On the Bartnik mass of non-negatively curved CMC spheres
aa r X i v : . [ m a t h . DG ] F e b ON THE BARTNIK MASS OF NON-NEGATIVELY CURVED CMC SPHERES
ALBERT CHAU AND ADAM MARTENS
Abstract.
Let g be a smooth Riemannian metric on S and H > m B ( S , g, H ) assuming that the Gauss curvature K g is non-negative.Our upper bound approaches the Hawking mass m H ( S , g, H ) when either g becomes round or else H → + ,the bound is zero for H sufficiently large, and in any case the bound is not more than r/ m H ( S , g, m B ( S , g, H ) as well in the case when g is arbitrary and H is sufficiently largedepending on g . Introduction
Let g be a smooth metric and H a smooth function on S . The Bartnik mass [2] of the triple ( S , g, H )is defined as(1) m B ( S , g, H ) := inf ( M,γ ) { m ADM ( M, γ ) : (
M, γ ) is an admissible extension of ( S , g, H ) } where m ADM ( M, γ ) is the ADM mass [1] and admissibility means that M = [0 , ∞ ) × S and γ is a smoothasymptotically flat Riemannian metric having non-negative scalar curvature such that γ | ∂M = g , and ∂M hasmean curvature H and is outer minimizing in M . The Bartnik mass associates a notion of quasi-local massto a closed compact 3 dimensional region Ω of an asymptotically flat space-like hypersurface in a Lorentzian4-manifold, where ∂ Ω = S and g and H are respectively the metric and mean curvature induced fromthe associated Riemannian metric g Ω on Ω. Assuming the dominant energy condition and time symmetricsetting, g Ω will have non-negative scalar curvature, and if ( M, γ ) is further required to smoothly extend(Ω , g Ω ) in (1) as in the original formulation in [2], then m B ( S , g, H ) will be non-negative by the positivemass theorem [13].Another such notion of quasi-local mass associated to the triple ( S , g, H ) is the Hawking mass:(2) m H ( S , g, H ) := r Area ( S , g )16 π (cid:18) − ˆ S H (cid:19) . The proof of the Riemman Penrose inequality in [7] implies that the Bartnik mass is bounded below by theHawking mass thus providing a positive lower bound for the Bartnik mass when m H ( S , g, H ) > − ∆ g + K g has positive first eigenvalue and H = 0, then theBartnik mass actually attains the above lower bound as given by the Riemann Penrose inequality so thatone has m B ( S , g,
0) = m H ( S , g,
0) = p Area ( S ) / π. In [5] the authors of this paper extended the above result to the degenerate case when the operator − ∆ g + K g has zero first eigenvalue. In [4, 11, 8] various upper bounds for Bartnik mass were obtained under theassumption that the Gauss curvature K g is strictly positive and that H is a positive constant. In [12],assuming the Gauss curvature K g is strictly positive and that H is a positive function, Miao-Xie adaptedthe method in [14] and its variation in [10] to construct admissible extensions with zero scalar curvature andADM mass arbitrarily close to p Area ( S ) / π from above, thus establishing the bound m B ( S , g, H ) ≤ p Area ( S ) / π . In Theorem 1.1 below, assuming the Gauss curvature K g is non-negative and that H is a positive constantwe obtain an upper bound for the Bartnik mass where the bound is not more than p Area ( S ) / π andis in fact equal to zero for all H sufficiently large, while the bound approaches the Hawking mass as g converges smoothly to a round metric and the Hakwing mass is non-negative. Theorem 1.1 follows in part Research partially supported by NSERC grant no. AND A. MARTENS from Theorem 1.2 which also implies an upper bound for the Bartnik mass for an arbitrary metric g provided H is sufficiently large depending on g , where again the upper bound equals zero for H sufficiently large. Theorem 1.1.
Let g a smooth Riemannian metric on S with K g ≥ and H > a constant. Then if D = D ( g, H ) is the constant in Defnintion 2 we have m B ( S , g, H ) ≤ max (cid:20) min (cid:18) r √ D (cid:20) − r H D ) (cid:21) , r (cid:19) , (cid:21) (3) where r = p Area ( S , g ) / π . Moreover, D ( g, H ) satisfies D → if either H → + or g converges smoothlyto a round metric. Theorem 1.2.
Let g a smooth Riemannian metric on S and H > a constant. Suppose there exists a ( g, H ) admissible path g ( t ) as in Definition 2. Then if D = D ( g, H ) is the constant in Defnintion 2 we have m B ( S , g, H ) ≤ max (cid:18) r √ D (cid:20) − r H D ) (cid:21) , (cid:19) (4) Remark 1.1.
The expression r √ D h − r H D ) i approaches the Hawking mass m H ( S , g, H ) as D → . Remark 1.2.
Given any smooth metric g , it is proven in Proposition 2.1 that a ( g, h ) admissible path existswhen either K g ≥ or else when H is sufficiently large depending on g . Acknowledgement . The authors would like to thank Pengzi Miao for helpful comments and his interest inthis work. 2.
Admissible paths and the definition of D ( g, H )Given a smooth metric g on S we define r g := p Area ( S , g ) / πK g := Gauss curvature of g Now we define the following notions of admissible paths of metrics starting from g , and the correspondingconstant D ( g, H ) in Theorem 1.1. Definition 1.
Given a smooth metric g on S , a smooth path of metrics ζ = g ( t ) for t ∈ [0 ,
1] is called g - admissible ifa) g (0) = g and g (1) is round (has constant curvature),b) tr g g ′ ≡ t . Definition 2.
Given a smooth metric g on S and a constant H >
0. A g -admissible path ζ = g ( t ) is called( g, H )- admissible if there exists a positive constant C > C H K g ( t ) (1 + C √ t ) − t | g ′ | (1 + C √ t ) + C > S × [0 , . Defining C ζ to be the infemum of all such constants C , we define D ( g, H ) as(6) D ( g, H ) := inf ( g, H )-admissible paths ζ C ζ . The following proposition confirms the existence of admissible paths when K g ≥ H is sufficientlylarge depending on g , in which cases lower bounds for D ( g, H ) are also provided. Proposition 2.1.
Let g be a smooth metric and H > . N THE BARTNIK MASS OF NON-NEGATIVELY CURVED CMC SPHERES 3 If K g ≥ then ( g, H ) -admissible paths ζ = g ( t ) exist, one can be chosen to satsify K g ( t ) + H H − q t | g ( t ) ′ | g ( t ) > on S × [0 , , and for any such path we have the estimate: (7) D ( g, H ) ≤ max S × [0 , q t | g ′ | g q K g ( t ) + H H − q t | g ′ | g If H is sufficiently large depending on g then ( g, H ) -admissible paths exist. In particular, a g -admissible path ζ = g ( t ) (independent of H ) can always be chosen to satisfy (8) 1 − t | g ( t ) ′ | g ( t ) > on S × [0 , , and if we then choose any constant C satisfying (9) C > max S × [0 , q t | g ′ | g − q t | g ′ | g , then ζ will be ( g, H ) -admissible provided H satisfies (10) H ≥ max S × [0 , s K g ( t ) (1 + C √ t ) C t | g ′ | (1 + C √ t ) − C (where the definition of C guarantees the positivity of the denominator) and in this case we have theestimate D ( g, H ) ≤ C .Proof. Given any smooth metric g , we may write g = e w ( x ) g ∗ for a round metric g ∗ with area 4 π . Theconstruction in [9] (Proposition 1.1 and Lemma 1.2) shows that given any smooth non-increasing function α ( t ) : [0 , → [0 ,
1] with α (0) = 1 and α (1) = 0, a function a ( t ) and a family of diffeomorphisms ξ t : S → S can be chosen so that ζ = h ( t ) := ξ ∗ t ( e α ( t ) w ( x )+ a ( t ) g ∗ ) is a g -admissible path. In particular, choosing α ( t ) = 1in a neighborhood of t = 0 ensures that a ( t ) must be constant, that ξ t remains constant by its construction,and thus h remains constant for all t > h c ( t ) for t ∈ [0 , h c ( t ) = ( g < t ≤ e − c h ( c log t + 1) e − c ≤ t ≤ h ( t ) is assumed to be constant in a neighborhood of t = 0 it follows that h c ( t )is smooth on S × [0 ,
1] for all c >
0. By the facts that h ( t ) itself is admissible, and that h c ( t ) is just are-parametrization of h ( t ) we see that h c ( t ) is g admissible as in Definition 1. We now proceed to completethe proof of each part of the Proposition separately. Part 1 . In this case we shrink c if necessary so that0 < c < s min S × [0 , K h c ( t ) + H H max S × [0 , | h ′ ( s ) | h ( s ) which givesmax S × [0 , t | h ′ c ( t ) | h c ( t ) = max S × [ e − /c , t | h ′ c ( t ) | h c ( t ) = max S × [0 , c | h ′ ( s ) | h ( s ) < min S × [0 , K h c ( t ) + H H (11)and thus allows us to define(12) C := max S × [0 , q t | h ′ c | h c q K hc ( t ) + H H − q t | h ′ c | g ALBERT CHAU AND A. MARTENS which in turn implies 0 ≤ K h c ( t ) C H − t | h ′ c | h c (1 + C √ t ) + C on S × [0 , h c ( t ) a ( g, H ) admissible path and also establishingthe estimate for D ( g, H ) in (7). Part 2 . In this case, by the calculation in (11) we see that we may shrink c if necessary so that the g -admissible path h c ( t ) also satisfies (8). If we take C as in (9), then − t | h ′ c | h c + (1 + C √ t ) − C > S × [0 , H satsifies (10) we have K h c ( t ) (1 + C √ t ) − C H − t | h ′ c | h c + (1 + C √ t ) − C > S × [0 , h c ( t ) a ( g, H )-admissible path and also establishing the stated estimate for D ( g, H ). (cid:3) Remark 2.1.
The construction of the g -admissible path ζ = h ( t ) from [9] (Proposition 1.1 and Lemma 1.2)referred to in the proof of Proposition 2.1 implies that if g is arbitrarily close to a fixed round metric on S (in each C k norm relative to a fixed metric),then | g ( t ) ′ | will be arbitrarily close to on S × [0 , . if K g ≥ , then K h ( t ) > for all t > .It follows that if either g is arbitrarily close to a fixed round metric on S , or H is arbitrarily large, theinequality C H K g ( t ) (1 + C √ t ) − t | g ′ | (1 + C √ t ) + C > on S × [0 , will be satisfied for C > arbitrily small, and thus D ( g, H ) from definition (2) will be arbitrarily close to . Proposition 2.2.
Let g be a smooth metric and H > a constant such that there exists a ( g, H ) -admissiblepath. Then given any ǫ > , a ( g, H ) -admissible path ζ = g ( t ) can be chosen so that C ζ ≤ D ( g, H ) + ǫ and also g ( t ) = g (1) for all t ∈ [1 − θ, for some θ > . Moreover,Proof. Let ǫ > g, H )-admissible path ζ = g ( t ) satisfying (5) for some C ≤ D ( g, H ) + ǫ . Namely,(13) 4 C H K g ( t ) (1 + C √ t ) − t | g ′ | (1 + C √ t ) + C > S × [0 , g, H )-admissible paths g θ ( t ) satisfying g θ ( t ) = g θ (1) for t ∈ [1 − θ,
1] sothat for all θ > C but with g ( t ) replaced with g θ ( t ). For each θ ∈ (0 , / σ θ : [0 , → [0 ,
1] satisfying σ θ ( t ) = t − θ , ∀ t ∈ [0 , − θ ] σ θ ( t ) = 1 , ∀ t ∈ [1 − θ, ≤ σ ′ θ ( t ) ≤ − θ , ∀ t ∈ [0 , . Such a function can be constructed by mollification as discussed in [4]. Then the path ζ θ = g θ ( t ) := g ( σ θ ( t ))satisfies g θ ( t ) = g (1) for all t ∈ [1 − θ, t | g ′ θ ( t ) | g θ ( t ) = 2 tσ ′ θ ( t ) | g ′ ( σ θ ( t )) | g ( σ θ ( t )) = σ ′ θ ( t ) tσ θ ( t ) (2 σ θ ( t ) | g ′ ( σ θ ( t )) | g ( σ θ ( t )) ) ≤ tσ θ ( t )(1 − θ ) (2 σ θ ( t ) | g ′ ( σ θ ( t )) | g ( σ θ ( t )) )(14) N THE BARTNIK MASS OF NON-NEGATIVELY CURVED CMC SPHERES 5 and thus letting s = σ θ ( t ), we have4 C H K g θ ( t ) (1 + C √ t ) − t | g ′ θ ( t ) | g ( σ θ ( t )) (1 + C √ t ) + C ≥ C H K g ( s ) (1 + √ t p σ θ ( t ) C √ s ) − ts (1 − θ ) (2 s | g ′ ( s ) | g ( s ) )(1 + √ t p σ θ ( t ) C √ s ) + C (15)On the other hand, as t/σ θ ( t ) → t ∈ [0 ,
1] as θ → θ > t ∈ [0 , g θ ( t ) a ( g, H ) − admissiblepath. (cid:3) collar metrics on ( S × [0 , , γ ) and an extension result The approach used in [9] to constructing admissible extensions of a given metric g on S is to firstconstruct a suitable collar metric γ on S × [0 ,
1] which extends g , then glue this isometrically to a Riemannianmanifold/extension ( S × ( b, ∞ ) , e g ) for 0 < b < t g ( t ) be a g -admissiblepath of metrics. Let M = ( S × [0 , , γ ) where γ = E ( t ) g ( t ) + Φ( t ) dt . Write h ( t ) = E ( t ) g ( t ) to simplify our notation to γ = h ( t ) + Φ( t ) dt . Mean Curvature:
Fix any t ∈ (0 , S × { t } as a submanifold of (Σ , γ )is H t = tr h ( t ) ( ρ ) where ρ = h N, II i γ . Here N = − t ) ∂∂t is the unit normal and II is the second fundamentalform. To calculate this, let E , E be a local coordinate frame on S × { t } . Then ρ ( t ) ij = h N, II ( E i , E j ) i γ = γ ab N a (( ˜ ∇ E i E j ) ⊥ ) b = γ tt N t (( ˜ ∇ E i E j ) ⊥ ) t = γ tt − Φ( t ) ˜Γ tij = γ tt − Φ( t ) ( − γ tt γ ij ; t ) = 12Φ( t ) h ij ; t . Since h ( t ) = E ( t ) g ( t ), we have˙ h = E ′ ( t ) g ( t ) + E ( t ) ˙ g ( t ) = ⇒ tr h ˙ h = E − tr g ˙ h = 2 E ′ ( t ) E − ( t ) . Note that we used tr g ˙ g ≡ t := S × { t } , the foliating sphere attime t we have(16) H t = tr h ( t ) ρ ( t ) = 1Φ( t ) tr h ( ˙ h ) = 1Φ( t ) E ′ ( t ) E − ( t ) . Scalar curvature:
The scalar curvature of γ is given by the formula [9]:(17) R γ = 2 K h ( t ) + Φ − (cid:20) − tr h h ′′ −
14 (tr h h ′ ) + 34 | h ′ | h + ∂ t ΦΦ tr h h ′ (cid:21) . By some basic calculations, we havetr h h ′ = 2 E − E ′ , and | h ′ | h = E − [2( E ′ ) + E | g ′ | g ] . Using this, we have −
14 (tr h h ′ ) + 34 | h ′ | h = − E − ( E ′ ) + 32 E − ( E ′ ) + 34 | g ′ | g = 12 E − ( E ′ ) + 34 | g ′ | g . We know that K h ( t ) = E ( t ) − K g ( t ) . and we also havetr h h ′′ = 2 E − E ′′ + tr g g ′′ and 0 = [(tr g g ′ )] ′ = tr g g ′′ − | g ′ | g = ⇒ tr g g ′′ = | g ′ | g . ALBERT CHAU AND A. MARTENS
This gives R γ = 2 K h ( t ) + Φ − (cid:20) − tr h h ′′ −
14 (tr h h ′ ) + 34 | h ′ | h + ∂ t ΦΦ tr h h ′ (cid:21) = 2 E − K g ( t ) + Φ − (cid:20) − E − E ′′ − tr g g ′′ + 12 E − ( E ′ ) + 34 | g ′ | g + 2 E − E ′ ∂ t ΦΦ (cid:21) = Φ − (cid:20) E − K g ( t ) Φ( t ) − E − E ′′ − | g ′ | g + 12 E − ( E ′ ) + 2 E − E ′ ∂ t ΦΦ (cid:21) = E − Φ − (cid:20) K g ( t ) Φ( t ) − E ′′ − E | g ′ | g + 12 E − ( E ′ ) + 2 E ′ ∂ t ΦΦ (cid:21) . (18) Bartnik Mass:
Recall that the Hawking mass of a foliating sphere Σ t is given by m H (Σ t , h ( t ) , H t ) = r | Σ t | π (cid:18) − π ˆ Σ t H t (cid:19) Now using the facts that (1) the path g ( t ) has constant pointwise area form, and (2) scaling a metric bya constant factor scales the area by the same constant factor, we have | Σ t | = 4 πr g E ( t ) for each t ∈ [0 , H t = E ′ ( t ) E ( t ) − Φ( t ) . Then m H (Σ ) = r | Σ | π (cid:18) − π ˆ Σ H (cid:19) = r g − r g H ! . Similarly,(19) m H (Σ ) = r | Σ | π (cid:18) − π ˆ Σ H (cid:19) = r g p E (1)2 − r g E ′ (1) E (1) − ! . After constructing suitable collars, we will use the following gluing/extension result from [4] to produceadmissible extensions with control on their ADM mass.
Proposition 3.1. [Proposition 2.1 in [4] ] Consider a smooth metric γ = f ( t ) g ( t ) + dt on a cylinder ( S × [0 , , γ ) . Suppose (1) γ has positive scalar curvature (2) g ( t ) = g ∗ (the standard round metric) and f ′ ( t ) > for all s ∈ [ a, for some < a < S × { } has positive mean curvature H (4) m H ( S × { } , f (1) g (1) , H ) ≥ .Then given any ǫ > there exists a smooth rotationally symetric metric e γ on the manifold with boundary S × [ a, ∞ ) such that (a) for sufficiently large c > a , ( S × ( c, ∞ ) , e γ ) is isometric to an exterior region of the Schwarzschildmanifold of mass m := m H ( S × { } , f (1) g (1) , H ) + ǫ : ( S × (2 m, ∞ ) , r g ∗ + 11 − mr dr )(b) e γ = γ on S × [ a, ( a + 1) / and thus in particular we have m B ( S × { } , f (0) g (0) , H ) ≤ m H ( S × { } , f (1) g (1) , H ) N THE BARTNIK MASS OF NON-NEGATIVELY CURVED CMC SPHERES 7 Proof of Theorem 1.2
Theorem 1.2 will follow immediately from the following Theorem, the defninition of D ( g, H ) in Definition2, Proposition 2.2. Theorem 4.1.
Let g be a Riemannian metric on S and H > a constant and let ζ = g ( t ) be a ( g, H ) admissible path, so that in particular, (20) 4 C H K g ( t ) (1 + C √ t ) − t | g ′ | (1 + C √ t ) + C > on S × [0 , for some C > . Suppose further that g ( t ) = g (1) for t < sufficiently close to .Then m B ( S , g, H ) ≤ max r g √ C " − r g H C ) , ! . Proof.
Let g, H and g ( t ) be as in the Theorem. Consider the smooth Riemannian 3 manifold with boundary f M = ( S × [0 , , γ ) where γ = (1 + 2 Hs ) g (cid:18) s A (cid:19) + ds where C is defined in the Theorem and A = C H . We will show that R γ ≥ f M and that the meancurvature of S × { t } in f M is positive and approaches H as t →
0. We will do this after changing variablesto t = s / (4 A ) on M = ( S × (0 , γ takes the form γ = E ( t ) g ( t ) + Φ( t ) dt where E ( t ) = 1 + C √ t and Φ( t ) = A/ √ t . In these coordinates, we will show that R γ ≥ M while themean curvature H t of the foliating spheres in M are positive and approach H as t → Claim 1 (mean curvature): H t > is positive and approaches H as t → : By (16) and our definitions of E and Φ we have H t = E ′ ( t )Φ( t ) E ( t ) = C √ t ( C H √ t )(1 + C √ t ) = H C √ t Clearly H t > t ∈ (0 ,
1] and H t → H uniformly as t ց Claim 2 (scalar curvature): R γ ≥ on M .From our definition of E and Φ we have − E − E ′′ + 2 E − E ′ ∂ t ΦΦ = E − (cid:20) − (cid:18) − C t − / (cid:19) + 2 (cid:18) Ct − / (cid:19) (cid:18) − At − / At − / (cid:19)(cid:21) ≡ . and it follows from (18) that R γ ≥ E − K g ( t ) + Φ − (cid:20) − | g ′ | g + 12 E − ( E ′ ) (cid:21) = 2(1 + C √ t ) − K g ( t ) + 4 H tC (cid:20) − | g ′ | g + 12 (1 + C √ t ) − C t (cid:21) = (1 + C √ t ) − H C (cid:18) C H K g ( t ) (1 + C √ t ) − t | g ′ | (1 + C √ t ) + C (cid:19) . ≥ S × [0 ,
1] by (20).
Hawking mass of (Σ := S × { } ): From (19) and our definition of E and Φ we have m H (Σ ) = r | Σ | π (cid:18) − π ˆ Σ H (cid:19) = r g p E (1)2 − r g E ′ (1) E (1) − ! = r g √ C − r g H C ) ! . ALBERT CHAU AND A. MARTENS
In conclusion, we have estabished that the metric γ = (1 + C A s ) g (cid:18) s A (cid:19) + ds on ( S × [0 , , γ ), introduced at the beginning of the proof, satisfies the hypothesis of Proposition 3.1 provided H < √ Cr g , in which case we conclude m B ( S , g, H ) ≤ m H (Σ ) = r g √ C − r g H C ) ! . Now suppose H ≥ √ Cr g and thus c := H r g − ≥ C > f M = ( S × [0 , , γ ) where γ = (1 + 2 Hs ) g (cid:18) s A (cid:19) + ds and A = c H , the exact same proof as above show that γ satisfies Claims 1 & 2 while our definition of c gives m H (Σ ) = r g √ c − r g H c ) ! = 0 , and again using Proposition 3.1 we conclude m B (Σ ) ≤ (cid:3) Proof of Theorem 1.1 proof of Theorem 1.1. If K g ≥ H > g, H ) will satisfy the hypothesis ofTheorem 1.2 by part 1 of Proposition 2.2. Moreover, in this case the constant D ( g, H ) → H → + or else g converges smoothly to a round metric as pointed out in remark 2.1. Theorem 1.1 then follows fromTheorem 5.1 below. (cid:3) Theorem 5.1.
Let g be a Riemannian metric on S with K g ≥ and let H > be a constant. Then m B ( S , g, H ) ≤ r/ Proof.
Fix any ǫ ≤ M = ( S × [0 , , γ )where γ = E ( t ) g ( t ) + Φ( t ) dt where g ( t ) is an g -admissible path, E ( t ) = (1 + ǫt ) andΦ( t ) = At + ǫH t ≤ φ ( t ) : ≤ t ≤ A + ǫH + 1 : ≤ t for a constant A to be chosen sufficiently large below. As described in the proof of Proposition 2.1 the g -admissible path can be taken to have the form ξ ∗ t ( e α ( t ) w ( x )+ a ( t ) g ∗ ) where α ( t ) : [0 , → [0 ,
1] is smoothand non-increasing, g = w ( x ) g ∗ for a round metric g ∗ with area 4 π , and ξ t is a family of diffeomorphismsso that in particular, choosing α ( t ) constant for t sufficiently close to 1 implies that g ( t ) is also constant for t > t = 1. Moreover, we can also bound K g ( t ) from below linearly in t as K g ( t ) ≥ ct on S × [0 ,
1] for some c >
0. To see why this is true, recall the relationship of Gauss curvature of conformalmetrics: K g ( t ) ◦ ξ t = K e α ( t ) w ( x ) − a ( t ) g ∗ = e a ( t ) K e α ( t ) w ( x ) g ∗ = e a ( t ) − α ( t ) w ( x ) (1 − α ( t )∆ ∗ w ) . We know that ∆ ∗ w ≤ S as K g (0) ≥
0. Letting B = inf t,x e a ( t ) − α ( t ) w ( x ) gives K g ( t ) ◦ ξ t ≥ B (1 − α ( t )) = B (1 − α ′ (0) t + O ( t )) & Bα ′ (0) t. N THE BARTNIK MASS OF NON-NEGATIVELY CURVED CMC SPHERES 9
Now (18) implies that in order to ensure R γ ≥ ct Φ( t ) − C + 14 E − ( E ′ ) + 4 E ′ ∂ t ΦΦ ≥ ct Φ( t ) − C + ǫ ∂ t ΦΦ ≥ C := max S × [0 ,
1] 14 | g ′ | g . To this end, let δ = min (cid:8) , ǫ C (cid:9) . Then since δ ≤ / t ) ≤ Φ( δ )for all t ∈ [0 , δ ]. If we choose A ≥ C/H we may then estimate for all t ∈ [0 , δ ] as ∂ t Φ( t )Φ( t ) ≥ A Φ( δ ) = AAδ + ǫH ≥ AA ǫ C + ǫH = (cid:18) Cǫ (cid:19) A + C A + CH = Cǫ . thus implying R γ ≥ t ∈ [0 , δ ]. Now if we further choose A ≥ q Ccδ , then for t ∈ [ δ, ct Φ( t ) = ct (cid:16) Aδ + ǫH (cid:17) ≥ cδ A ≥ cδ r Ccδ = C thus implying R γ ≥ t ∈ [ δ, A is sufficiently large (as described), we have R γ ≥ , H t of the sphere S × t in M is given by H t = E ′ ( t ) E ( t ) − Φ( t )which is clearly positive by our choice of E ( t ) and Φ( t ), and is equal to ǫ (1+ ǫt ) − ( At + ǫH ) for t ≤ / H as t → E ( t ) and Φ( t ) the Hawking mass of Σ = S × m H (Σ ) = r g p E (1)2 − r g E ′ (1) E (1) − ! = r g p (1 + ǫ )2 − r g ǫ ǫ )( A/ ǫ/H + 1) ! (22)Noting that in the above construction, ǫ could have been chosen arbitrarily small while A could have bechosen arbitrarily large, it follows from (22) and Proposition 3.1 that m B ( S , g, H ) ≤ r g / (cid:3) References
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