Remarks on curvature behavior at the first singular time of the Ricci flow
aa r X i v : . [ m a t h . DG ] M a y REMARKS ON CURVATURE BEHAVIOR AT THE FIRST SINGULARTIME OF THE RICCI FLOW
NAM Q. LE AND NATASA SESUM ∗ Abstract.
In this paper, we study curvature behavior at the first singular time of so-lution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold M , ∂∂t g ij = − R ij for t ∈ [0 , T ). If the flow has uniformly bounded scalar curvature anddevelops Type I singularities at T , using Perelman’s W -functional, we show that suit-able blow-ups of our evolving metrics converge in the pointed Cheeger-Gromov sense toa Gaussian shrinker. If the flow has uniformly bounded scalar curvature and developsType II singularities at T , we show that suitable scalings of the potential functions inPerelman’s entropy functional converge to a positive constant on a complete, Ricci flatmanifold. We also show that if the scalar curvature is uniformly bounded along the flowin certain integral sense then the flow either develops a type II singularity at T or it canbe smoothly extended past time T . Introduction
The Ricci flow and previous results.
Let M be a smooth, compact n-dimensionalRiemannian manifold without boundary and equipped with a smooth Riemannian metric g ( n ≥ g ( t ) ( 0 ≤ t < T ) be a one-parameter family of metrics on M . The Ricciflow equation on M with initial metric g ∂∂t g ( t ) = − g ( t )) , (1.1) g (0) = g . has been introduced by Hamilton in his seminal paper [8]. It is a weakly parabolic systemof equations whose short time existence was proved by Hamilton using the Nash-Moserimplicit function theorem in the same paper and after that simplified by DeTurck [4]. Thegoal in the analysis of (1.1) is to understand the long time behavior of the flow, possiblesingularity formation or convergence of the flow in the cases when we do have a long timeexistence. In general, the behavior of the flow can serve to give us more insights about thetopology of the underlying manifold. One of the great successes is the resolution of thePoincar´e Conjecture by Perelman. In order to discuss those things we have to understandwhat happens at the singular time and also what the optimal conditions for having asmooth solution are.In [10] Hamilton showed that if the norm of Riemannian curvature | Rm | ( g ( t )) staysuniformly bounded in time, for all t ∈ [0 , T ) with T < ∞ , then we can extend the ∗ : Partially supported by NSF grant 0905749. ∗ flow (1.1) smoothly past time T . In other words, either the flow exists forever or thenorm of Riemannian curvature blows up in finite time. This result has been extended in[24] and [26], assuming certain integral bounds on the Riemannian curvature. Namely, if R T R M | Rm | α dvol g ( t ) dt ≤ C , for some α ≥ n +22 then the flow can be extended smoothlypast time T . Throughout the paper, we denote dvol g the Riemannian volume densityon ( M, g ). On the other hand, in [23] Hamilton’s extension result has been improvedby the second author and it was shown that if the norm of Ricci curvature is uniformlybounded over a finite time interval [0 , T ), then we can extend the flow smoothly past time T . In [24] this has been improved even further. That is, if Ricci curvature is uniformlybounded from below and if the space-time integral of the scalar curvature is bounded,say R T R M | R | α dvol g ( t ) dt ≤ C for α ≥ n +22 , where R is the scalar curvature, then Wangshowed that we can extend the flow smoothly past time T . The requirement on Riccicurvature in [24] is rather restrictive. Ricci flow does not in general preserve nonnegativeRicci curvature in dimensions n ≥
4. See Knopf [15] for non-compact examples startingin dimension n = 4 and B¨ohm and Wilking [2] for compact examples starting in dimen-sion n = 12. Without assuming the boundedness from below of Ricci curvature, Ma andCheng [18] proved that the norm of Riemannian curvature can be controlled provided thatone has the integral bounds on the scalar curvature R and the Weyl tensor W from theorthogonal decomposition of the Riemannian curvature tensor. Their bounds are of theform R T R M ( | R | α + | W | α ) dvol g ( t ) dt ≤ C , for some α ≥ n +22 . In [27] it has been proved thatthe scalar curvature controls the K¨ahler Ricci flow ∂∂t g i ¯ j = − R i ¯ j − g i ¯ j starting from anyK¨ahler metric g .1.2. Main results.
The above results, in particular that in [27], support the belief thatthe scalar curvature should control the Ricci flow in the Riemannian setting as well. In [5],Enders, M¨uller and Topping justified this belief for Type I Ricci flow, that is, they provedthe following theorem.
Theorem 1.1 (Enders, M¨uller, Topping) . Let M be a smooth, compact n-dimensionalRiemannian manifold equipped with a smooth Riemannian metric g and g ( · , t ) be a solutionto the Type I Ricci flow equation (1.1) on M . Assume there is a constant C so that sup M | R ( · , t ) | ≤ C , for all t ∈ [0 , T ) and T < ∞ . Then we can extend the flow past time T . Their proof was based on a blow-up argument using Perelman’s reduced distance andpseudolocality theorem.Assume the flow (1.1) develops a singularity at
T < ∞ . Throughout the paper, we usethe following Definition 1.1.
We say that (1.1) has a Type I singularity at T if there exists a constant C > such that for all t ∈ [0 , T )(1.2) max M | Rm ( · , t ) | · ( T − t ) ≤ C. Otherwise we say the flow develops Type II singularity at T . Moreover, the flow thatsatisfies (1.2) will be referred to as to the Type I Ricci flow.
URVATURE BEHAVIOR AT THE FIRST SINGULAR TIME OF THE RICCI FLOW 3
In this paper, we also use a blow-up argument to study curvature behavior at thefirst singular time of the Ricci flow. We deal with both Type I and II singularities. Assumethat the scalar curvature is uniformly bounded along the flow. If the flow develops Type Isingularities at some finite time T then by using Perelman’s entropy functional W , we showthat suitable blow-ups of our evolving metrics converge in the pointed Cheeger-Gromovsense to a Gaussian shrinker. Theorem 1.2.
Let M be a smooth, compact n -dimensional Riemannian manifold ( n ≥ )and g ( · , t ) be a solution to the Ricci flow equation (1.1) on M . Assume there is a constant C so that sup M | R ( · , t ) | ≤ C , for all t ∈ [0 , T ) and T < ∞ . Assume that at T we havea type I singularity and the norm of the curvature operator blows up. Then by suitablerescalings of our metrics, we get a Gaussian shrinker in the limit. A simple consequence of the proof of previous theorem is following result, which is thesame to the one proved by Naber in [19]. The difference is that instead of the reduceddistance techniques used by Naber, we use Perelman’s monotone functional W . Corollary 1.1.
Let M be a smooth, compact n -dimensional Riemannian manifold ( n ≥ )and g ( · , t ) be a solution to the Ricci flow equation (1.1) on M . If the flow has type Isingularity at T , then a suitable rescaling of the solution converges to a gradient shrinkingRicci soliton. We also have the following consequence.
Corollary 1.2.
Let M be a smooth, compact n -dimensional Riemannian manifold ( n ≥ )and g ( · , t ) be a Type I solution to the Ricci flow equation (1.1) on M . There exists a δ > so that if | R | ( g ( · , t )) ≤ C for all t ∈ [0 , T ) , then vol g ( t ) ( M ) ≥ δ , for all t ∈ [0 , T ) . In [19] it has been proved that in the case of type I singularity, a suitable rescaling of theflow converges to gradient shrinking Ricci soliton. In [5], it has been recently showed thatthe limiting soliton represents a singularity model, that is, it is nonflat. The open questionis whether using Perelman’s W -functional, one can produce in the limit a singularity model( nonflat gradient shrinking Ricci solitons). We prove some interesting estimates on theminimizers of Perelman’s W -functional which can be of independent interest.On the other hand, if the flow develops Type II singularities at some finite time T , thenwe show that suitable scalings of the potential functions in Perelman’s entropy functionalconverge to a positive constant on a complete, Ricci flat manifold which is the pointedCheeger-Gromov limit of a suitably chosen sequence of blow-ups of our original evolvingmetrics. Theorem 1.3.
Let M be a smooth, compact n -dimensional Riemannian manifold ( n ≥ )and g ( · , t ) be a solution to the Ricci flow equation (1.1) on M . Assume there is a constant C so that sup M | R ( · , t ) | ≤ C , for all t ∈ [0 , T ) and T < ∞ . Assume that at T we havea type II singularity and the norm of the curvature operator blows up. Let φ i be as in theproof of Theorem 1.2 (see , e.g, (3.10)). Then by suitable rescalings of our metrics and φ i ,we get as a limit of φ i a positive constant on a complete, Ricci flat manifold. NAM Q. LE AND NATASA SESUM ∗ We believe that previous theorem may play a role in proving the nonexistence of typeII singularities if the scalar curvature is uniformly bounded along the flow. We are stillinvestigating that.For a precise definition of φ i , see Section 3.There has been a striking analogy between the Ricci flow and the mean curvature flow fordecades now. About the same time when Hamilton proved that the norm of the Riemanniancurvature under the Ricci flow must blow up at a finite singular time, Huisken [13] showedthat the norm of the second fundamental form of an evolving hypersurface under the meancurvature flow must blow up at a finite singular time. In [16] the authors showed that theanalogue of Wang’s result holds for the mean curvature flow as well, namely if the secondfundamental form of an evolving hypersurface is uniformly bounded from below and if themean curvature is bounded in certain integral sense, then we can smoothly extend theflow. In the follow-up paper [17] the authors show that if one only has the uniform boundon the mean curvature of the evolving hypersurface, then the flow either develops a typeII singularity or can be smoothly extended. In the case the dimension of the evolvinghypersurfaces is two they show that under some density assumptions one can smoothlyextend the flow provided that the mean curvature is uniformly bounded. Finally, we notethat, in contrast to the lower bound on the scalar curvature (2.3), at the first singular timeof the mean curvature flow, the mean curvature can either tend to ∞ (as in the case of around sphere) or −∞ as in some examples of Type II singularities [1].If we replace the pointwise scalar curvature bound in Theorem 1.1 with an integral boundwe can prove the following theorems. Theorem 1.4. If g ( · , t ) solves (1.1) and if (1.3) Z M | R | α ( t ) dvol g ( t ) ≤ C α for all t ∈ [0 , T ) where α > n/ and T < ∞ , then either the flow develops a type IIsingularity at T or the flow can be smoothly extended past time T . Remark 1.1.
The condition on α in Theorem 1.4 is optimal. Let ( S n , g ) be the spaceform of constant sectional curvature 1. The Ricci flow on M = S n with initial metric g has the solution g ( t ) = (1 − n − t ) g . Therefore T = n − is the maximal existencetime. We can rewrite g ( t ) = 2( n − T − t ) g and compute Z M | R | α ( t ) dvol g ( t ) = vol g ( t ) ( M )( n T − t ) ) α = vol g (0) ( M ) (2( n − T − t )) n/ ( n T − t ) ) α = vol g (0) ( M )2 n/ − α ( n − n/ n α T − t ) α − n/ . Hence R M | R | α ( t ) dvol g ( t ) tends to ∞ as t → T if and only if α > n/ . Theorem 1.5. If g ( · , t ) is as above, then if we have the following space-time integral bound, (1.4) Z T Z M | R | α ( t ) dvol g ( t ) dt ≤ C α URVATURE BEHAVIOR AT THE FIRST SINGULAR TIME OF THE RICCI FLOW 5 for α ≥ n +22 , then the flow either develops a type II singularity at T or can be smoothlyextended past time T . Remark 1.2.
The condition on α in Theorem 1.5 is optimal . As in Remark 1.1 considerthe Ricci flow on the round sphere. Following the computation in Remark 1.1 we get Z T Z M | R | α dvol g ( t ) dt = vol g (0) ( M )2 n/ − α ( n − n/ n α Z T T − t ) α − n/ dt, and therefore the integral is ∞ if and only if α ≥ n +22 . For the mean curvature flow, similar results to Theorem 1.5 have been obtained bythe authors [17].The rest of the paper is organized as follows. In Section 2 we will give some necessarypreliminaries. Section 3 is devoted to the statements and proofs of Theorems 1.2, 1.3 andCorollary 1.2. In section 4 we prove Theorems 1.4 and 1.5.
Acknowledgements : The authors would like to thank John Lott for helpful conversa-tions during the preparation of this paper.2.
Preliminaries
In this section, we recall basic evolution equations during the Ricci flow and thedefinition of singularity formation. Then we recall Perelman’s entropy functional W andprove one of its properties concerning the µ -energy, Lemma 2.1. The nonpositivity of the µ -energy turns out to be very crucial for the proof of Theorem 1.1.2.1. Evolution equations and singularity formation.
Consider the Ricci flow equa-tion (1.1) on [0 , T ). Then, the scalar curvature R and the volume form vol g ( t ) evolve bythe following equations(2.1) ∂∂t R = ∆ R + 2 | Ric | and(2.2) ∂∂t vol g ( t ) = − Rvol g ( t ) . Because | Ric | ≥ R n , the maximum principle applied to (2.1) yields(2.3) R ( g ( t )) ≥ min M R ( g (0))1 − min M R ( g (0)) tn . If T < + ∞ and the norm of the Riemannian curvature | Rm | ( g ( t )) becomes unbounded as t tends to T , we say the Ricci flow develops singularities as t tends to T and T is a singulartime. It is well-known that the Ricci flow generally develops singularities.If a solution ( M, g ( t )) to the Ricci flow develops singularities at T < + ∞ , then accordingto Hamilton [10], we say that it develops a Type I singularity ifsup t ∈ [0 ,T ) ( T − t ) max M | Rm ( · , t ) | < + ∞ , NAM Q. LE AND NATASA SESUM ∗ and it develops a Type II singularity ifsup t ∈ [0 ,T ) ( T − t ) max M | Rm ( · , t ) | = + ∞ . Clearly, the Ricci flow of a round sphere develops Type I singularity in finite time. Theexistence of type II singularities for the Ricci flow has been recently established by Gu andZhu [7], proving the degenerate neckpinch conjecture of Hamilton [10].Finally, by the curvature gap estimate for Ricci flow solutions with finite time singularity(see, e.g., Lemma 8.7 in [3]), we have(2.4) max x ∈ M | Rm ( x, t ) | ≥ T − t ) . Perelman’s entropy functional W and the µ -energy. In [21] Perelman has in-troduced a very important functional, the entropy functional W , for the study of the Ricciflow,(2.5) W ( g, f, τ ) = (4 πτ ) − n/ Z M [ τ ( R + |∇ f | ) + f − n ] e − f dvol g , under the constraint (4 πτ ) − n/ R M e − f dvol g = 1. The functional W is invariant under theparabolic scaling of the Ricci flow and invariant under diffeomorphism. Namely, for anypositive number α and any diffeomorphism ϕ , we have W ( αϕ ∗ g, ϕ ∗ f, ατ ) = W ( g, f, τ ) . Perelman showed that if ˙ τ = − f ( · , t ) is a solution to the backwards heat equation(2.6) ∂f∂t = − ∆ f + |∇ f | − R + n τ , and g ( · , t ) solves the Ricci flow equation (1.1) then(2.7) ddt W ( g ( t ) , f ( t ) , τ ) = (2 τ ) · (4 πτ ) − n/ Z M | R ij + ∇ i ∇ j f − g ij τ | e − f dvol g ( t ) ≥ . We see that W is constant on metrics g with the property that R ij + ∇ i ∇ j f − g ij τ = 0 , for a smooth function f . These metrics are called gradient shrinking Ricci solitons andappear often as singularity models, that is, limits of blown up solutions around finite timesingularities of the Ricci flow.Let g ( t ) be a solution to the Ricci flow (1.1) on ( −∞ , T ). We call a triple ( M, g ( t ) , f ( t ))on ( −∞ , T ) with smooth functions f : M → IR a gradient shrinking soliton in canon-ical form if it satisfies(2.8) Ric( g ( t )) + ∇ g ( t ) ∇ g ( t ) f ( t ) − T − t ) g ( t ) = 0 and ∂∂t f ( t ) = | f ( t ) | g ( t ) . Perelman also defines the µ -energy(2.9) µ ( g, τ ) = inf { f | (4 πτ ) − n/ R M e − f dvol g =1 } W ( g, f, τ ) , URVATURE BEHAVIOR AT THE FIRST SINGULAR TIME OF THE RICCI FLOW 7 and shows that(2.10) ddt µ ( g ( · , t ) , τ ) ≥ (2 τ ) · (4 πτ ) − n/ Z M | R ij + ∇ i ∇ j − g ij τ | e − f dvol g ( t ) ≥ , where f ( · , t ) is the minimizer for W ( g ( · , t ) , f, τ ) with the constraint on f as above. Notethat µ ( g, τ ) corresponds to the best constant of a logarithmic Sobolev inequality. Adjustingsome of Perelman’s arguments to our situation we get the following Lemma whose proofwe will include below for reader’s convenience. Lemma 2.1 ( Nonpositivity of the µ -energy ). If g ( t ) is a solution to (1.1) for all t ∈ [0 , T ) , then µ ( g ( t ) , T − t ) ≤ , for all t ∈ [0 , T ) . Proof.
We are assuming the Ricci flow exists for all t ∈ [0 , T ). Fix t ∈ [0 , T ). Define˜ g ( s ) = g ( t + s ), for s ∈ [0 , T − t ). Pick any ¯ τ < T − t . Let τ = ¯ τ − ε with ε > p ∈ M . We use normal coordinates about p on ( M, ˜ g ( τ )) to define(2.11) f ( x ) = ( | x | ε if d ˜ g ( τ ) ( x, x ) < ρ , ρ ε elsewherewhere ρ > dvol ˜ g ( τ ) ( x ) = 1 + O ( | x | )near p . We compute Z M (4 πε ) − n/ e − f dvol ˜ g ( τ ) = Z | x |≤ ρ (4 πε ) − n/ e −| x | / ε (1 + O ( | x | )) dx + O ( ε − n/ e − ρ / ε )= Z | y |≤ ρ / √ ε (4 π ) − n/ e −| y | / (1 + O ( ε | y | )) dy + O ( ε − n/ e − ρ / ε )The second term goes to zero as ε → Z R n (4 π ) − n/ e −| y | / dy = 1 . If we write the integral as e C , then C → ε →
0. And f = f + C then satisfies theconstraint R M (4 πε ) − n/ e − f dvol ˜ g ( τ ) = 1.We solve the equation (2.6) backwards with initial value f at τ . Then W (˜ g ( τ ) , f ( τ ) , ¯ τ − τ )= Z | x |≤ ρ [ ε ( | x | ε + R ) + | x | ε + C − n ](4 πε ) − n/ e −| x | / ε − C (1 + O ( | x | )) dx + Z M − B ( p,ρ ) ( ρ o ε + εR + C − n )(4 πε ) − n/ e − r / ε − C = I + II,
NAM Q. LE AND NATASA SESUM ∗ where I = e − C R | x |≤ ρ ( | x | ε − n )(4 πε ) − n/ e −| x | / ε (1 + O ( | x | )) dx and II contains all theremaining terms. It is obvious that II → ε → I = e − C Z | y |≤ ρ / √ ε ( | y | − n )(4 π ) − n/ e −| y | / (1 + O ( ε | y | )) dy → Z R n ( | y | − n )(4 π ) − n/ e −| y | / dy = 0 as ε → . Therefore W (˜ g ( τ ) , f ( τ ) , ¯ τ − τ ) → τ → ¯ τ . By the monotonicity of µ along the flow, µ ( g ( t ) , ¯ τ ) = µ (˜ g (0) , ¯ τ ) ≤ W (˜ g (0) , f (0) , ¯ τ ) ≤ W (˜ g ( τ ) , f ( τ ) , ¯ τ − τ ) . Let τ → ¯ τ , we get µ ( g ( t ) , ¯ τ ) ≤
0. Since ¯ τ < T − t is arbitrary we get µ ( g ( t ) , T − t ) ≤ . (cid:3) Uniform bound on scalar curvature
In this section, we prove Theorems 1.2, 1.3 and Corollary 1.2.
Proof of Theorem 1.2.
By our assumptions, there exists a sequence of times t i → T so that Q i := max M × [0 ,t i ] | Rm | ( x, t ) → ∞ as i → ∞ . Assume that the maximum is achieved at( p i , t i ) ∈ M × [0 , t i ]. Define a rescaled sequence of solutions(3.1) g i ( t ) = Q i · g ( t i + t/Q i ) . We have that(3.2) | Rm( g i ) | ≤ M × [ − t i Q i ,
0] and | Rm( g i ) | ( p i ,
0) = 1 . By Hamilton’s compactness theorem [9] and Perelman’s κ -noncollapsing theorem [21] wecan extract a pointed subsequence of solutions ( M, g i ( t ) , q i ), converging in the Cheeger-Gromov sense to a solution to (1.1), which we denote by ( M ∞ , g ∞ ( t ) , q ∞ ), for any sequenceof points q i ∈ M . In particular, if we take that sequence of points to be exactly { p i } , wecan guarantee the limiting metric is nonflat. The limiting metric has a sequence of niceproperties that we discuss below. Since | R ( g i ( t )) | = | R ( g ( t i + tQ i )) | Q i ≤ CQ i → , our limiting solution ( M ∞ , g ∞ ( t )) is scalar flat, for each t ∈ ( −∞ , R ∞ := R ( g ∞ ) evolves by ∂∂t R ∞ = ∆ R ∞ + 2 | Ric( g ∞ ) | , we have that Ric( g ∞ ) ≡
0, that is, the limiting metric is Ricci flat. We will get a Gaussianshrinker by using Perelman’s functional µ defined by (2.9). Recall that (see the computa-tion in [14]) ddt µ ( g ( t ) , τ ) ≥ τ · (4 πτ ) − n/ Z M | Ric + ∇∇ f − g τ | e − f dvol g ( t ) , URVATURE BEHAVIOR AT THE FIRST SINGULAR TIME OF THE RICCI FLOW 9 where f ( · , t ) is the minimizer realizing µ ( g ( t ) , τ ), and τ = T − t . In this Theorem 1.2, we take s, v ∈ [ − , with s < v . Then, by (3.2), g i ( s ) and g i ( v )are defined for i sufficiently large. Then, by the invariant property of µ under the parabolicscaling of the Ricci flow, one has, for s < v ∈ [ − , µ ( g i ( v ) , Q i ( T − t i ) − v ) − µ ( g i ( s ) , Q i ( T − t i ) − s )= µ ( g ( t i + vQ i ) , T − t i − vQ i ) − µ ( g ( t i + sQ i ) , T − t i − sQ i )= Z t i + vQi t i + sQi ddt µ ( g ( t ) , T − t ) dt ≥ Z t i + vQi t i + sQi Z M τ (4 πτ ) − n/ · | Ric + ∇∇ f − g τ | e − f dvol g ( t ) dt = 2 Z vs Z M m i ( r )(4 πm i ( r )) − n/ | Ric( g i ( r )) + ∇∇ f − g i m i ( r ) | e − f dvol g i ( r ) dr (3.3)where, for simplicity, we have denoted(3.4) m i ( r ) = Q i ( T − t i ) − r. Since we are assuming the flow develops a type I singularity at T , we have(3.5) lim i →∞ Q i ( T − t i ) = a < ∞ . Thus, by (2.4), one has for r ∈ [ − , i →∞ m i ( r ) = a − r > . By Lemma 2.1 and by the monotonicity of µ ( g ( t ) , T − t ) (see (2.10)), we have(3.7) µ ( g (0) , T ) ≤ µ ( g ( t ) , T − t ) ≤ . Estimate (3.7) implies that there exists a finite lim t → T µ ( g ( t ) , T − t ) which has as a conse-quence that the left hand side of (3.3) tending to zero as i → ∞ . Letting i → ∞ in (3.3)and using (3.6), we get(3.8) lim i →∞ Z vs Z M ( a − r )[4 π ( a − r )] − n/ | Ric ( g i ) + ∇∇ f − g i a − r ) | e − f dvol g i ( r ) dr = 0 . We would like to say that we can extract a subsequence so that f ( · , t i + rQ i ) convergessmoothly to a smooth function f ∞ ( r ) on ( M ∞ , g ∞ ( r )), which will then be a potentialfunction for a limiting gradient shrinking Ricci soliton g ∞ . In order to do that, we need toget some uniform estimates for f ( · , t i + rQ i ). The equation satisfied by f ( t i + rQ i ) in (3.3) is(3.9) ( T − t i − rQ i )(2∆ f − |∇ f | + R ) + f − n = µ ( g ( t i + rQ i ) , T − t i − rQ i ) . Let f i ( · , r ) = f ( · , t i + rQ i ) . Then[ Q i ( T − t i ) − r ](2∆ g i ( r ) f i ( r ) − (cid:12)(cid:12) ∇ g i ( r ) f i ( r ) (cid:12)(cid:12) + R ( g i ( r ))) + f i ( r ) − n = µ ( g i ( r ) , Q i ( T − t i ) − r ) . ∗ Define φ i ( · , r ) = e − f i ( · ,r ) / . This function φ i ( · , r ) satisfies a nice elliptic equation(3.10) [ Q i ( T − t i ) − r ]( − g i ( r ) + R ( g i ( r ))) φ i = 2 φ i log φ i + ( µ ( g i ( r ) , Q i ( T − t i ) − r ) + n ) φ i . Recall that, in this Theorem 1.2, we consider r ∈ [ − , r whenever no possible confusion may arise.Our first estimates are uniform global W , estimates for φ i ( r ) as shown in the following Lemma 3.1.
There exists a uniform constant C so that for all r ∈ [ − , and all i , onehas Z M φ i ( · , r ) dvol g i ( r ) + Z M |∇ g i ( r ) φ i ( · , r ) | dvol g i ( r ) ≤ C ( Q i ( T − t i ) − r ) n/ ≤ ˜ C. Proof.
Note that φ i ( r ) satisfies the L -constraint Z M [4 πm i ( r )] − n/ ( φ i ( r )) dvol g i ( r ) = 1and is in fact smooth [22]. Here, we have used m i ( r ) = Q i ( T − t i ) − r .To simplify, let F i ( r ) = φ i ( r ) c i ( r ) where c i ( r ) = [4 πm i ( r )] n/ . Then Z M ( F i ( r )) dvol g i ( r ) = 1and the equation for F i ( r ) becomes m i ( r )( − g i ( r ) + R ( g i ( r ))) F i ( r ) = 2 F i ( r ) log F i ( r ) + ( µ ( g i ( r ) , m i ( r )) + n + 2 log c i ( r )) F i ( r ) . Introduce µ i ( r ) = µ ( g i ( r ) , m i ( r )) + n + 2 log c i ( r ) . Then − ∆ g i ( r ) F i = 12 m i ( r ) F i log F i + ( µ i ( r )4 m i ( r ) − R ( g i ( r ))) F i . Multiplying the above equation by F i ( r ) and integrating over M , we get(3.11) Z M |∇ g i F i | dvol g i ( r ) = 12 m i ( r ) Z M F i log F i dvol g i ( r ) + Z M ( µ i ( r )4 m i ( r ) − R ( g i )) F i dvol g i ( r ) . Because R M ( F i ( r )) dvol g i ( r ) = 1, by Jensen’s inequality for the logarithm, Z M F i log F i dvol g i ( r ) = n − Z M F i log F n − i dvol g i ( r ) ≤ n −
24 log Z M F n − i dvol g i ( r ) = n −
24 log Z M F nn − i dvol g i ( r ) . (3.12)On the other hand, we recall the following Sobolev inequality due to Hebey-Vaugon [12](see also Theorem 5.6 in Hebey [11]) URVATURE BEHAVIOR AT THE FIRST SINGULAR TIME OF THE RICCI FLOW 11
Theorem 3.1 (Hebey-Vaugon) . For any smooth, compact Riemannian n-manifold ( M, g ) , n ≥ such that | Rm ( g ) | ≤ Λ , |∇ g Rm ( g ) | ≤ Λ , inj ( M,g ) ≥ γ one has a uniform constant B ( n, Λ , Λ , γ ) so that, for any u ∈ W , ( M )(3.13) (cid:18)Z M | u | nn − dvol g (cid:19) n − n ≤ C ( n ) Z M |∇ u | dvol g + B ( n, Λ , Λ , γ ) Z M u dvol g . By Perelman’s noncollapsing result, Theorem 3.1 applies to (
M, g i ( r )) with uniformconstants Λ , Λ , γ , independent of r ∈ [ − ,
0] and i . In particular, letting u = F i ( r ) in(3.13), we find that(3.14) Z M ( F i ( r )) nn − dvol g i ( r ) ≤ C ( n ) (cid:18)Z M (cid:12)(cid:12) ∇ g i ( r ) F i ( r ) (cid:12)(cid:12) dvol g i ( r ) (cid:19) nn − + B ( n, Λ , Λ , γ ) . Combining (3.11), (3.12) and (3.14), we obtain(3.15) Z M |∇ g i F i | dvol g i ( r ) ≤ n − m i ( r ) log Z M F nn − i dvol g i ( r ) + Z M ( µ i ( r )4 m i ( r ) − R ( g i )) F i dvol g i ( r ) ≤ n − m i ( r ) log C ( n ) (cid:18)Z M |∇ F i | dvol g i ( r ) (cid:19) nn − + B ( n, Λ , Λ , γ ) ! + Z M ( µ i ( r )4 m i ( r ) − R ( g i )) F i dvol g i ( r ) . Recall that R ( g i ( r )) is uniformly bounded by our scaling and furthermorelim i →∞ Q i ( T − t i ) = a ∈ [ 18 , ∞ ) . Thus, if r ∈ [ − , R M (cid:12)(cid:12) ∇ g i ( r ) F i ( r ) (cid:12)(cid:12) dvol g i ( r ) .Hence, we have a global uniform bound for R M (cid:12)(cid:12) ∇ g i ( r ) φ i ( r ) (cid:12)(cid:12) dvol g i ( r ) because φ i ( r ) = c i ( r ) F i ( r ). (cid:3) Now, elliptic L p theory gives uniform C ,α estimates for φ i ( r ) on compact sets [6]. Toget higher order derivative estimates on φ i ( r ), in order to be able to conclude that for asuitably chosen sequence of points q i around which we decide to take the limit we have f ∞ ( r ) = − φ ∞ ( r ), for a smooth function f ∞ ( r ) (where f ∞ ( r ) is the limit of f i ( r ) and φ ∞ ( r ) is the limit of φ i ( r )), it is crucial to prove that { φ i ( r ) } stay uniformly bounded frombelow on compact sets around q i . ∗ In (3.8), take s = −
10 and v = 0. For each i , let r i ∈ [ − ,
0] be such that( a − r i )[4 π ( a − r i )] − n/ (cid:12)(cid:12)(cid:12)(cid:12) Ric ( g i ( r i )) + ∇∇ f ( t i + r i Q i ) − g i a − r i ) (cid:12)(cid:12)(cid:12)(cid:12) e − f ( t i + riQi ) dvol g i ( r i ) ≤ ( a − r )[4 π ( a − r )] − n/ (cid:12)(cid:12)(cid:12)(cid:12) Ric ( g i ( r )) + ∇∇ f ( t i + rQ i ) − g i a − r ) (cid:12)(cid:12)(cid:12)(cid:12) e − f ( t i + rQi ) dvol g i ( r ) , for all r ∈ [ − , q i ∈ M at which the maximum of φ i ( r i ) over M has beenachieved and denote also by ( M ∞ , g ∞ ( t ) , q ) the smooth pointed Cheeger-Gromov limit ofthe rescaled sequence of metrics ( M, g i ( t ) , q i ), defined as above. Take any compact set K ⊂ M ∞ containing q . Let ψ i : K i → K be the diffeomorphisms from the definition ofCheeger-Gromov convergence of ( M, g i , q i ) to ( M ∞ , g ∞ , q ) and K i ⊂ M . Following theprevious notation, consider the functions F i ( r i ), φ i ( r i ) and denote them for simplicity by F i and φ i , respectively. We will also denote the metric g i ( r i ) shortly by g i . Lemma 3.2.
For any α ∈ (0 , , there is a uniform constant C ( α ) so that (3.16) k F i k C ,α ( M ) ≤ C ( α ) . Proof.
The proof is via boostrapping and rather standard for the equation satisfied by F i (3.17) − ∆ g i F i = 12 m i ( r ) F i log F i + ( µ i ( r )4 m i ( r ) − R ( g i )) F i . The reason that bootstrapping works is simple. If F i is uniformly bounded in L p ( K i ),where K i ∈ M is a compact set, then F i log F i is uniformly bounded in L p − δ ( K i ) for any δ >
0. The standard local parabolic estimates will give us (3.16) which will be independentof a compact set since we have uniform global W , bound on F i . (cid:3) Let us now discuss how to get higher order derivatives estimates for F i . Covariantlydifferentiating (3.17), commuting derivatives, and noting that − ∆ g i ∂ l F i = − ∂ l ∆ g i F i − Ric( g i ) lk g kpi ∂ p F i we get(3.18) − ∆ g i ∂ l F i = 12 m i ( r ) ∂ l F i log F i + ( 2 + µ i ( r )4 m i ( r ) − R ( g i )) ∂ l F i − ∂ l R ( g i ) F i − Ric( g i ) lk g kpi ∂ p F i . The major obstacle in applying L p theory to get uniform C ,α estimates for ∂ l F i is the term ∂ l F i log F i . This emanates from the potential smallness of | F i | , though we have alreadyhad a nice uniform upper bound on it. Thus, to proceed further, we need to bound | F i | uniformly from below. Equivalently, we will prove in Lemma 3.3 that φ i stays uniformlybounded from below on K i .As the first step, we bound φ i ( q i ) from below. This is simple. If we apply the maximum URVATURE BEHAVIOR AT THE FIRST SINGULAR TIME OF THE RICCI FLOW 13 principle to (3.9) we obtain min M f i ≤ C , where f i = f i ( r i ), for a uniform constant C .This can be seen as follows. Define α i = Q i ( T − t i ). At the minimum of f i , we have f i − nα i − r i = µ ( g i ( r i ) , α i − r i ) α i − r i − R ( g i ( r i )) − g i ( r i ) f i ≤ µ ( g i ( r i ) , α i − r i ) α i − r i − R ( g i ( r i )) . Thus,(3.19) f i ≤ n + µ ( g i ( r i ) , α i − r i ) − R ( g i ( r i ))( α i − r i ) ≤ n + µ ( g i ( r i ) , α i − r i ) + CQ i [ Q i ( T − t i ) − r i ] ≤ C, where we have used the fact that R ( · , t ) ≥ − C on M , for all t ∈ [0 , T ) (see (2.3)). Thisimplies φ i ( q i ) ≥ δ > i , with a uniform constant δ .Let K ⊂ M ∞ and K i ⊂ M be compact sets as before. Also recall that m i ( r i ) = Q i ( T − t i ) − r i . Lemma 3.3.
For every compact set K ⊂ M ∞ there exists a uniform constant C ( K ) sothat φ i ≥ C ( K ) , on K i , for all i. Proof.
Assume the lemma is not true and that there exist points P i ∈ K i so that φ i ( P i ) ≤ /i → i → ∞ . Assume ψ i ( P i ) converge to a point P ∈ K . Then φ ∞ ( P ) = 0. Takea smooth function η ∈ C ∞ ( M ∞ ), compactly supported in K \{ P } . Then ψ ∗ i η ∈ C ∞ ( M ),compactly supported in K i \{ P i } . Multiplying (3.10) by ψ ∗ i η , assuming lim i →∞ r i = r , andthen integrating by parts, we get Z M m i ( r i ) · (4 ∇ φ i ∇ ( ψ ∗ i η )+ R i φ i ψ ∗ i η ) − φ i ψ ∗ i η ln φ i − nφ i ψ ∗ i η − µ ( g i , m i ( r i )) φ i ψ ∗ i η ) dvol g i ( r i ) = 0 . Letting i → ∞ , using that φ i C ,α → φ ∞ locally, ψ ∗ i η → η smoothly as i → ∞ , lim i →∞ R ( g i ) =0, and a − r := lim i →∞ m i ( r i ) ≡ lim i →∞ ( Q i ( T − t i ) − r i ) < ∞ , one finds that Z M ∞ (4( a − r ) ∇ φ ∞ ∇ η − ηφ ∞ ln φ ∞ − nφ ∞ η − µ ( g ∞ , a − r ) ηφ ∞ ) dvol g ∞ ( r ) = 0 . Proceeding in the same manner as in Rothaus [22] we can get that φ ∞ ≡ P . Using the connectedness argument, φ ∞ ≡ M ∞ . Thatcontradicts φ ∞ ( q ) ≥ δ > (cid:3) Having Lemma 3.3 and C ,α uniform estimates on φ i , we see that the right hand sideof (3.18) is uniformly bounded in L ( K i ). Because log F i is uniformly bounded on K i , wecan bootstrap (3.18) to obtain C ,α estimates for |∇ g i F i | . Hence, one has uniform C ,α estimates for F i on K i . In terms of φ i , one has that(3.20) k φ i k C ,α ( K i ) ≤ C ( K, α )( Q i ( T − t i ) − r i ) n/ . One can differentiate (3.18) again and obtain all higher order derivative estimates on φ i and therefore all higher order derivative estimates on f i = f i ( r i ) = − φ i . However, for ∗ our purpose, C ,α estimates suffice.Then, using (3.8), for s = −
10 and v = 0,lim i →∞ a − r i )(4 π ( a − r i )) − n/ Z M | Ric( g i ( r i )) + ∇∇ f i − g i ( r i )2( a − r i ) | e − f i dvol g i ( r i ) ≤ lim i →∞ Z − Z M ( a − r )[4 π ( a − r )] − n/ | Ric( g i ) + ∇∇ f i − g i a − r ) | e − f i dvol g i ( r ) dr = 0 . By Lemma 3.3 and (3.8), applying Arzela-Ascoli theorem on f i results inRic ∞ + ∇∇ f ∞ − g ∞ a − r ) = 0 . Since Ric ∞ ≡
0, we get g ∞ = 2( a − r ) ∇∇ f ∞ , and therefore M ∞ is isometric to a standard Euclidean space R n ; see, e.g., Proposition 1.1in [20]. It is now easy to see that(3.21) f ∞ = | x | a − r ) , that is, the limiting manifold ( R n , g ∞ , q ∞ ) is a Gaussian shrinker. (cid:3) In Corollary 1.2 we claim that if our solution is Type I and if it has uniformly boundedscalar curvature, then the volume can not go to zero.
Proof of Corollary 1.2.
Inspecting the proof of Theorem 1.2 and noting that our solution g ( t ) is of type I, we see that, if we rescale the metrics by g j = Q j g ( T + tQ j ) then we also geta Gaussian shrinker. Now, using Perelman’s pseudolcality theorem ([21], Theorem 10.3)and Theorem 1.2 we get | Rm | g ( t ) ≤ Q j ( εr ) , in B g ( T − ( εr )2 Qj ) ( q j , εr p Q j ) , for all t ∈ [ T − ( εr ) Q j , T ) and all j ≥ j , for sufficiently large j . Here r is arbitrary and ε is a small number in Perelman’s pseudolocality theorem. This tells us all metrics g ( t ), for t ≥ T − ( εr ) Q j are uniformly equivalent to each other in a fixed ball B g ( T − ( εr )2 Qj ) ( q j , εrQ j ) andtherefore, vol g ( t ) ( M ) ≥ vol g ( t ) ( B g ( T − ( εr )2 Qj ) ( q j , εrQ j )) ≥ C ( j ) > . (cid:3) Proof of Theorem 1.3.
We will use many estimates and arguments that we have developedin the proof of Theorem 1.2. Assume the flow does develop a type II singularity at T .Then we can pick a sequence of times t i → T and points p i ∈ M as in [10] so thatthe rescaled sequence of solutions ( M, g i ( t ) := Q i g ( t i + t/Q i ) , p i ), converges in a pointedCheeger-Gromov sense to a Ricci flat, nonflat, complete, eternal solution ( M ∞ , g ∞ ( t ) , p ∞ ). URVATURE BEHAVIOR AT THE FIRST SINGULAR TIME OF THE RICCI FLOW 15
Here Q i := max M × [0 ,t i ] | Rm | ( x, t ) → ∞ as i → ∞ . The reasons for getting Ricci flat metricare the same as in the proof of Theorem 1.2. Define α i := ( T − t i ) Q i . Since we are assuming type II singularity occurring at T , we may assume that for a chosensequence t i we have lim i →∞ α i = ∞ .By Lemma 2.1 and the monotonicity of µ we have | µ ( g ( t ) , T − t ) | ≤ C for all t ∈ [0 , T ).Let f i ( · , s ) be a smooth minimizer realizing µ ( g ( t i + s/Q i ) , T − t i − s/Q i ) = µ ( g i ( s ) , α i − s ) = inf W ( g ( t i + sQ i ) , f, T − t i − sQ i )over the set of all smooth functions f satisfying[4 π ( T − t i − sQ i )] − n/ Z M e − f dvol g ( t i + sQi ) = 1 . Then f i = f i ( · , s ) satisfies(3.22) 2∆ g i ( s ) f i − |∇ g i ( s ) f i | + R i + f i − nα i − s = µ ( g i ( s ) , α i − s ) α i − s . In terms of φ i ( x, s ) = e − f i ( x,s ) / this is equivalent to(3.23) − g i ( s ) φ i ( s ) + R ( g i ( s )) φ i ( s ) = 2 φ i ( s ) log φ i ( s ) α i − s + ( µ ( g i ( s ) , α i − s ) + n ) φ i ( s ) α i − s , with(3.24) Z M ( φ i ( s )) dvol g i ( s ) = (4 π ( α i − s )) n/ . In what follows, we fix s = 0. Define ˜ φ i ( · ) := φ i ( · , β i , where(3.25) β i := max M (cid:0) φ i ( x,
0) + (cid:12)(cid:12) ∇ g i (0) φ i ( x, (cid:12)(cid:12)(cid:1) . This choice of β i gives us uniform C estimates for ˜ φ i on M . Thus, we can apply L p theoryto get uniform C ,α estimates for ˜ φ i on compact sets around the points where the maximain (3.25) are achieved. To be more precise, we proceed as follows.Take q i ∈ M at which this maximum in (3.25) has been achieved and denote alsoby ( M ∞ , g ∞ ( t ) , q ) the smooth pointed Cheeger-Gromov limit of the rescaled sequence ofmetrics ( M, g i ( t ) , q i ), defined as above. Lemma 3.1, Theorem 3.1 and standard elliptic L p estimates applied to (3.23) yield the estimates on β i in terms of the W , norm of φ i ,with respect to metric g i (0), that is, there exists a uniform constant C so that for all i , β i ≤ Cα n/ i , which implies(3.26) log β i ≤ C log α i + C , ∗ for some uniform constants C and C . This can be proved the same way we obtained(3.20) in Theorem 1.2. After dividing (3.23) by β i we get(3.27) − g i (0) ˜ φ i + R ( g i (0)) ˜ φ i = 2 ˜ φ i · log ˜ φ i + log β i α i + ( µ ( g ( t i ) , T − t i ) + n ) ˜ φ i α i . Since (
M, g i ( t ) , q i ) converges in the pointed Cheeger-Gromov sense to ( M ∞ , g ∞ ( t ) , q ), and (cid:13)(cid:13)(cid:13) ˜ φ i (cid:13)(cid:13)(cid:13) C ( M,g i (0)) is uniformly bounded, we can get uniform C ,α estimates for ˜ φ i on compactsets around points q i . By Arzela-Ascoli theorem ˜ φ i converges uniformly in the C norm oncompact sets around points q i to a smooth function ˜ φ ∞ . We will show below that ˜ φ ∞ ( · ) isa positive constant.Indeed, if we apply the maximum principle to (3.22), similarly as in the proof of Theo-rem 1.2, we obtain min M f i ( · , ≤ C , for a uniform constant C . This implies log β i ≥ − C for a uniform constant C . In particular, there is a uniform constant δ > i , one has(3.28) β i ≥ δ > . This together with (3.26) and the lim i →∞ α i = ∞ implies(3.29) lim i →∞ log β i α i = 0 . If we multiply (3.27) by any cut off function η i = ψ ∗ i η (where η is any cut off functionon M ∞ and ψ i is a sequence of diffeomorphisms from the definition of Cheeger Gromovconvergence) and integrate by parts we get4 Z M ∇ ˜ φ i ∇ η i dvol g i (0) = − Z M R ( g i (0)) ˜ φ i η i dvol g i (0) + 2 Z M η i ˜ φ i · log ˜ φ i + log β i α i dvol g i (0) − µ ( g ( t i ) , T − t i ) + nα i Z M η i ˜ φ i dvol g i (0) . If we let i → ∞ in the previous identity, using (3.29), the lim i →∞ α i = ∞ , R ( g i (0)) → φ i C → ˜ φ ∞ , and uniform bounds on µ ( g ( t ) , T − t ) we obtain Z M ∇ ˜ φ ∞ ∇ η dvol g ∞ (0) = 0 . This means ∆ ˜ φ ∞ = 0 in the distributional sense. By Weyl’s theorem, ˜ φ ∞ is a harmonicfunction on M ∞ . Since ( M ∞ , g ∞ (0)) is a complete, Ricci flat manifold and φ ∞ ≥
0, by thetheorem of Yau [25], ˜ φ ∞ = C ∞ is a constant function on M ∞ . At the same time, from thedefinition of ˜ φ i , we get, for x in compact sets around points q i (3.30) 1 = lim i →∞ (cid:16) ˜ φ i ( x ) + (cid:12)(cid:12)(cid:12) ∇ g i (0) ˜ φ i ( x ) (cid:12)(cid:12)(cid:12)(cid:17) = ˜ φ ∞ ( x ) + (cid:12)(cid:12)(cid:12) ∇ g ∞ (0) ˜ φ ∞ ( x ) (cid:12)(cid:12)(cid:12) ≡ C ∞ . This implies, in particular C ∞ ≡ > . (cid:3) URVATURE BEHAVIOR AT THE FIRST SINGULAR TIME OF THE RICCI FLOW 17 Integral bounds on scalar curvature
In this section we will prove Theorem 1.4 and Theorem 1.5. Observe that Theorem 1.1is a special case of Theorem 1.4 when α = ∞ in the case we deal with type I singularitiesonly. A crucial ingredient in our arguments is the following result. Theorem 4.1. (Enders-M¨uller-Topping, Theorem 1.4 [5] ) Let g ( t ) be the solution to a TypeI Ricci flow (1.1) on [0 , T ) and suppose that the flow develops a Type I singularity at T .Then for every sequence λ j → ∞ , the rescaled Ricci flows ( M, g j ( t )) defined on [ − λ j T, by g j ( t ) := λ j g ( T + tλ j ) subconverge, in the Cheeger-Gromov sense, to a normalized nontrivialgradient shrinking soliton in canonical form on ( −∞ , .Proof of Theorem 1.4. The proof is by contradiction. Assume the flow develops a type Isingularity at p ∈ M at T < ∞ . Consider any sequence λ j → ∞ and define g j ( t ) := λ j g ( T + tλ j ) where t ∈ [ − λ j T, M, g j ( t ) , p ) de-fined on [ − λ j T,
0) subconverge, in the Cheeger-Gromov sense, to a normalized nontrivialgradient shrinking soliton ( M ∞ , g ∞ ( t ) , p ∞ ) in canonical form on ( −∞ , Z M | R ( g j ( t )) | α dvol g j ( t ) = 1 λ α − n/ j Z M (cid:12)(cid:12)(cid:12)(cid:12) R ( g ( T + tλ j )) (cid:12)(cid:12)(cid:12)(cid:12) α dvol g ( T + tλj ) ≤ C α λ α − n/ j → . Thus our limiting solution ( M ∞ , g ∞ ( t ) , p ∞ ) is scalar flat. Arguing as in the proof ofTheorem 1.1, we see that M ∞ is isometric to a standard Euclidean space R n . However,this contradicts the nontriviality of M ∞ . (cid:3) Proof of Theorem 1.5.
By H¨older inequality, it suffices to consider the case α = n +22 . Thenour integral bound is invariant under the usual parabolic scaling of the Ricci flow.The proof is by contradiction. Assume the flow develops a type I singularity at p ∈ M at T < ∞ . Consider any sequence λ j → ∞ and define g j ( t ) := λ j g ( T + tλ j ) where t ∈ [ − λ j T, M, g j ( t ) , p ) defined on[ − λ j T,
0) subconverge, in the Cheeger-Gromov sense, to a normalized nontrivial gradientshrinking soliton ( M ∞ , g ∞ ( t ) , p ∞ ) in canonical form on ( −∞ , Z − Z M | R ( g j ( t ) | α dvol g j ( t ) dt = Z TT − λj Z M | R ( g ( s )) | α dvol g ( s ) ds Since R T R M | R ( g ( t )) | α dvol g ( t ) dt < ∞ , letting j → ∞ , we obtain(4.1) Z − Z M ∞ | R ( g ∞ ( t ) | α dvol g ∞ ( t ) dt ≤ lim j →∞ Z TT − λj Z M | R ( g ( s )) | α dvol g ( s ) ds = 0 , which implies R ( g ∞ ( t )) ≡ M ∞ , for t ∈ [ − , M ∞ , g ∞ ( t ))is scalar flat. Arguing as in the proof of Theorem 1.1, we see that M ∞ is isometric to astandard Euclidean space R n . However, this contradicts the nontriviality of M ∞ . (cid:3) ∗ References [1] Angenent, S.B., Velzquez, J.J.L..
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E-mail address : [email protected] Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA
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