aa r X i v : . [ m a t h . DG ] A ug RICCI EXPANDERS AND TYPE III RICCI FLOW
LI MA
Abstract.
In this paper, we study how to get the Ricci expandersfrom W + -functional through the heat kernel estimate of the conjugateheat equation to the type III singularity of Ricci flow. The Gaussianupper and lower bounds are established for the related heat kernel inaccordance to the interesting work of Cao-Zhang for the type I Rici flow. Mathematics Subject Classification 2000 : 53Cxx,35Jxx
Keywords : Ricci expanders, type III singularity,heat kernel esti-mates, Ricci flow Introduction
In this paper, we study the question how to get the Ricci expandersfrom W + -functional through the heat kernel estimate of the conjugate heatequation to the type III singularity of Ricci flow ([6]). As we shall see, thegradient estimate is an important step in solving the Hamilton conjecture.Given a complete non-compact Riemannian manifold ( M, g ) of dimensionn. Recall that a family ( g ( t )) of Riemannian metrics on M n is called a Ricciflow if g ( t ) satisfies the following Ricci flow equation(1) ∂ t g ij ( t ) = − R ij ( g ( t )) , on M, with g (0) = g . We shall assume the Ricci flow ( M, g ( t )) a type III singularityin the sense that the flow is k -non-collapsed on all scales for some constant k > , ∞ ) with thecurvature bound(2) | Rm ( g ( t )) | ≤ AA + t for some uniform constant A >
0. Then the conjugate heat equation asso-ciated to the Ricci flow ( g ( t )) is(3) u t = − ∆ u + Ru where ∆ is the Laplacian-Beltrami operator of the evolving metric g ( t ). Notethat the adjoint equation to the conjugate heat equation associated to Ricciflow is(4) u t = ∆ u. The research is partially supported by the National Natural Science Foundation ofChina 10631020 and SRFDP 20090002110019.
We remark that the maximum principle is true for this heat equation asso-ciated to the Ricci flow (
M, g ( t )) (see [13]). Since we are studying the heatequation on complete non-compact Riemannian manifold, the heat kernelis restricted to the minimal fundamental solution G = G ( z, l ; x, t ), l < t ,to (3) (or u ( x, t ) == G ( z, l ; x, t ) to (4)). Note here that we have used animportant observation that the fundamental solution to the heat equation(4) can also be regards as the fundamental solution to the conjugate heatequation (3). As R.Hamilton [6] expected, the type III singularity of Ricciflow gives an expanding soliton. For the W + -functional(5) W + ( g, G, σ ) = Z M [ σ ( |∇ u | + R ) − f + n ] udv g , σ = t − T > , with G = e − f / (4 πσ ) n/ , introduced by M.Freldman, T.Ilmanen, L.Ni [5] (following the W-functionalof G.Perelman), we have ddt W + ( g ( t ) , G ( t ) , t − T ) = Z M t − T ) | Rc + D f + g t − T ) | dv g ≥ . Fix T = 0. Hence one may get an Ricci expander by studying W + along theheat kernel and the limit of some normalization of g ( t ) as t ∞ .Our main result is below. Theorem 1.
Let ( M, g ( t )) , t ∈ [0 , ∞ ) , be a non-flat, type III k-non-collapsedRicci flow with non-negative Ricci curvature and for some A > , | Rm ( g ( t )) | ≤ AA + t , t > . Then at any point x ∈ M , a sequence of times τ k → ∞ , and a sequence ofre-scaled metrics g k ( x, s ) = τ − k g ( x, sτ k ) such that the pointed Ricci flow sequence ( M, x , g k ) converges to a non-flatgradient expanding Ricci soliton in the sense of Cheeger-Gromov sense. This result is used in [8], where part of Hamilton’s conjecture has beenproved and a full study about the singularities of Ricci flow with positiveRicci pinching condition is presented. Related work by us using Yamabeflow is [9].We now make some remarks about the proof. At the first step, we shouldmake sure that the W + -functional is well-defined. This will be achievedby obtaining gradient estimate of the positive solution u and the upperand lower bounds of u in space-time. According [14] (see also the work ofCao-Hamilton [1], both following the idea of R.Hamilton), for any positivesolution u to (4) on M × [0 , T ], we have(6) |∇ log u ( x, t ) | ≤ r t s log Mu ( x, t ) ICCI EXPANDERS AND TYPE III RICCI FLOW 3 for M = sup M × [0 ,T ] u and ( x, t ) ∈ M × [0 , T ] and moreover, for any δ > t < t ≤ T , and x, y ∈ M :(7) u ( y, t ) ≤ C u ( x, t ) / (1+ δ ) M δ/ (1+ δ ) e C d ( x,y,t ) /t where C and C are positive constants depending only on δ , d ( x, y, t ) is thedistance between x and y in the metric g ( t ). We shall follow the recentinteresting works of Zhang [14], Cao-Zhang [2] to derive the desired heatkernel estimates and the result is presented in section 2. Our main heatkernel estimate for the Ricci flow in Theorem 1 is the same result as Theorem3.1 in [2]. In section 3 we give the argument of Theorem 1.2. Heat kernel estimate along the Ricci flow
Let us see some analytic parts of the Type III singularity of the Ricci flow(M,g(t)).1. (M,g(t)) has a space-time doubling property. Namely, the distancesof two points x, y ∈ M at two different times t > s > s/t (which is comparable in the sense that it is bigger than a smallconstant). In fact, let γ ( τ ; g ( t )) be the minimizing geodesic connecting x and y in the metric g ( t ). Then d ( x, y ; t ) = L ( γ ; g ( t )). Then by using0 ≤ Rc ( g ( t )) ≤ AA + t , we get 0 ≥ ddt d ( x, y ; t ) = − Z Rc ( γ ′ , γ ′ ) dτ ≥ − AA + t d ( x, y ; t ) , where the derivative is in Lipschitz sense. The latter implies that( s/t ) A ≤ d ( x, y ; s ) d ( x, y ; t ) ≤ .
2. Similarly, local volume comparable property is also true. In fact, wehave0 ≥ ddt Z B ( x, √ t ; t ) dv g ( t ) = − Z B ( x, √ t ; t ) Rdv g ( t ) ≥ − AA + t Z B ( x, √ t ; t ) dv g ( t ) . Upon integration we know that the volumes of the balls ( x, √ t ; t ) in termsof the metric g ( t ) are comparable for t , t , t ∈ [ s, t ].3. According to the result of E.Hebey we know that the following Sobolevinequality holds for the k-non-collapsed ( M, g ( t )) with Ric ( g ( t )) ≥
0. Namelyfor all v ∈ H ( B ( x, r ; t )), we have( Z | v | n/ ( n − ) ( n − /n ≤ c n r | B ( x, r ; t ) | /n Z [ |∇ v | + r − v ] dv g ( t ) . We shall choose r = c √ t for c ∈ (0 , | Ric | ≤ AA + t and the k-non-collapsing property, we know that | B ( x, r ; t ) | ≥ kA − n t n/ . LI MA
Hence, we obtain the following uniform Sobolev inequality along the Ricciflow. For all v ∈ H ( B ( x, √ t ; t )), we have( Z | v | n/ ( n − ) ( n − /n ≤ c n A k /n Z [ |∇ v | + t − v ] dv g ( t ) . In the following we try to get the Gaussian upper and lower bounds forthe heat kernel to the heat equation. The general program for this work isin three steps below. 1). We derive a weaker on-diagonal upper bound u ( x, t ) ≤ const.f ( t )for some increasing function in t. Here u ( x, t ) = G ( x , x, t ). The methodis the Moser iteration and in this step, the uniform Sobolev inequality plays aimportant role. 2). We derive the Gaussian upper bound by the exponentialweight method due to E.B.Davies. 3). We derive a on-diagonal lower boundat some point and then we obtain the full Gaussian lower bound by usingthe gradient estimate obtained by Zhang [14] and Cao-Hamilton [1], bothpapers follow the idea of the work of R.Hamilton for standard heat equation.However, such a machinery can be go through in our setting. So we shallprove the Gaussian upper and lower bound of the heat kernel in a new way(which is also observed by X.Cao and Zhang [2]).Once we have the Gaussian upper and lower bound of the heat kernel, weimmediately see the W + -functional is well-defined along the Ricci flow.Let us now go to the detail. Note that if G = G ( z, l ; x, t ), l < t , is thefundamental solution of u l = − ∆ u + Ru along the Ricci flow, then as a function of ( x, t ), p ( x, t ; x , l ) := G ( x , l ; x, t ), l < t , is the fundamental solution to the heat equation u t = ∆ u. Let u = u ( x, t ) be a positive solution to the heat equation (4) in the region Q στ := { ( y, s ) ∈ M × [ τ − ( σr ) , τ, d ( y, x ; s ) ≤ σr ] } . Here r = √ t/ > σ ∈ [1 , p ≥ ∂ t u p ≤ ∆ u p . Choose a non-negative smooth function φ : [0 , ∞ ) → [0 ,
1] such that | φ ′ | ≤ σ − r , φ ′ ≤ , and φ ( ρ ) = 1 for 0 ≤ ρ ≤ r and φ ( ρ ) = 0 for ρ ≥ σr . Then we choose asmooth non-negative function η such that | η ′ | ≤ σ − r , η ′ ≥ , and η ( ρ ) = 1 for τ − r ≤ s ≤ τ and φ ( s ) = 0 for s ≥ τ − ( σr ) .Define ξ = φ ( d ( x, y ; s )) η ( s ). ICCI EXPANDERS AND TYPE III RICCI FLOW 5
Set w = u p and using wξ as a testing function to the above differentialinequality we deduce that Z ∇ w · ∇ ( wξ ) dv g ( s ) ds ≤ − Z ∂ s wwξ dv g ( s ) ds. Note that the left hand side is Z |∇ ( wξ ) | dv g ( s ) ds − Z | ξ | w dv g ( s ) ds and the right hand side is Z w ξ ∂ s dv g ( y,s ) ds − Z ( wξ ) Rdv g ( s ) ds − Z ( wξ ) dv g ( y,τ ) , Using R ≥ φ and η we know that the latter is boundedby c ( σ − r Z w dv g ( s ) ds − Z ( wξ ) dv g ( y,τ ) . Re-arranging the above relations we get Z |∇ ( wξ ) | dv g ( s ) ds + 12 Z ( wξ ) dv g ( y,τ ) ≤ c ( σ − r Z w dv g ( s ) ds. Note that the Sobolev inequality gives us( Z | wξ | n/ ( n − ) ( n − /n ≤ c n A k /n Z [ |∇ ( wξ ) | + r − ( wξ ) ] dv g ( t ) . by Holder inequality Z ( wξ ) /n ) dv g ( s ) ≤ Z ( wξ ) n/ ( n − dv g ( s ) ) ( n − /n ( Z ( wξ ) dv g ( s ) ) /n . Then using the trick as in [4] we get that Z Q r ( x,τ ) w θ ≤ c ( k, A )( 1( σ − r Z Q σr ( x,τ ) w ) θ with θ = 1 + 2 /n . Choose σ = 2 , σ i = 2 − i X − j, p = θ i , and we find that sup Q r/ ( x,τ ) u ≤ c ( k, A ) r n +2 Z Q r ( x,τ ) u dv g ( s ) ds. Using the general trick of Li-Schoen [10], we then find the L mean valueinequality in the form below:sup Q r/ ( x,τ ) u ≤ c ( k, A ) r n +2 Z Q r ( x,τ ) udv g ( s ) ds. LI MA
Using u ( x, t ) = G ( x , x, t ), r = √ t and the fact R M udv g ( s ) = 1, we obtain G ( x , x, t ) ≤ c ( k, A ) t n/ . Next we prove the lower bound by the trick of Perelman [11]. This argu-ment is borrowed from [2]. Let u = u ( x, t ) = G ( x, t ; x , t ) for t < t . Claimthat for some uniform constant C >
0, we have, for t < t , G ( x , t ; x , t ) ≥ Cτ n/ e − √ τ R t t √ t − tR ( x ,s ) ds where τ := t − t .Following G.Perelman, we set u = (4 πτ ) − n/ e − f . Using Perelman’s differential Harnack inequality for the fundamental solu-tion we have that for γ ( t ) = x , we have − ∂ t f ( x , t ) ≤ R ( x , t ) − τ f ( x , t ) . Then for any t < t < t we can integrate the above inequality to obtain f ( x , t ) √ t − t ≤ f ( x , t ) √ t − t + 12 Z t t √ t − sR ( x , s ) ds. We remark that by the asymptotic formula for G we know that for t ap-proaches to t , f ( x , t ) stays bounded since G ( x , t ; x , t )( t − t ) n /
2) isbounded between two positive constants. Then for any t ≤ t , f ( x , t ) ≤ √ t − t Z √ t − sR ( x , s ) ds. Hence, G ( x , t ; x , t ) ≥ c (4 πτ ) − n/ e − √ t − t R √ t − sR ( x ,s ) ds . Using the assumption | R ( x, s ) | ≤ A/ ( t − s ), we then know that G ( x , t ; x , t ) ≥ c (4 πτ ) − n/ . In summarize, we have obtained the below.
Lemma 2.
Let ( M, g ( t )) , t ∈ [0 , ∞ ) , be a k-non-collapsed Ricci flow withbounded curvature | Rm | ≤ AA + t , and non-negative Ricci curvature. Thenthere exist positive constant C and c which depends only on k, n, A , suchthat for all x, x ∈ M , t > , we have G ( x , x, t ) ≤ C t n/ , and G ( x , x , t ) ≥ C t n/ , With this Lemma to replace Theorem 2.1 in [2], we can get the sameresult as Theorem 3.1 in [2], which is enough for our use of W + -functional. ICCI EXPANDERS AND TYPE III RICCI FLOW 7 Blow-down for the Type III Ricci flow
Choose suitable time sequence τ k → ∞ and point sequence x k ∈ M forthe blowing up metrics as in Hamilton [6]. Consider the pointed Ricci flow( M, g k , x k ) with g k ( s ) := τ − k g ( · , t k + sτ k ) . Let, for s ∈ [1 ,
4] and for x = x k , u k = u k ( x, s ) := τ n/ k G ( x, sτ k ; x , τ k ) . Then u k satisfies ∂ s u k = − ∆ g k u k + R ( g k ) u k . Recall that f k is defined by the relation(4 πs ) − n/ e − f k = u k . Using the upper bound for u k , we know that − f k = log u k + n πs ) ≤ C for all k = 1 , , ... and s ∈ [1 , R M u k dg k = 1, by our uniform boundsfor u , we know that there is a limit u ∞ of u k as k → ∞ . Note also that W + k ( s ) ≤ C − n for all k = 1 , , ... and s ∈ [1 , W + k ( s ) = W + ( g, u, sτ k ) ≤ C − n, where u = u ( x, t ) = G ( x, t ; x , τ k ), 0 ≤ t ≤ τ k . Note that using theasymptotic behavior of u (see [3]) we have W + ( g, G ( x, t ; x , τ k ) , sτ k ) ≤ W + ( g, G ( x, t ; x , τ k +1 ) , sτ k ) + ◦ k (1) , and using the increasing property we have W + ( g, G ( x, t ; x , τ k +1 ) , sτ k ) ≤ W + ( g, G ( x, t ; x , τ k +1 ) , sτ k +1 ) . Hence there exists the limit W ∞ ( s ) for the sequence W + k ( s ) as k → ∞ .Then we can get an expanding Ricci soliton similar to Zhang did in thecase for shrinking soliton [14]. Since the argument is almost the same, weomit the detailed proof. Thus we have completed the proof of Theorem 1. Acknowledgement : The author would also like to thank IHES, France forhost and the K.C.Wong foundation for support in 2010.
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Department of Mathematical Sciences, Tsinghua University, Peking 100084,P. R. China
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