On the Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space

0.23 MB·

by  Yong Wei

ON THE MINKOWSKI-TYPE INEQUALITY FOR OUTWARD
MINIMIZING HYPERSURFACES IN SCHWARZSCHILD SPACE
YONGWEI
7
1
0
Abstract. UsingtheweaksolutionofInversemeancurvatureflow,weprovethesharp
2
Minkowski-typeinequalityforoutwardminimizinghypersurfacesinSchwarzschildspace.

n
a
J

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1 1. Introduction

] The Schwarzschild space is an n-dimensinal (n ≥ 3) manifold (Mn,g) with boundary
G
∂M, which is conformal to Rn\D , with the metric
D r0
h. g (x) = 1+ m|x|2−n n−42 δ , x ∈ Rn\D , (1.1)
t ij 2 ij r0
a (cid:16) (cid:17)
m wherem > 0 is a constant, r0 = (m2)n−12. Thecoordinate sphereSr0 = ∂Dr0 is the horizon
[ of the Schwarzschild space and is outward minimizing. Equivalently, M = [s ,∞)×Sn−1
0
and
1
1
4v g = 1−2ms2−nds2+s2gSn−1, (1.2)
96 where s0 is the unique positive solution of 1 − 2ms20−n = 0 and gSn−1 is the canonical
4 round metric on the unit sphere Sn−1. In this paper, we will denote
0
f(x) = 1−2ms2−n, for any x = (s,θ)∈ Mn, (1.3)
.
1
0 whichiscalledthepotentialpfunctionof(Mn,g). Asiswellknown,theSchwarzschildspace
7 is asymptotically flat, and is static in the sense that the potential function f satisfies
1
: ∇2f = fRic, ∆f = 0, (1.4)
v
Xi whereRic is the Ricci tensor of (Mn,g), ∇,∇2 and ∆ are gradient, Hessian and Laplacian
operator with respect to the metric g on Mn. It can be easily checked that the spacetime
r
a metric gˆ = −f2dt2 +g on Mn ×R solves the vacuum Einstein equation. In particular,
(1.4) implies that (Mn,g) has constant zero scalar curvature R.
Let Ω be a bounded domain with smooth boundary in (Mn,g). Then there exists a
smooth hypersurface Σ such that either ∂Ω = Σ, or ∂Ω = Σ ∪ ∂M, i.e., either Σ is
null-homologous or Σ is homologous to the horizon ∂M. Σ is called outward minimizing
if whenever E is a domain containing Ω then |∂E| ≥ |∂Ω|. Σ is called strictly outward
minimizingifequality impliesthat∂E = ∂Ω. Theoutwardminimizingassumptionimplies
that thehypersurfaceis mean-convex (i.e. themean curvatureH ≥ 0). Themain resultof
thispaperisthefollowing Minkowski-typeienquality foroutwardminimizinghypersurface
in Schwarzschild space.
2010 Mathematics Subject Classification. 53C44, 53C42.
Key words and phrases. Inverse mean curvature flow, Minkowski inequality, Outward minimizing,
Schwarzschild space.
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2 YONGWEI
Theorem 1.1. Let Ω be a bounded domain with smooth outward minimizing boundary in
the Schwarzschild space (Mn,g). Assume either
(1) n < 8, or
(2) n ≥ 8, Σ = ∂Ω\∂M is homologous to the horizon and has H > 0.
Then
n−2
1 |Σ| n−1
fHdµ≥ −2m, (1.5)
(n−1)ω ω
n−1 ZΣ (cid:18) n−1(cid:19)
where ω isthe area of the unit sphere Sn−1 ⊂ Rn, and |Σ|isthe area of Σ with respect to
n−1
the induced metric. Moreover, equality holds in (1.5) if and only if Σ is a slice {s}×Sn−1.
For strictly mean convex and star-shaped hypersurface in Schwarzschild space, the
inequality(1.5)wasobtainedbyBrendle-Hung-Wang[3]asthelimitcaseoftheirinequality
in Anti-de Sitter-Schwarzschild space. Our result does not require the hypersurface to be
star-shaped. The inequality (1.5) is a natural generalization of the classical Minkowski
inequality for convex hypersurface Σ in Rn, which states that
Hdµ ≥ (n−1)ωn−11|Σ|nn−−12. (1.6)
n−1
ZΣ
This was proved by Guan-Li [7] for mean convex and star-shaped hypersurfaces using the
smooth solution of inverse mean curvature flow (IMCF). Huisken (cf. Theorem 6 in [7])
recently applied the weak solution of IMCF to show that the inequality (1.6) also holds
for outward minimizing hypersurfaces in Rn. The proof of this result was also given by
Freire-Schwartz [6]. By letting m → 0, the Schwarzschild metric reduces to the Euclidean
metric g = ds2+s2gSn−1 and the potential function f approaches to 1. Thus Theorem 1.1
generalizes the Minkowski inequality (1.6) of Huisken and Freire-Schwartz for outward-
minimizing hypersurfaces in Rn.
To prove Theorem 1.1, we use the standard procedure in proving geometric inequalities
using the hypersurface curvature flows (see e.g.,[3, 6, 7, 9]). We will employ the weak
solution of IMCF, which was developed by Huisken-Ilmanen in [9] and was applied to
prove the Riemannian Penrose inequality for asymptotically flat 3-manifold with nonneg-
ative scalar curvature. The weak solution of IMCF has also been applied in many other
problems, see for example [1, 2, 6, 12]. In our case, if Σ is homologous to the horizon,
then starting from Σ, there exists the weak solution of IMCF which is given by the level
sets Σ = ∂Ω = ∂{u < t} of a proper locally Lipschitz function u : Ωc → R+. Each Σ
t t t
is C1,α away from a closed singular set Z of Hausdorff dimension at most n−8 and Σ
t
will become C1,α close to a large coordinate sphere as t → ∞. On each Σ we define the
t
following quantity
Q(t) = |Σt|−nn−−21 fHdµt+2(n−1)mωn−1 , (1.7)
(cid:18)ZΣt (cid:19)
where |Σ | is the area of Σ . Note that Q(t) is well-defined, because each Σ is C1,α
t t t
with small singular set, the weak mean curvature of Σ can be defined as a locally L1
t
function using the first variation formula for area. We will prove that Q(t) is monotone
non-increasing along the weak solution of IMCF. If Σ is null-homologous, we fill-in the
region W bounded by the horizon ∂M to obtain a new manifold M˜ and run the weak
IMCF in M˜ with initial condition Σ. When the flow Σ = ∂Ω nearly touches the horizon
t t
MINKOWSKI-TYPE INEQUALITY IN SCHWARZSCHILD SPACE 3
∂M, we jump to the strictly minimizing hull F of the union Ω ∪W. Assume that n < 8,
t
we show that
|∂F| ≥ |Σ |, fHdµ ≥ fHdµ,
t t
ZΣt Z∂F
which implies that Q(t) does not increase during the jump. Then we restart the flow from
∂F. The restriction n< 8 on the dimension in this case comes from that we need to jump
to the strictly minimizing hull F before we restart the flow, and ∂F is only known to be
smooth, more precisely C1,1 for n < 8.
Finally we estimate the limit of Q(t) as t → ∞:
1
lim Q(t)= (n−1)ωn−1. (1.8)
n−1
t→∞
The main inequality (1.5) follows easily from the monotonicity and (1.8). To complete
the proof of Theorem 1.1, we need to show the rigidity of the inequality. If the equality
holds in (1.5), from the proof of the monotonicity in §4 we know that Σ is homologous
to the horizon and Σ is umbilic a.e. for almost all t. We will use the property that a
t
hypersurface to be umbilic is invariant under the conformal change of the ambient metric
and the lower semicontinuity to show that Σ is a sphere, and each Σ ,t ≥ t for some large
t 0
time t , is a disjoint union of spheres in Rn \D with respect to the Euclidean metric.
0 r0
SinceΣandΣ ,t ≥ t aretotally umbilic, weprovethatthiscanonlyoccurifΣiscentered
t 0
at the origin, and Σ is the smooth expanding sphere solution to the IMCF. This proves
t
the main theorem.
The rest of this paper is organized as follows. In §2, we review some properties of the
weak solution ofIMCF.For moredetail, wereferthereaderstoHuisken-Ilmanen’soriginal
paper [9]. In §3, we show how to derive the monotonicity of Q(t) in the case that the flow
is smooth. In §4, we use the approximation argument to show the monotonicity of Q(t)
under the weak IMCF. In the last section, we estimate the limit of Q(t) as t → ∞ and
complete the proof of Theorem 1.1.
Acknowledgments. The author would like to thank Ben Andrews, Gerhard Huisken,
Pei-Ken Hung and Hojoo Lee for their suggestions and discussions, and Haizhong Li, Mu-
Tao Wang for their interests and comments. The author was supported by Ben Andrews
throughout his Australian Laureate Fellowship FL150100126 of the Australian Research
Council.
2. Weak solution of IMCF
Let (Mn,g) be the Schwarzschild space. The classical solution of IMCF is a smooth
family x :Σ×[0,T) → M of hypersurfaces Σ = x(Σ,t) satisfying
t
∂x 1
= ν, x ∈ Σ , (2.1)
t
∂t H
where H,ν are the mean curvature and outward unit normal of Σ , respectively. If the
t
initial hypersurface is star-shaped and strictly mean convex, the smooth solution of (2.1)
exists for all time t ∈ [0,∞), and the flow hypersurfaces Σ converge to large coordinate
t
sphere in exponentially fast, see [13, 18]. In general, without some special assumption on
the initial hypersurface, the smoothness may not be preserved, the mean curvature may
tend to zero at some points and the singularities develop. See for example the thin torus
in Euclideanspace ([9, §1]), i.e. theboundaryof an ǫ-neighborhoodof alarge roundcircle.
4 YONGWEI
The mean curvature is positive on this thin torus, so the smooth solution of (2.1) exists
for at least a short time. By deriving the upper bound of the mean curvature along the
flow, we can see that the torus will steadily fatten up and the mean curvature will become
negative in the donut hole in finite time.
In [9], Huisken-Ilmanen used the level-set approach and developed the weak solution of
IMCF to overcome this problem. The evolving hypersurfaces are given by the level-sets
of a scalar function u:Mn → R via
Σ = ∂{x ∈ M :u(x) < t}.
t
Whenever u is smooth with non vanishing gradient ∇u6= 0, the flow (2.1) is equivalent to
the following degenerate elliptic equation
∇u
div = |∇u|. (2.2)
M
|∇u|
(cid:18) (cid:19)
Using the minimization principle and elliptic regularization, Huisken-Ilmanen proved the
existence, uniquess, compactness and regularity properties of the weak solution of (2.2).
Theexistence resultonly requiremildgrowth assumptionon theunderlyingmanifold, and
applies in particular to the Schwarzchild space here. We summaries their results in the
following.
Theorem2.1([9]). LetΩbeaboundeddomain withsmooth boundary inthe Schwarzschild
space (Mn,g) with n < 8 and Σ = ∂Ω\∂M. In case that Σ is null-homologous, we fill-in
the region W bounded by the horizon. Then there exists a proper, locally Lipschitz function
u≥ 0 on Ωc = M\Ω, called the weak solution of IMCF with initial condition Σ, satisfying
(a) u| = 0, lim u = ∞. For t > 0, Σ = ∂{u < t} and Σ′ = ∂{u > t} define
Σ x→∞ t t
increasing families of C1,α hypersurfaces.
(b) The hypersurfaces Σ (resp. Σ′) minimize (resp. strictly minimize) area among
t t
hypersurfaces homologous to Σ in the region {u ≥ t}. The hypersurface Σ′ =
t
∂{u > 0} strictly minimizes area among hypersurfaces homologous to Σ in Ωc.
(c) For t >0, we have
Σ → Σ as s ր t, Σ → Σ′ as s ց t (2.3)
s t s t
locally in C1,β in Ωc, β < α. The second convergence also holds as s ց 0.
(d) For almost all t > 0, the weak mean curvature of Σ is defined and equals to |∇u|,
t
which is positive and bounded for almost all x ∈ Σ .
t
(e) For each t > 0, |Σ | = et|Σ′|, and |Σ |= et|Σ| if Σ is outward minimizing.
t t
For n ≥ 8, the regularity and convergence in (a),(c) are also true away from a closed
singular set Z of dimension at most n−8 and disjoint from Ω¯.
Note that in [8], Heidusch proved the optimal C1,1 regularity for the level sets Σ
t
and Σ′ away from the singular set Z. The property (b) says that Ω = {u < t} and
t t
Ω′ = int{u ≤ t} are minimizing hull and strictly minimizing hull in {u > t}. Here we call
t
a set E a minimizing hull in G if E minimizes area on the outside in G, that is, if
|∂∗E ∩K|≤ |∂∗F ∩K|
for any F of locally finite perimeter containing E such that F\E ⊂⊂G, and any compact
setK containing F\E. Here∂∗F denotes thereducedboundaryof asetF of locally finite
perimeter. E is called a strictly minimizing hull if equality implies that F ∩G = E ∩G.
MINKOWSKI-TYPE INEQUALITY IN SCHWARZSCHILD SPACE 5
Define E′ to be the intersection of all strictly minimizing hulls in G that contain E. Up
to a set of measure zero, E′ may be realised by a countable intersection, so E′ itself is a
strictly minimizing hull and open. We call E′ the strictly minimizing hull of E in G.
The existence result of weak IMCF in Theorem 2.1 was proved using a minimization
principle (see [9, §1]), together with the elliptic regularization. Consider the following
perturbed equation
∇uǫ
div = |∇uǫ|2+ǫ2 (2.4)
M
|∇uǫ|2+ǫ2!
p
on a large domain Ω = {v < L} defined using a subsolution v of (2.2), with Dirichilet
L p
boundary condition uǫ = 0 on Σ and uǫ = L−2 on the boundary ∂Ω \Σ. This equation
L
(2.4) has the geometric interpretation that the downward translating graph
uǫ(x) t
Σˆǫ := graph −
t ǫ ǫ
(cid:18) (cid:19)
solves the smooth IMCF (2.1) in the manifold M ×R of one dimension higher. Using the
compactness theorem to pass the solutions of (2.4) to limits as ǫ → 0, we obtain a family
i
of cylinders in M ×R, which sliced by M ×{0} gives a family of hypersurfaces weakly
solving (2.2). Similar techniques to show existence of weak solutions of geometric flows
have been used by various authors, cf. [5, 11, 14, 15, 17].
From the argument in [9, §3], we find that there exits a sequence of smooth function
ui = uǫi such that ui → u locally uniformly in Ωc to a function u ∈ C0,1(Ωc). ui and u
are uniformly bounded in C0,1(Ωc). For a.e. t ≥ 0, the hypersurfaces Σˆi := Σˆǫi converges
t t
to the cylinder Σˆ := Σ ×R locally in C1,α away from the singular set Z ×R. Moreover,
t t
as in [9, §5], the mean curvature H of Σˆi converges to the weak mean curvature H of
Σˆit t Σˆt
the cylinder Σˆ locally in L2 sense for a.e. t ≥ 0. Precisely,
t
φH2 → φH2 , a.e., t ≥ 0 (2.5)
ZΣˆit Σˆit ZΣˆt Σˆt
for any cut-off function φ ∈ C0(Ωc ×R). The weak second fundamental form A exists
c Σˆt
on Σˆ in L2 and the lower semicontinuity implies
t
|A |2 ≤ liminf |A |2 < ∞ (2.6)
ZΣˆt Σˆt i→∞ ZΣˆit Σˆit
for a.e. t ≥ 0. Slicing this families Σˆi by M ×{0}, we obtain Σi = Σˆi ∩(M ×{0}) and
t t t
Σ = Σˆ ∩(M ×{0}). Since Σˆi solves the smooth IMCF, its mean curvature in M ×R is
t t t
∇uǫi
HΣˆit = divM  |∇uǫi|2+ǫ2i  = q|∇uǫi|2+ǫ2i. (2.7)
q 
The mean curvature of Σi considered as a hypersurface in M is
t
∇uǫi
H =div
Σit M |∇uǫi|
(cid:18) (cid:19)
|∇uǫi|2+ǫ2i ∇uǫi |∇uǫi|2+ǫ2i
=H + ·∇
Σˆitq |∇uǫi| |∇uǫi|2+ǫ2 q |∇uǫi| 
i
q  
6 YONGWEI
=H |∇uǫi|2+ǫ2i −ǫ2∇uǫi ·∇HΣˆit. (2.8)
Σˆitq |∇uǫi| i |∇uǫi|3HΣˆi
t
Since the limit function u of uǫi has |∇u| > 0 a.e. on Σt, using the weak convergence of
∇H /H (as in the proof of Lemma 5.2 in [9]), we have that the second term on the
Σˆi Σˆi
t t
right hand side of (2.8) converges to zero locally in L2 sense as ǫ → 0. Thus, the mean
i
curvature H of the sliced hypersurface Σi converges to the weak mean curvature H of
Σit t Σt
Σ locally in L2 sense for a.e. t ≥ 0.
t
3. The smooth case
As we mentioned in §1, the key step to prove Theorem 1.1 is to show the monotonicity
of Q(t) defined in (1.7) along the weak IMCF. In this section, we firstly show how to
derive the monotonicity of Q(t) in the smooth case. Let Σ be a smooth solution of the
t
IMCF (2.1). It’s well known that the following evolution equations for the area form dµ
and mean curvature H of Σ in (Mn,g) hold.
t
Lemma 3.1.
∂ dµ =dµ , (3.1)
t t t
1 1
∂ H =−∆ − |A|2+Ric(ν,ν) (3.2)
t Σ
H H
(cid:0) (cid:1)
Employing the above two evolution equations, we can derive the monotonicity of Q(t)
in the smooth case.
Theorem 3.2. Let Σ be a smooth solution of the IMCF (2.1). For any 0 < t < t < T,
t 1 2
if Σ is homologous to the horizon for all t ∈ [t ,t ], then
t 1 2
Q(t ) ≤ Q(t )
2 1
with equality holds if and only if each Σ is totally umbilic for t ∈ [t ,t ]. If Σ is null-
t 1 2 t
homologous for all t ∈ [t ,t ], then
1 2
Q˜(t ) ≤ Q˜(t )
2 1
with equality holds if and only if each Σ is totally umbilic for t ∈ [t ,t ], and Q(t) is
t 1 2
strictly decreasing in time t ∈ [t ,t ], where
1 2
Q˜(t) := |Σt|−nn−−12 fHdµt.
ZΣt
Proof. The case that Σ is homologous to horizon has been treated in [3, 13]. For conve-
t
nience of readers, we include the proof here. Using the evolution equations (3.1)–(3.2),
d
fHdµ = (∂ fH +f∂ H +fH)dµ
t t t t
dt
ZΣt ZΣt
1 f
= h∇f,νi−f∆ − |A|2 +Ric(ν,ν) +fH dµ
Σ t
H H
ZΣt(cid:18) (cid:19)
1 (cid:0) n−2(cid:1)
≤ h∇f,νi− (∆ f +fRic(ν,ν))+ fH dµ , (3.3)
Σ t
H n−1
ZΣt(cid:18) (cid:19)
MINKOWSKI-TYPE INEQUALITY IN SCHWARZSCHILD SPACE 7
where we used |A|2 ≥ H2/(n−1) in the last inequality. Combining the identity
∆ f = ∆f −∇2f(ν,ν)−Hν ·∇f
Σ
and the static equation (1.4), we have
∆ f +fRic(ν,ν)= −Hν ·∇f. (3.4)
Σ
Substituting (3.4) into (3.3) yields that
d n−2
fHdµ ≤ fH +2h∇f,νi dµ . (3.5)
t t
dt n−1
ZΣt ZΣt(cid:18) (cid:19)
If equality holds in (3.5), then |A|2 = H2/(n−1) and Σ is totally umbilical.
t
If Σ is homologous to the horizon for all t ∈ [t ,t ], then denote Ω denote the region
t 1 2 t
bounded by Σ and the horizon ∂M. Applying the divergence theorem and noting that
t
∆f = 0 on M, we get
h∇f,νidµ = ∆f + ∇f ·ν = m(n−2)ω
t ∂M n−1
ZΣt ZΩt Z∂M
which is a constant. Thus we obtain
d n−2
fHdµ +2(n−1)mω ≤ fHdµ +2(n−1)mω . (3.6)
t n−1 t n−1
dt n−1
(cid:18)ZΣt (cid:19) (cid:18)ZΣt (cid:19)
If Σ = ∂Ω is null-homologous for all t ∈ [t ,t ], we have
t t 1 2
h∇f,νidµ = ∆f = 0.
t
ZΣt ZΩt
Then
d n−2
fHdµ ≤ fHdµ . (3.7)
t t
dt n−1
ZΣt ZΣt
Thus the theorem follows directly from (3.6)–(3.7) and the evolution equation of the area
|Σ |
t
d
|Σ | = |Σ |.
t t
dt
(cid:3)
4. The monotonicity
Firstly, we prove the following lemma which was inspired by Lemma A.1 of [6].
Lemma 4.1. Suppose that Ω is a smooth bounded domain in (Mn,g) and Σ= ∂Ω\∂M.
Let u : Ωc → R be a smooth proper function with u| = 0. Let t > 0, Ω = {u ≤ t} and
+ Σ t
Φ :(0,t) → R be Lipschitz and compactly supported in (0,t). Then ϕ =Φ◦u: Ω → R
+ t +
satisfies
− f∇ϕ·νHdv = ϕ 2∇f ·νH +fH2−f|A|2 dv , (4.1)
g g
ZΩt ZΩt
(cid:0) (cid:1)
where ν,H,A denote the unit outward normal, mean curvature and second fundamental
form of the level sets of u, ∇ be the gradient operator on (Mn,g) and ∇ϕ·ν = g(∇ϕ,ν).
8 YONGWEI
Proof. The Sard’s theorem implies that the level set Σ = {x ∈ Ωc : u(x) = s} is regular
s
(∇u 6= 0 on Σ ) for a.e. s > 0. Let U ⊂ Ωc be the open subset where ∇u 6= 0. For any
s
regular level set Σ with outward unit normal ν, in Σ ∩U the variation vector field along
s s
Σ is ∇u/|∇u|2 and ν = ∇u/|∇u|. By the second variation formula for area, we have
s
1 1
− ν ·∇H = ∆ |∇u|−1+ |A|2+Ric(ν,ν) (4.2)
|∇u| Σs |∇u|
(cid:0) (cid:1)
in Σ ∩U, where ∆ denotes the Laplacian operator with respect to the induced metric
s Σs
on Σ . We multiply (4.2) by f and integrate over Σ .
s s
f f
− ν ·∇Hdµ = f∆ |∇u|−1+ |A|2+Ric(ν,ν) dµ
|∇u| s Σs |∇u| s
ZΣs ZΣs ZΣs
1 (cid:0) (cid:1)
= ∆ f +f|A|2+fRic(ν,ν) dµ ,
|∇u| Σs s
ZΣs
(cid:0) (cid:1)
where we used the divergence theorem in the second equality. Applying the identity (3.4),
we obtain
f 1
− ν ·∇Hdµ = −Hν ·∇f +f|A|2 dµ . (4.3)
s s
|∇u| |∇u|
ZΣs ZΣs
(cid:0) (cid:1)
As Σ is regular for a.e. s > 0, the coarea formula and (4.3) imply that
s
t f
fϕν ·∇Hdv = Φ(s) ν ·∇Hdµ ds
g s
|∇u|
ZΩt Z0 ZΣs
t 1
= Φ(s) Hν ·∇f −f|A|2 dµ ds
s
|∇u|
Z0 ZΣs
(cid:0) (cid:1)
= ϕ Hν ·∇f −f|A|2 dv . (4.4)
g
ZΩt
(cid:0) (cid:1)
We firstly assume that Φ ∈ C1. Then in the open subset U = {x ∈ Ωc : |∇u| 6= 0}, we
have
div(fϕHν)= ϕ∇f ·νH +f∇ϕ·νH +fϕν ·∇H +fϕHdiv ν
= ϕ∇f ·νH +fΦ′◦u|∇u|H +fϕν ·∇H +fϕHdiv ν, (4.5)
where div is the divergence operator on (Mn,g). Since ϕ is compactly supported in Ω ,
t
integrating (4.5) yields that
− f∇ϕ·νHdv = − div(fϕHν)dv
g g
ZΩt ZΩt
+ (ϕ∇f ·νH +fϕν ·∇H +fϕHdiv ν)dv
g
ZΩt
= ϕ 2H∇f ·ν −f|A|2 dv + fϕHdiv νdv , (4.6)
g g
ZΩt ZΩt
(cid:0) (cid:1)
where we used the divergence theorem and (4.4).
We now deal with the last term in (4.6). Since Σ is regular for a.e. s > 0, the co-area
s
formula and the first variation formula for area imply that
t fϕH
fϕHdiv(ν)dv = div(ν)dµ ds
g s
|∇u|
ZΩt Z0 ZΣs
MINKOWSKI-TYPE INEQUALITY IN SCHWARZSCHILD SPACE 9
t fϕH
= div νdµ ds
|∇u| Σs s
Z0 ZΣs
t fϕH
= div ν dµ ds
Σs |∇u| s
Z0 ZΣs (cid:18) (cid:19)
t fϕH2
= dµ ds
s
|∇u|
Z0 ZΣs
= fϕH2dv , (4.7)
g
ZΩt
whereinthesecondequality weusedthefactthatdiv(ν) = div ν onΣ ∩U. Substituting
Σs s
(4.7) into (4.6) yields that
− f∇ϕ·νHdv = ϕ 2∇f ·νH +fH2−f|A|2 dv (4.8)
g g
ZΩt ZΩt
(cid:0) (cid:1)
for Φ ∈ C1. Since Lipschitz function can be approximated by C1 function up to a set of
measure zero (see [19, p.32]), we conclude that (4.8) also holds for Lipschitz function Φ
by approximation. (cid:3)
4.1. The case that Σ is homologous to horizon.
Lemma 4.2. Let Ω be a smooth bounded domain in the Schwarzschild space (Mn,g).
Suppose that the boundary ∂Ω = Σ∪∂M and Σ is outward minimizing. Let {Σ } be the
t
weak solution of IMCF in Ωc = M \Ω with initial data Σ. Then for all 0 < t¯< t,
n−2 t
fHdµ ≤ fHdµ+ fHdµ +2(n−1)mω ds (4.9)
t s n−1
n−1
ZΣt ZΣt¯ Zt¯ (cid:18)ZΣs (cid:19)
Proof. As in the discussion in §2, the weak solution of IMCF u∈ C0,1(Ωc) can be approxi-
loc
mated bysmoothproperfunctionsu locally uniformlyinΩc,withC1,α convergence ofthe
i
level sets Σi away from the singular set Z and L2 convergence of the weak mean curvature
s
H := H of level sets for a.e. s > 0. Moreover, we can show that H converges to the
i Σi i
s
mean curvature H of the weak solution Σ of IMCF in locally L2 sense in any domain Ω .
s t
In fact, by the coarea formula and (2.7)–(2.8),
t φH
φH = i dµids
i |∇u | s
ZΩit Z0 ZΣis i
= t φ |∇ui|2+ǫ2i −ǫ2∇ui·∇HΣˆis dµids,
|∇u |2 i |∇u |4H s
Z0 ZΣis i i Σˆis !
where φ ∈ C0(Ω ), Ωi = {x ∈ Ωc : u (x) < t} and Ω = {x ∈ Ωc : u(x) < t}. By the
c t t i t
fact that |∇u| > 0 a.e. on Σ and the weak convergence of ∇H /H , we have that
s Σˆi Σˆi
s s
φH → φH as i→ ∞. Similarly we have the convergence of φH2 → φH2.
Ωit i Ωt Ωit i Ωt
R R R R
10 YONGWEI
For any nonnegative Lipschitz function Φ ∈ Lip(0,t) with compact supportin (0,t) and
ϕ = Φ◦u , by (4.1) we have
i i
− f∇ϕ ·ν H dv = ϕ 2∇f ·ν H +fH2−f|A |2 dv
i i i g i i i i i g
ZΩt ZΩt
(cid:0) n−2 (cid:1)
≤ ϕ 2∇f ·ν H + fH2 dv ,
i i i n−1 i g
ZΩt (cid:18) (cid:19)
where we used |A |2 ≥ H2/(n−1) in the last inequality. Taking the limit of i → ∞, and
i i
using the convergence of u and H , we obtain that
i i
n−2
− f∇ϕ·νHdv ≤ ϕ 2∇f ·νH + fH2 dv (4.10)
g g
n−1
ZΩt ZΩt (cid:18) (cid:19)
As u is the weak solution of IMCF, we have H = |∇u| a.e. in Ω . Also note that by
t
Rademacher’s theorem, the Lipschitz function Φ is differentiable a.e. in (0,t). From the
coarea formula and (4.10), we have
t
− Φ′(s) fHdµ ds =− Φ′(s)∇u·νfHdv = − f∇ϕ·νHdv
s g g
Z0 ZΣs ZΩt ZΩt
n−2
≤ ϕ 2∇f ·νH + fH2 dv
g
n−1
ZΩt (cid:18) (cid:19)
n−2
= ϕ 2∇f ·ν + fH |∇u|dv
g
n−1
ZΩt (cid:18) (cid:19)
t n−2
= Φ(s) 2∇f ·ν + fH dµ ds. (4.11)
s
n−1
Z0 ZΣs(cid:18) (cid:19)
For any 0 < t¯< t and 0 < δ < (t−t¯)/2, define Φ by
0, on [0,t¯]
(s−t¯)/δ, on [t¯,t¯+δ]
Φ(s) = 
1, on [t¯+δ,t−δ]


 (t−s)/δ, on [t−δ,t]

The left hand side of (4.11) is equal to
1 t 1 t¯+δ
fHdµ ds− fHdµ ds
s s
δ δ
Zt−δZΣs Zt¯ ZΣs
Sincefora.e. s> 0,thelevelsetΣi ofu converges toΣ inC1,α awayfromthesingularset
s i s
Z of Hausdorff dimension at most n−8 with L2 convergence of the weak mean curvature,
taking the limits δ → 0 in (4.11), we find that for a.e. 0 < t¯< t
t n−2
fHdµ − fHdµ ≤ 2∇f ·ν + fH dµ ds. (4.12)
t t¯ n−1 s
ZΣt ZΣt¯ Zt¯ ZΣs(cid:18) (cid:19)
To show (4.12) holds for all pair of 0 < t¯< t, we use the C1,β convergence (2.3) and
the weak convergence of mean curvature. For any t > 0, we can find a sequence of time
t ր t such that 0 < t¯< t satisfies (4.12), then Σ → Σ in C1,β away from the singular
i i ti t
set Z as i → ∞ by (2.3). As the weak mean curvature of Σ equals to |∇u| a.e. and is
ti