The Einstein-Lambda flow on product manifolds
TThe Einstein– Λ flow on product manifolds David Fajman and Klaus Kröncke
July, 2016
Abstract
We consider the vacuum Einstein flow with a positive cosmological constant Λ on spatial manifoldsof product form M = M × M . In dimensions n = dim M ≥ we show the existence of continuousfamilies of recollapsing models whenever at least one of the factors M or M admits a RiemannianEinstein metric with positive Einstein constant. We moreover show that these families belong to largercontinuous families with models that have two complete time directions, i.e. do not recollapse. Com-plementarily, we show that whenever no factor has positive curvature, then any model in the productclass expands in one time direction and collapses in the other. In particular, positive curvature of onefactor is a necessary criterion for recollapse within this class. Finally, we relate our results to the in-stability of the Nariai solution in three spatial dimensions and point out why a similar construction ofrecollapsing models in that dimension fails. The present results imply that there exist different classesof initial data which exhibit fundamentally different types of long-time behavior under the Einstein– Λ flow whenever the spatial dimension is strictly larger than three. Moreover, this behavior is related tothe spatial topology through the existence of Riemannian Einstein metrics of positive curvature. A fundamental open problem in the mathematical study of cosmological models is the determination ofthe long-time behavior of the Einstein flow (
Λ = 0 ) or the Einstein- Λ flow ( Λ > ) on closed spatialmanifolds M , where Λ denotes the cosmological constant. By Einstein flow we refer to the form of theEinstein equations when a foliation of spacetime is chosen. In that case the equations appear as a sys-tem of PDEs that determines the evolution of geometric tensor fields and is in this way similar to othergeometric flows. This term has been coined in earlier works (cf. [AnMo11]). In the
Λ = 0 case it iswell-known that the long-time behavior depends on the topology of M . If M is three-dimensional and ofpositive Yamabe type Y + , it is conjectured that its maximal globally hyperbolic development (MGHD)recollapses, i.e. is future and past geodesically incomplete [BaGaTi86] ( Closed Universe Recollapse con-jecture ). This behavior is proven for the class of spherically symmetric solutions [BuRe96, He02]. Inthe general case, the problem is open. In the negative Yamabe class Y − it is known that in an openneighborhood of initial data corresponding to the Milne model, this solution serves as an attractor of theEinstein flow [AnMo11]. It is conjectured that under additional restrictions the global geometry of theMilne model (future completeness) is prototypical for the behavior of the Einstein flow in the classes Y − or Y (but the spacetime has to be non-flat) in three spatial dimensions, i.e. that the spacetimes admitCMC foliations where the mean curvature takes values in ( −∞ , and that this foliation covers the entirespacetime [Re96]. (Cf. also [An01] for a discussion of the recollapse conjecture.)For the Einstein- Λ flow with Λ > on closed manifolds of dimension n = 3 (i.e. n denotes the spatialdimension) the explicit solutions and their stability analysis suggest that the behavior of the flow is lesstopology-dependent and in particular recollapsing models are to our knowledge not known to exist (inthe vacuum case). All known models possess at least one complete time direction, in the case of deSitterspacetime, which has spatial topology S , both time directions are complete – one of which can be viewn Mathematics Subject Classification.
Key words and phrases.
General Relativity, Einstein metrics, Recollapse, positive cosmological constant. a r X i v : . [ g r- q c ] N ov s future contracting, the other one as future expanding. These regions merge at a minimal surface. ForCauchy hypersurfaces in the negative or zero Yamabe class homogeneous models are past incompleteand future complete. All those models are future nonlinear stable [Ri08]. In particular in an open neigh-borhood of initial data induced by those background solutions, the Einstein- Λ flow is well understoodtowards the expanding direction. Λ flow on product manifolds In this paper we prove that for any spatial dimension n ≥ there exist open sets of initial data, whoseMGHDs recollapse under the vacuum Einstein- Λ flow. This implies in particular that the behavior ofthe Einstein- Λ flow in three spatial dimensions differs substantially from the situation in any higher di-mension. Moreover, we show that within the class of product manifolds, which we consider, positivecurvature of at least one factor is necessary for recollapse. This will be made precise in the following.The explicit models, which we construct are of product form (cid:0) M , g (cid:1) = (cid:0) I × M × M , − dt + a ( t ) g + b ( t ) g (cid:1) , (1.1)where g and g are Einstein metrics, i.e. Ric[ g i ] = λ i g i for i = 1 , and constants λ i . We in-troduce x = log a and y = log b . Initial data for spacetimes of this form at t = 0 is given by ( M × M , λ , λ , ˙ x (0) , ˙ y (0)) . We assume x (0) = y (0) = 0 without loss of generality. Note thatother values for x and y can be encoded in the constants λ and λ . We show that if for both factors M i , λ i is positive, then there exists a continuous 1-parameter family ofinitial data such that this family decomposes into two open sets, with recollapsing and non-recollapsingMGHDs, respectively, and one initial data set which necessarily corresponds to an unstable spacetime.An explicit condition for the family of initial data reveals that a necessary criterion for recollapse is acertain disequilibrium between the two factors. We make this precise in the following theorem. Theorem 1.1.
Let
Λ = n ( n − and let M , M be closed manifolds of dimension dim M i = n i ≥ , n + n = n . Consider initial data of the form x (0) = y (0) = ˙ x (0) = ˙ y (0) = 0 such that n ( n −
1) = n λ + n λ , (1.2) holds. Then the following cases occur.1. If λ > n and λ < ( n − n/n , the MGHD recollapses, i.e. is future- and past incomplete, I = ( t − , t + ) for t − < < t + . Both singularities are of Kasner-type, i.e. one factor collapses andthe other factor has unbounded volume as t → t ± . The mean curvature H → ±∞ as t → t ± .The Kretschmann scalar is unbounded towards both singularities. In particular, the MGHDs are C -inextendible.2. If λ = n and λ = ( n − n/n , MGHD is (cid:32) R × M × M , − dt + cosh (cid:18)(cid:114) nn t (cid:19) · g + g (cid:33) , (1.3) and in particular future and past complete. . If n ( n − /n < λ < ( n − or equivalently n − < λ < n the MGHD is future- and pastcomplete. In both time directions, both factors expand exponentially with asymptotic growth ratesdepending only on the dimensions n and n .Furthermore, the behavior is stable in the following sense. Initial data induced by any recollapsingsolution in the class above has an open neighborhood in the initial-data manifold, such that the MGHDof each element in this neighborhood recollapses. Similarly, initial data induced by any complete solution(except the critical solution) has an open neighborhood with elements that give rise to complete MGHDs.The critical solution (1.3) is unstable.Remark . The case λ = λ = n − corresponds to the background solution − dt + cosh( t ) · ( g + g ) . (1.4)If λ < λ we divide into three subcases that relate to the previous cases by interchanging λ and λ (and n and n ) in the statements of the theorem. Remark . Relation (1.2) provides a simple interpretation for the case λ , λ > . Then the firstcondition in the theorem concerns initial data for which the curvatures of both factors differ substantiallywhile the last condition corresponds to initial data where both factors are of equal or almost equal Ricci-curvature. We conclude that for the case of positive curvature recollapse corresponds to a disequilibriumin curvature between the factors. For product manifolds with two non-positive factors (i.e. flat or negatively curved), all spacetimes of theform (1.1) have a uniform behavior – one time direction is complete while the other one is incomplete.
Theorem 1.4.
Every spacetime of the form (1.1) with λ , λ ≤ is complete in one time direction andincomplete in the other.Remark . The previous theorem implies that within the class given by (1.1) positive curvature of atleast one factor is a necessary criterion for recollapse.
We have mentioned that an analogous construction of recollapsing models in 3+1 dimensions fails. If weconsider the product manifold S × S , there is, however, an interesting connection to the present study.The Nariai solution − dt + 13 g S + cosh ( √ t ) dx , (1.5)where g S denotes the metric of constant curvature , is known to be unstable [Be09-1, Be09-2] andshares similar features with the critical solution in Theorem 1.1, which has one constant factor and oneexpanding factor (as Nariai does). However, the instability of this critical solution is obvious since itmarks the border between two different regimes of initial data, in particular, it is unstable already in thisclass of product manifolds. We provide an elementary proof of the instability of the Nariai solution in ourformalism. Theorem 1.6 (cf. [Be09-1, Be09-2]) . The Nariai solution is unstable in the class of product manifolds ofthe type (1.1) on S × S , i.e. there are arbitrarily small perturbations with an incomplete MGHD. Here, open is meant in the sense of ( g, K ) ∈ H (cid:96) × H (cid:96) − spaces with (cid:96) = (cid:96) ( n ) sufficiently large, where ( g, K ) denote thespatial metric and second fundamental form, respectively. .5 Discussion The results presented in this paper for the case where at least one factor has positive curvature concernso far only one-parameter families of initial data and their neighborhood in the initial data manifold. Acomplete analysis of the 3-dimensional space of initial data with this product structure with techniquessimilar to those used in [AnHe07] is clearly desirable. However, in view of the implications for openneighborhoods in the full initial data manifold our results already establish the existence of differentstable types of behavior for the Einstein- Λ flow in higher dimensions ( dim M = n ≥ ). A partialresult by the authors, which concerns only even spatial dimensions has been discussed in further detail in[FaKr15]. We proceed with proving Theorems 1.1, 1.4 and 1.6. We begin by presenting the Einstein equations forthe product ansatz (1.1) and recall a modified version of the Hawking singularity theorem by Anderssonand Galloway [AnGa04], which applies to solutions of the Einstein equations with positive cosmologicalconstant. This theorem allows to conclude incompleteness towards one time direction for a large set ofmodels and is relevant for certain proofs below. We then close with the proofs of Theorems 1.1, 1.4 and1.6.
We set
Λ = n ( n − and consider solutions of the form (1.1). Then, recall x = log( a ) and y = log( b ) , theEinsteins equations imply the following system of ODEs: x (cid:48)(cid:48) = n − λ e − x − n ( x (cid:48) ) − n x (cid:48) y (cid:48) y (cid:48)(cid:48) = n − λ e − y − n ( y (cid:48) ) − n x (cid:48) y (cid:48) (2.1)and the Hamiltonian constraint reduces to n ( n −
1) = n λ e − x + n λ e − y + ( n − n ( x (cid:48) ) + ( n − n ( y (cid:48) ) + 2 n n x (cid:48) y (cid:48) , (2.2)where x (cid:48) denotes the time derivative of x . The equations are a direct consequence of [DoUe05, Propo-sition 2.5] using the ansatz (1.1) for the metric. A straightforward computation shows that the meancurvature H , the second fundamental form K and the norm of the tracefree part Σ of the latter are givenby H = − n x (cid:48) − n y (cid:48) ,K = − x (cid:48) · g − y (cid:48) · g , | Σ | = n n n · ( x (cid:48) − y (cid:48) ) , (2.3)respectively. We cite below the generalized version of the Hawking singularity theorem from [AnGa04] and derive acorollary suitable for the present purpose.
Proposition 2.1 (cf. [AnGa04, Proposition 3.4]) . Let ( M n +1 , g ) be a n + 1 -dimensional globally hyper-bolic Lorentzian manifold and suppose Ric ( X, X ) ≥ − n | g ( X, X ) | (2.4)4 or all timelike vectors X . Assume furthermore that there exists a hypersurface M whose expansionsatisfies θ ≤ θ < − n . Then, ( M , g ) is future geodesically incomplete. Corollary 2.2.
Let ( M , g ) be a MGHD of Einstein’s equations with cosmological constant Λ = n ( n − > . If M contains a Cauchy hypersurface M whose induced metric has scalar curvature R ( g ) < R < ,then M is geodesically incomplete either to the future or to the past.Proof. Let g be the induced metric on M and as above, let K be the second fundamental form of M and Σ = K − n H its tracefree part. By the constraint equations, n ( n −
1) = R ( g ) − | K | + H = R ( g ) − | Σ | + n − n H ≤ R + n − n H , (2.5)so that θ = H ≥ ( n ( n − − R ) nn − > n . (2.6)After time reflection, if necessary, we may conclude θ ≤ θ < − n (2.7)and now future incompleteness follows from the above singularity theorem. Proof of Theorem 1.1.
We first prove Case 1. By assumptions on the λ i , y (cid:48)(cid:48) (0) = n − λ < and x (cid:48)(cid:48) (0) = n − λ > nn > . Then, x is monotonically increasing: If t > is the first time such that x (cid:48) ( t ) = 0 , we get x (cid:48)(cid:48) ( t ) > nn > (if λ ≤ , we even get x (cid:48)(cid:48) ( t ) ≥ n ) which implies x (cid:48) ( t ) < for t ∈ ( t − ε, t ) and thus contradicts the assumption. Analogously, one shows that y is monotonicallydecreasing.Now we show that as long as y is uniformly bounded, its derivatives exist and x and its derivativescannot blow up in finite time. If y (cid:48) ≤ − and y ≥ Y , we obtain by monotonicity, the evolution equationand x (cid:48) y (cid:48) < that y (cid:48)(cid:48) ≥ − C ( y (cid:48) ) (2.8)which after dividing by y (cid:48) and integration in turn implies log | y (cid:48) | ≤ C (1 − Y ) , (2.9)i.e. y (cid:48) is bounded. Observe that by the evolution equation, we get x (cid:48)(cid:48) ≤ n + Cx (cid:48) as long as y (cid:48) > − C which implies the assertion on x .Next we show that y is unbounded. If not, it must converge to a limit by monotonicity, so we have y ( ∞ ) = lim t →∞ y ( t ) . Then we get either a sequence of t i → ∞ such that y (cid:48)(cid:48) ( t i ) = 0 or lim t →∞ y (cid:48)(cid:48) ( t ) =0 . The first case cannot occur, since then y (cid:48)(cid:48) ( t i ) = n − λ e − y ( t ) ≥ nn > . In the latter case, wenecessarily get lim x (cid:48) ( t ) = ∞ by the evolution of y . On the other hand, by the constraint equation, n x (cid:48)(cid:48) − n y (cid:48)(cid:48) = n ( n − n ) − n λ e − x + n λ e − y − n ( x (cid:48) ) + n ( y (cid:48) ) ≤ C − n ( x (cid:48) ) + n ( y (cid:48) ) (2.10)From the constraint and adding the evolution equations, we also easily obtain n ( x (cid:48)(cid:48) + ( x (cid:48) ) ) + n ( y (cid:48)(cid:48) + ( y (cid:48) ) ) = n, (2.11)from which we get − n ( x (cid:48) ) − n y (cid:48)(cid:48) + n − n ( y (cid:48) ) ≤ C − n ( x (cid:48) ) + n ( y (cid:48) ) (2.12)5nd therefore, since n > (if n = 1 , the above bound implies λ < which in turn contradicts n = 1 ), we get a bound on x (cid:48) if y (cid:48) and y (cid:48)(cid:48) are bounded. This yields the contradiction.By monotonicity of x and y and the unboundedness of y from below there is a constant C > and atime t > such that the differential inequality n x (cid:48)(cid:48) + n y (cid:48)(cid:48) ≤ − C − ( n x (cid:48) + n y (cid:48) ) (2.13)holds for all t ≥ t . By ODE comparison, z ( t ) = n x (cid:48) ( t ) + n y (cid:48) ( t ) ≤ −√ C tan( D + √ C ( t − t )) for t ≥ t and some D ∈ R . Therefore, z → −∞ as t approaches some t + . As x (cid:48) > , y (cid:48) → −∞ . We alsohave y ( t ) ≤ n (cid:82) t z ( s ) ds → −∞ as t → t + , which implies that y diverges. In addition, because x (cid:48)(cid:48) = n − λ e − x − zx (cid:48) ≥ − zx (cid:48) , (2.14)we obtain x ( t ) ≥ (cid:82) t exp( − (cid:82) s z ( r ) dr ) ds → ∞ as t → t + .It remains to show that the Kretschmann scalar is unbounded at the singularity: Let be the indexreferring to the t -coordinate and i, j and a, b be indices referring to coordinates on M , M respectively.One can check that the coefficients with an odd number of zeros vanish and therefore, the Kretschmannscalar satisfies | ˜ R | = ˜ R µνλσ ˜ R µνλσ ≥ (2.15)because all summands are nonnegative. Now it can be checked that ˜ R ij = − ∂ t aa ˜ g ij , ˜ R ab = − ∂ t bb ˜ g ab , (2.16)so that | ˜ R | ≥ (cid:18) n ∂ t aa (cid:19) + n (cid:18) ∂ t bb (cid:19) = n ( x (cid:48)(cid:48) + ( x (cid:48) ) ) + n ( y (cid:48)(cid:48) + ( y (cid:48) ) ) (2.17)where as above x = log( a ) , y = log( b ) . Now, since x → ∞ as t → t + , we also get x (cid:48)(cid:48) + ( x (cid:48) ) → ∞ ,which shows that the Kretschmann scalar must blow up. This finishes the proof in Case 1. In Case 2, itis easy to see that the metric given in the statement of the theorem is the MGHD of the given initial data.Let us now consider Case 3. By assumption, y (cid:48)(cid:48) (0) = n − λ ∈ (0 , (2.18)and x (cid:48)(cid:48) (0) = n − λ ∈ (1 , nn ) . (2.19)Both x and y are strictly monotonically increasing. We have x (cid:48) ( t ) > for small t . Let t be the first time,where x (cid:48) ( t ) = 0 . Then we obtain x (cid:48)(cid:48) ( t ) = n − λ e − x ( t ) > , (2.20)as long as y (cid:48) ( t ) exists. Thus x (cid:48) < on ( t − ε, t ) , which causes the contradiction. Analogously oneshows that y is strictly monotonically increasing. From the evolution equations and monotonicity wededuce n − n λ − n λ − ( n x (cid:48) + n y (cid:48) ) ≤ n x (cid:48)(cid:48) + n y (cid:48)(cid:48) ≤ n − ( n x (cid:48) + n y (cid:48) ) . (2.21)The solution of the corresponding ODE, z (cid:48) = n − z , is z ( t ) = n tanh( nt + c ) , where c ∈ R . By theinitial conditions this implies < n x (cid:48) + n y (cid:48) < n for all t > . Due to the positivity of x (cid:48) and y (cid:48) thesestatements hold for x (cid:48) and y (cid:48) individually. In particular, x and y exist for all times, which finishes theproof of Case 3. 6inally, recollapse of MGHDs of small perturbations of the initial data induced by the models followsby the Cauchy stability of the Einstein equations in combination with Theorem 2.1. As the backgroundsolutions have Cauchy surfaces of strictly positive and strictly negative mean curvature, this also holdsfor the MGHDs of small perturbations. Then the singularity theorem yields incompleteness. This idea ofproof is explained in further detail originally in [Ri09]. Stability in the expanding direction follows fromthe results in [Ri08]. Remark . With some more effort, we can determine the expansion rate of the warping factors of thesolutions appearing in Case 3: By the above, we have n x (cid:48) ( t ) + n y (cid:48) ( t ) > C > for all t ≥ t and n x ( t ) + n y ( t ) > C t for all t ≥ t and C > . Using this, we can improve the above estimate.Denoting z = n x (cid:48) + n y (cid:48) as above, we get n − Ce − Ct − z ≤ z (cid:48) ≤ n − z . (2.22)from which we obtain the uniform bounds ≥ z − n ≥ − Ce − Ct . After integrating, ≥ n x + n y − nt ≥ − C. (2.23)Similarly, we obtain for w := n x (cid:48) − n y (cid:48) the estimate n ( n − n ) − Ce − Ct − zw ≤ w (cid:48) ≤ n ( n − n ) + Ce − Ct − zw (2.24)which implies because of the estimate on n − z that | w − ( n − n ) | ≤ Ce − Ct . Again by integrating, C ≥ n x − n y − ( n − n ) t ≥ C . (2.25)Finally, we obtain from (2.23) and (2.25) that sup {| x − t | , | y − t |} ≤ C . It is immediate that the warpingfunctions satisfy C e t ≤ a ( t ) ≤ C e t and C e t ≤ b ( t ) ≤ C e t for t ≥ . Proof of Theorem 1.4. If λ , λ ≤ , we are going to show that all corresponding solutions are geodesi-cally incomplete in one direction and geodesically complete in the other. Let us first remark that the casewhere R ( g ) ≡ | Σ | ≡ corresponds to the solution − dt + e t g which is geodesically incomplete in thepast and geodesically complete in the future. Therefore we assume R ( g ) − | Σ | ≤ R < from now on.By the constraint equation, we have n ( n −
1) = R ( g ) − | K | + H = R ( g ) − | Σ | + n − n H ≤ R + n − n H , (2.26)As above, we get | θ | ≥ θ > n . If θ ≤ − θ < − n , we obtain geodesic incompleteness by Proposition2.1. If θ ≥ θ > n , we have the differential inequality ∂ t θ ≤ n − n θ (2.27)so that we have the upper bound θ ≤ n · e t − A e t + A = n − | A | e − t (1 + | A | e − t ) ≤ n + n | A | e − t (2.28)where A = n − θ n + θ . Again by the constraint equation, ≤ − R ( g ) + | Σ | = n − n ( θ − n ) ≤ Ce − t . (2.29)In particular, since R ( g ) = n λ e − x + n λ e − y , both factors x, y grow linearly, in particular, they donot converge to −∞ . Thus, they are geodesically complete.7 emark . Since θ → n and | Σ | → exponentially, one concludes that x (cid:48) , y (cid:48) → exponentially and sup {| x − t | , | y − t |} ≤ C as in Remark 2.3. Remark . We expect that the generic type of singularity that occurs in the geodesically incompletedirection is of Kasner type, i.e. one of the factors goes to ∞ while the other goes to −∞ as t → t + . Thisalso implies a blow-up of the Kretschmann scalar as in the proof of Theorem 1.1. Proof of Theorem 1.6.
We have the data n = 2 , n = 1 , λ = λ , λ = 0 , so the evolution equations are x (cid:48)(cid:48) = 3 − λe − x − x (cid:48) ) − x (cid:48) y (cid:48) y (cid:48)(cid:48) = 3 − ( y (cid:48) ) − x (cid:48) y (cid:48) (2.30)and the constraint equation reduce to λe − x + 2( x (cid:48) ) + 4 x (cid:48) y (cid:48) . (2.31)We demand y (cid:48) (0) = 0 such that we obtain a one-parameter family of solutions by prescribing λ and x (cid:48) (0) such that λ + ( x (cid:48) (0)) (2.32)by the constraint. Recall that we generally require x (0) = 0 . For our purposes, we additionally assume λ > throughout. By using the constraint, the equation on x decouples as x (cid:48)(cid:48) = 3 − λe − x − x (cid:48) ) − x (cid:48) y (cid:48) = 12 (3 − λe − x − x (cid:48) ) ) . (2.33)Now we show that the MGHD is future incomplete, if λ < and x (cid:48) (0) < . At first, x (cid:48)(cid:48) (0) = 3 − λ − x (cid:48) (0) = (3 − λ − ( x (cid:48) (0)) ) − x (cid:48) (0) < , (2.34)which shows that both x and x (cid:48) are monotonically decreasing for small time. In addition, as long as x and x (cid:48) are monotonically decreasing, we have x (cid:48)(cid:48) = 12 (3 − λe − x − x (cid:48) ) ) ≤
12 (3 − λ − x (cid:48) (0))) ≤ x (cid:48)(cid:48) (0) < , (2.35)which shows that this monotonicity property holds for all time and x and x (cid:48) are unbounded. In particular,for large time, we get x (cid:48)(cid:48) ≤ −
32 ( x (cid:48) ) . (2.36)This shows that x (cid:48) → −∞ as t approaches some t + , since the comparison ODE is solved by z ( t ) = c +3 t .By integrating, one obtains the estimate x ( t ) ≤ C + log( C − t ) (2.37)for some constants C , C > which shows that also x → −∞ as t → t + . Remark . We are also able to show that this singularity is of Kasner type, i.e. y ( t ) → ∞ as t → t + .At first note that y is monotonically increasing for small time. In fact, it is monotonically increasing forall t > since y (cid:48)(cid:48) ( t ) = 3 > as long as y (cid:48) ( t ) = 0 . But then we have an estimate y (cid:48)(cid:48) ≥ − ( y (cid:48) ) + 2 | z | y (cid:48) (2.38)and since z → ∞ as t → t + this shows that y (cid:48) → ∞ as t → t + and because the integral of z diverges,we get that also y diverges as t → t + . 8 emark . A similar analysis as in the proof of Theorem 1.1 shows that every MGHD is complete in theother time-direction and arguments as in Remark 2.3 imply exponential growth of both warping functionsin the past.
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AVID F AJMAN F ACULTY OF P HYSICS , U
NIVERSITY OF V IENNA ,B OLTZMANNGASSE
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IENNA , A
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[email protected] LAUS K RÖNCKE F ACULTY OF M ATHEMATICS , U
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ERMANY [email protected]@uni-hamburg.de