g)v2]dV
Y(S3,[g]) = inf M g g, (2)
1
v∈W1,2(S3) R v6dV 3
M g
where R(g) is the scalar curvature of g.(cid:0)RIf Y(S3,(cid:1)[g]) > 0, there exists
a positive Green’s function G of the conformal Laplacian 8△ − R(g) at
g
any fixed point P ∈ S3. Consider the new metric G4g on S3 \{P}, it is
easily checked that (S3 \{P},G4g) is asymptotically flat with zero scalar
curvature. One basic fact about this construction is that the blowing-up
manifold (S3 \ {P},G4g), up to a constant scaling, depends only on the
conformal class [g]. Precisely, if one replaces g by another metric g¯∈[g] and
let G¯ be the Green’s function associated to g¯, then the metric G¯4g¯ differs
from G4g only by a constant multiple. Therefore, it is of interest to seek
conditions on [g] that determine whether (S3\{P},G4g) has a horizon.
Sofar,nosuchaconformalinvariantconditionhasbeenfound. However,
there are results where conditions in terms of a single metric are given. In
[1], Beig and O´ Murchadha studied the behavior of a critical sequence, i.e.
a sequence of metrics {g } on S3 converging to a metric g with zero scalar
n 0
curvature. They showed the blowing-up manifold (S3 \ {P},G4g ) has a
n n
horizon for sufficiently large n. Their idea was further explored by Yan [12].
Given ametric g onS3, assumingthediameter of(S3,g) ≤ D, thevolumeof
(S3,g) ≥ V and the Ricci curvature of g satisfies Ric(g) ≥ µg, Yan showed
that,foranyr > 3,thereexistsasmallpositivenumberδ = δ(µ,V,D,r) ≤ 1
2
such that, if R(g) > 0 and ||R(g)||Lr(S3,g) < δ, then the blowing-up manifold
(S3\{P},G4g) has a horizon.
One question arising from Yan’s theorem is whether a positive Ricci cur-
vature metric on S3 can produce a blowing-up manifold with a horizon, as
it is unclear whether Yan’s theorem could be applied when µ > 0. Another
motivation to this question is, as a positive Ricci curvature metric can be
2
deformedtothestandardmetricon S3 throughmetrics of positiveRiccicur-
vature, it is of potential interest to study how the horizon disappears in the
corresponding deformation of the blowing-up manifold if it exists initially.
In this paper, we focus on conformal classes of metrics with a positive
Riccicurvaturemetric. Ourmainresultistheobservationofavolumecondi-
tion which guarantees non-existence of horizons in the blowing-up manifold.
Throughout the paper, S3 denotes S3 with the standard metric of constant
curvature +1.
Theorem Let [g] be a conformal class of metrics on S3 which has a metric
of positive Ricci curvature. Consider
V (S3,[g]) = sup{Vol(S3,g¯) | Ric(g¯) ≥ 2g¯},
max
g¯∈[g]
where Vol(·) is the volume functional. If
1
V (S3,[g]) ≥ Vol(S3),
max
2
then the asymptotically flat manifold (S3\{P},G4g) has no horizon.
We also givean exampleof (S3,g) withRic(g) > 0where(S3\{P},G4g)
does have horizons.
2 Positive Ricci curvature and maximum volume
We first explain the volume assumption in the Theorem. Let Mn be a
smooth, connected, closed manifold of dimension n ≥ 3. Assume [g] is
a conformal class of metrics on Mn which has a metric of positive Ricci
curvature. One can define
V (Mn,[g]) = sup{Vol(Mn,g¯) | Ric(g¯)≥ (n−1)g¯}. (3)
max
g¯∈[g]
The following result relating V (Mn,[g]) and the Yamabe constant of
max
(Mn,[g]) was observed in [5].
Proposition 1 Let [g] be a conformal class of metrics on Mn which has a
metric of positive Ricci curvature. Then the Yamabe constant of (Mn,[g])
satisfies
Y(Mn,[g]) ≥ n(n−1)Vmax(Mn,[g])n2. (4)
3
Proof: By definition,
[c |∇v|2 +R(g¯)v2]dV
Y(Mn,[g]) = inf M n g¯ g¯ (5)
v∈W1,2(M) R M vn2−n2dVg¯ n−n2
(cid:16) (cid:17)
R
4(n−1)
for any g¯∈ [g], where c = .
n n−2
Now we assume Ric(g¯) ≥ (n−1)g¯. Then by a result of Ilias [7], which is
based on the isoperimetric inequality of Gromov [9], we have
n−2
[cn|∇v|2g¯+n(n−1)v2]dVg¯ ≥ vn2−n2dVg¯ n n(n−1)Vol(Mn,g¯)n2 (6)
ZM (cid:18)ZM (cid:19)
for any v ∈ W1,2(M). Note that R(g¯) ≥ n(n−1), hence
[c |∇v|2 +n(n−1)v2]dV
Y(Mn,[g]) ≥ inf M n g¯ g¯
v∈W1,2(M) R M vn2−n2dVg¯ n−n2
≥ n(n−1)Vol(Mn(cid:16),Rg¯)n2. (cid:17) (7)
Taking the supremum over g¯∈ [g] satisfying Ric(g¯)≥ (n−1)g¯, we have
Y(Mn,[g]) ≥ n(n−1)Vmax(Mn,[g])n2. (8)
2
As an immediate corollary, we see the assumption
1
V (S3,[g]) ≥ Vol(S3)
max
2
in the Theorem implies
2
Y(S3,[g]) ≥ 6 1 3 Vol(S3)32
2
(cid:18) (cid:19)
= Y(RP3,[g ]), (9)
0
where RP3 is the three dimensional projective space and g is the standard
0
metric on RP3 which has constant sectional curvature +1.
4
3 An upper bound of the Sobolev constant when
a horizon is present
One basic fact relating the conformal class [g] on S3 and the blowing-up
metric h = G4g on R3 = S3\{P} is
Y(S3,[g]) = 8S(h), (10)
whereS(h)istheSobolevconstantoftheasymptoticallyflatmanifold(R3,h)
[3]. Recall S(h) is defined by
|∇u|2 dV
S(h) = inf R3 h h . (11)
1
u∈W1,2(R3,h)(R( R3u6 dVh)3 )
The next proposition, which plays a key Rrole in the derivation of the Theo-
rem, was essentially established by Bray and Neves in [3] using the inverse
mean curvature flow technique [6]. As the statement of Bray and Neves is
different from what we need, we include the proof here.
Proposition 2 Lethbe acomplete metric onR3 suchthat (R3,h) isasymp-
totically flat. If (R3,h) has nonnegative scalar curvature and has a closed
minimal surface, then
1
S(h) < Y(RP3,[g ]). (12)
0
8
Proof: Since (R3,h) has a closed minimal surface, the outermost minimal
surface S in (R3,h), i.e. the closed minimal surface that is not enclosed by
any other minimalsurface[2], exists and consists of a finiteunion of disjoint,
embedded minimal two-spheres and projective planes. As our background
manifold is R3, S must consist of embedded minimal two-spheres alone,
furthermore each component of S necessarily bounds a three-ball.
We fixacomponentΣof S anddenotebyΩthethree-ballthatΣbounds
in R3. Let φ be the weak solution to the inverse mean curvature flow in
(R3\Ω¯, h) with initial condition Σ [6]. φ satisefies
φ≥ 0, φ| = 0, lim φ = ∞.
Σ
x→∞
Let Σ be the set ∂{u < t} for t > 0 and Σ be the starting surface Σ, then
t 0
the family of surfaces {Σ } satisfies the following properties [6]:
t
1. {Σ } consists of C1,α surfaces. For a.e. t, Σ has weak mean curvature
t t
H and H = |∇u| for a.e. x∈ Σ .
h t
5
2. |Σ | = et|Σ |, where |Σ | denotes the area of Σ .
t 0 t t
3. Since (R3,h) has nonnegative scalar curvature, Σ is connected and
R3\Ω¯ is simply connected, the Hawking quasi-local mass of Σ ,
t
|Σ | 1
m (Σ ) = t 1− H2dµ ,
H t
16π 16π
r (cid:18) ZΣt (cid:19)
is monotone increasing. Here dµ is the induced surface measure.
Now we restrict attention to functions u∈ W1,2(R3,h) that have the form
f(0) x ∈Ω
u(x) = (13)
f(φ(x)) x ∈R3\Ω
(cid:26)
for some C1 functions f(t) defined on [0,∞). By the coarea formula and
Property 1 above, we have
∞
|∇u|2dV = f′(t)2 Hdµ dt
h h
ZR3 Z0 (cid:18)ZΣt (cid:19)
∞
≤ f′(t)2 16π|Σ|(et −e2t)dt, (14)
Z0 q
where the inequality follows from Property 2, 3 and H¨older’s inequality.
Similarly, we have
∞
u6dV ≥ f(t)6 H−1dµ dt
h
ZR3 Z0 (cid:18)ZΣt (cid:19)
∞
≥ f(t)6e2t|Σ|2[16π|Σ|(et −e2t)]−12dt. (15)
Z0
Therefore,
R3|∇u|2hdVh ≤ (16π)23 0∞f′(t)2(et−e2t)12dt. (16)
1 1
R R3u6dVh 3 ∞f(Rt)6e2t(et −e2t)−12dt 3
0
(cid:0)R (cid:1) (cid:16) (cid:17)
R
To pick an optimal f(t) that minimizes the right side of (16), we consider
the half spatial Schwarzschild manifold
1
(M3,g ) = (R3\B (0),(1+ )4δ )
S 1 ij
|x|
6
and the quotient manifold (M˜3,g˜ ) obtained from (M3,g ) by identifying
S S
the antipodal points of {|x| = 1}. Up to scaling, (M˜3,g˜ ) is isometric
S
to (RP3 \ {Q},G4g ), the blowing-up manifold of (RP3,g ) by its Green
0 0 0
function at a point Q. Hence, the Sobolev constant S(g˜ ) of (M˜3,g˜ ) equals
S S
1Y(RP3,[g ]). On the other hand, S(g˜ ) is achieved by a function u that
8 0 S 0
is constant on each coordinate sphere {|x| = t} in M˜, and the level set of
the solution φ to the inverse mean curvature flow starting at {|x| = 1} in
0
(M,g ) is also given by coordinate spheres. Therefore, lifted as a function
S
on (M3,g ), u has the form of
S 0
u = f ◦φ
0 0 0
for some explicitly determined function f (t), and
0
S(g˜ ) = M |∇u0|2gSdVgS = (16π)23 0∞f0′(t)2(et −e2t)12dt , (17)
S 1 1
R( M u60 dVgS)3 0∞f0(Rt)6e2t(et −e2t)−12dt 3
R (cid:16) (cid:17)
R
where the second equality holds because the Hawking quasi-local mass re-
mains unchanged along the level sets of φ . Now consider u = f ◦ φ on
0 0
(R3,h). It was verified in [3] that u∈ W1,2(R3,h). Therefore, we have
S(h) ≤ R3|∇u|2h dVh ≤ (16π)23 0∞f0′(t)2(et −e2t)12dt
1 1
R( R3u6 dVh)3 0∞f0(Rt)6e2t(et−e2t)−21dt 3
R = S(cid:16)(Rg˜ )= 1Y(RP3,[g ]). (cid:17) (18)
S 0
8
To show the strict inequality, we assume S(h) = 1Y(RP3,[g ]). Then,
8 0
S(h) is achieved by u= f ◦φ. It follows from the Euler-Lagrange equation
0
of the Sobolev functional (11) that u satisfies
△ u+Cu5 = 0 on R3, (19)
h
where C = S(h)||u||−4 . However, u ≡ f (0) on Ω and f (0) 6= 0 (In-
L6(R3,h) 0 0
deed, up to a constant multiple, f0(t) = (2et −e2t)−12 [3]). Hence, C = 0,
which contradicts to the fact that u is not a constant. Therefore, the strict
inequality S(h) < 1Y(RP3,[g ]) holds. 2
8 0
Proof of the Theorem: Suppose(S3\{P},G4g)hasahorizon,thenitfollows
from (10) and Proposition 2 that
Y(S3,[g]) < Y(RP3,[g ]). (20)
0
7
On the other hand, the assumption V (S3,[g]) ≥ 1Vol(S3) implies
max 2
Y(S3,[g]) ≥ Y(RP3,[g ]) (21)
0
by (9), which is a contradiction. Hence, there are no horizons. 2
4 An example with horizons
In this section, we provide an example to show that there exist metrics on
S3 with positive Ricci curvaturesuch that the blowing-up manifolds dohave
horizons.
Our example comes from a 1-parameter family of left-invariant metrics
{g } on S3, commonly known as the Berger metrics. Precisely, we think S3
ǫ
as the Lie Group
z −w
SU(2) = : |z|2+|w|2 = 1 ,
w¯ z¯
(cid:26)(cid:18) (cid:19) (cid:27)
where the Lie algebra of SU(2) is spanned by
i 0 0 1 0 i
X = , X = , and X = .
1 0 −i 2 −1 0 3 i 0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Then {g } is defined by declaring X ,X ,X to be orthogonal, X to have
ǫ 1 2 3 1
length ǫ and X ,X to be unit vectors. Note that scalar multiplication on
2 3
eiθ 0
S3 ⊂ C2 correspondstomultiplication on theleft by matrices
0 e−iθ
(cid:18) (cid:19)
onSU(2), henceX isexactly tangenttothecirclefiberof theHopf fibration
1
π :S3 −→ S2 = S3/S1
and g shrinks the circle fiber as ǫ → 0. One fact of g for small ǫ is that
ǫ ǫ
all sectional curvature of (S3,g ) lies in the interval [ǫ2,4−3ǫ2] (see [10]), in
ǫ
particular g has positive Ricci curvature.
ǫ
Proposition 3 Let P ∈ S3 be a fixed point and G be the Green’s function
ǫ
of the conformal Laplacian of g at P. Then (S3\{P},G4g ) has a horizon
ǫ ǫ ǫ
for ǫ sufficiently small.
Proof: For each ǫ ∈ (0,1], we consider the rescaled metric g¯ = ǫ−2g and
ǫ ǫ
the Green’s function G¯ associated to g¯ at P. Then, with respect to g¯ ,
ǫ ǫ ǫ
X becomes a unit vector and X ,X have large length ǫ−1 as ǫ → 0. Let
1 2 3
8
U ⊂ S3 beafixedneighborhoodofP suchthatπ| isatrivialfiberation. Let
U
O be a fixed point in the product manifold S1×R2. By a scaling argument,
there exists a family of diffeomorphisms
Ψ : U −→ Ψ (U) ⊂ S1×R2,
ǫ ǫ
such that Ψ (P) = O ∈ Ψ (U), {Ψ (U)} forms an exhaustion family
ǫ ǫ ǫ 1≥ǫ>0
of S1 × R2 as ǫ → 0, and the push forward metrics gˆ = Ψ−1∗(g¯ | ) on
ǫ ǫ ǫ U
Ψ (U) converge in C2 norm on compact sets to a flat metric gˆ on S1 ×R2.
ǫ
Now fix another point Q ∈ Ψ (U) that is different from O and consider the
1
normalized function
G¯ ◦Ψ−1(x)
Gˆ (x) = ǫ ǫ (22)
ǫ G¯ ◦Ψ−1(Q)
ǫ ǫ
for x ∈ Ψ (U)\{O}. Then Gˆ satisfies
ǫ ǫ
8△ Gˆ −R(gˆ )Gˆ = 0 on Ψ (U)\{O}
gˆǫ ǫ ǫ ǫ ǫ . (23)
Gˆ = 1 at Q
ǫ
(cid:26)
Since Gˆ is positive and gˆ converges to gˆ as ǫ → 0, it follows from the
ǫ ǫ
Harnack inequality that Gˆ coverges to a positive function Gˆ on (S1×R2)\
ǫ
{O}inC2normonanycompactsetawayfrom{O}. Furthermore,Gˆ satisfies
△ Gˆ = 0 on (S1×R2)\{O}
gˆ . (24)
Gˆ = 1 at Q
(cid:26)
On the other hand, the fact that the geodesic ball in (S1 ×R2,gˆ) only has
quadraticvolumegrowthimplies(S1×R2,gˆ)doesnothaveapositiveGreen’s
function for the usualLapacian △ [4]. Therefore, Gˆ ≡ 1 on (S1×R2)\{O}.
gˆ
Hence, the metrics Gˆ4gˆ converge to gˆ in C2 norm on any compact set away
ǫ ǫ
from {O}. Now let V ⊂ S1 ×R2 be a small open ball containning O such
that ∂V is an embedded two sphere whose mean curvature vector computed
with respect to gˆ points towards O. Then, for suffiently small ǫ, the mean
curvaturevectorof∂V computedwithrespecttoGˆ4gˆ stillpointstowardsO.
ǫ ǫ
As (Ψ (U),Gˆ4gˆ ) is isometric to (U,G¯4g¯ ), the mean curvature vector of the
ǫ ǫ ǫ ǫ ǫ
boundary of Ψ−1(V) in (S3\{P},G¯4g¯ ) must point towards the blowing-up
ǫ ǫ ǫ
point P. On the other hand, as (S3 \{P},G¯4g¯ ) is asymptotically flat, its
ǫ ǫ
infinity is foliated by two shperes whose mean curvature vector points away
from P. Therefore, it follows from standard geometric measure theory that
thereexistsanembeddedminimaltwosphereinΨ (V),hence(S3\{P},G¯4g¯)
ǫ ǫ
has a horizon. 2
Acknowledgment I want to thank Justin Corvino and Rick Schoen for
helpful discussions.
9
References
[1] R. Beig and N. O´ Murchadha. Trapped surfaces due to concentration
of gravitational radiation. Phys. Rev. Lett., 66(19):2421–2424, 1991.
[2] Hubert L. Bray. Proof of the Riemannian Penrose inequality using the
positive mass theorem. J. Differential Geom., 59(2):177–267, 2001.
[3] Hubert L. Bray and Andr´e Neves. Classification of prime 3-manifolds
with Yamabe invariant greater than RP3. Ann. of Math. (2),
159(1):407–424, 2004.
[4] S. Y. Cheng and S. T. Yau. Differential equations on Riemannian
manifolds and their geometric applications. Comm. Pure Appl. Math.,
28(3):333–354, 1975.
[5] PengfeiGuanandGuofangWang. Conformaldeformationsofthesmall-
est eigenvalue of the ricci tensor. Max Planck Institute Preprint Nr.
43/2005, 2005.
[6] Gerhard Huisken and Tom Ilmanen. The inverse mean curvature
flow and the Riemannian Penrose inequality. J. Differential Geom.,
59(3):353–437, 2001.
[7] Sa¨ıd Ilias. Constantes explicites pour les in´egalit´es de Sobolev sur
les vari´et´es riemanniennes compactes. Ann. Inst. Fourier (Grenoble),
33(2):151–165, 1983.
[8] William Meeks, III, Leon Simon, and Shing Tung Yau. Embedded
minimal surfaces, exotic spheres, and manifolds with positive Ricci cur-
vature. Ann. of Math. (2), 116(3):621–659, 1982.
[9] Vitali D. Milman and Gideon Schechtman. Asymptotic theory of finite-
dimensional normed spaces, volume1200ofLecture Notes inMathemat-
ics. Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov.
[10] Peter Petersen. Riemannian geometry, volume 171 of Graduate Texts
in Mathematics. Springer-Verlag, New York, 1998.
[11] Richard Schoen and S. T. Yau. The existence of a black hole due to
condensation of matter. Comm. Math. Phys., 90(4):575–579, 1983.
[12] Yu Yan. The existence of horizons in an asymptotically flat 3-manifold.
Math. Res. Lett., 12(2-3):219–230, 2005.
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