Small curvature concentration and Ricci flow smoothing
aa r X i v : . [ m a t h . DG ] M a r SMALL CURVATURE CONCENTRATION AND RICCIFLOW SMOOTHING
PAK-YEUNG CHAN, ERIC CHEN, AND MAN-CHUN LEE
Abstract.
We show that a complete Ricci flow of bounded curva-ture which begins from a manifold with a Ricci lower bound, localentropy bound, and small local scale-invariant integral curvaturecontrol will have global point-wise curvature control at positivetimes. As applications, we obtain under similar assumptions acompactness result and a gap theorem for complete noncompactmanifolds with Ric ≥ Introduction
The Ricci flow deforms a metric g on a Riemannian manifold ( M n , g )according to the equation ∂∂t g = − g . Since its introduction by Hamilton [22], the Ricci flow has been used ina wide variety of settings to regularize metrics. One sense in which thisoccurs is described by Perelman’s pseudolocality theorem [42], whichhas played a crucial role in work on the short time existence of theRicci flow, especially in settings where the initial data lacks boundedcurvature or completeness [51, 26].
Theorem 1.1. [42, Theorem 10.1]
For any α > , there exist positiveconstant ε and δ such that if ( M n , g ( t )) , t ∈ [0 , εr ] is a solution to theRicci flow for some r > and in addition • R ≥ − r − on B ( x , r ) ; • | ∂ Ω | n ≥ (1 − δ ) c n | Ω | n − , for any open set in B ( x , r ) , where c n is the isoperimetric constant of R n ,then for any t ∈ [0 , ( εr ) ] and x ∈ B t ( x , εr ) | Rm | ( x, t ) ≤ αt − + ( εr ) − . Date : March 5, 2021.2010
Mathematics Subject Classification.
Primary 53C44 .
Key words and phrases.
Ricci flows, pseudolocality, Gap theorems.
Thus, Perelman’s pseudolocality tells us that given a lower Riccibound on an almost Euclidean region, we can deduce regularization inthe sense of curvature control along the Ricci flow for short times. SincePerelman’s work, many extensions have been developed in a variety ofsettings [14, 8, 50, 54].Related regularization results for the Ricci flow under critical L n/ bounds of Rm have previously been studied in [57, 37] assuming alsopointwise two-sided bounds on | Ric | , and in [52] assuming alternativelya supercritical k Ric k p , p > n/ L n/ bounds of Rm, but will instead doso in combination with a Ricci lower bound and control of the localentropy, a localization of Perelman’s entropy introduced by Wang [53].Below we state our main result, referring to the beginning of Section2 for most of the associated notation. Throughout, we will use a ∧ b todenote min { a, b } . Theorem 1.2.
For all
A, λ > , there are C ( n, A, λ ) , σ ( n, A ) and ˆ T ( n, A, λ ) > such that the following holds. Suppose ( M n , g ( t )) is acomplete Ricci flow of bounded curvature on [0 , T ] and for all p ∈ M ,all of the following conditions are satisfied: (a) Ric( g (0)) ≥ − λ ;(b) ν ( B ( p, , g (0) , ≥ − A ; (c) (cid:16) ´ B ( p, | Rm( g ) | n/ dµ (cid:17) /n ≤ ε for some ε < σ .Then we have for any p ∈ M and t ∈ (0 , T ∧ ˆ T ] , (1.1) | Rm | ( p, t ) ≤ C εt and inj g ( t ) ( p ) ≥ C − √ t. Moreover, we have (cid:16) ´ B t ( p, | Rm( g ( t )) | n/ dµ t (cid:17) /n ≤ C ε for all p ∈ M .In particular, the Ricci flow must exist up to ˆ T .Remark . In the statement above, we choose the scale 1 in the localentropy condition only for convenience.Theorem 1.2 is a smoothing result also based on an initial “almostEuclidean” assumption. However, instead of characterizing this usingthe isoperimetric constant as in Theorem 1.1, we instead use roughnon-collapsing and small curvature concentration.Using ideas related to those used to prove Theorem 1.2 and point-picking technique, we prove a gap result for steady and shrinking gra-dient Ricci solitons without assuming curvature boundedness.
Theorem 1.3.
For all
A > , there is ε ( n, A ) > such that if ( M n , g, f ) is a complete shrinking or steady gradient Ricci soliton sat-isfying (a) ν ( M, g ) ≥ − A ; (b) (cid:0) ´ M | Rm | n/ dµ (cid:1) /n ≤ ε ,then ( M, g ) is isometric to the standard Euclidean space R n . Gap results for gradient Ricci solitons have been previously studiedfor instance in [59, 20, 61] under global assumptions on the potentialfunction f , sometimes along with pointwise curvature control (see also[41, 15, 19, 7] ).Theorem 1.2 lends itself to several applications. First, we have agap result for Ricci-nonnegative Riemannian manifolds with k Rm k L n/ small. Corollary 1.1.
For all
A > , there is σ ( n, A ) > such that if ( M n , g ) is a complete noncompact manifold with (1) bounded curvature (2) Ric ≥ ; (3) ν ( M, g ) ≥ − A ; (4) (cid:0) ´ M | Rm | n/ dµ g (cid:1) /n ≤ σ .Then g is of Euclidean volume growth. Moreover, M n is diffeomorphicto R n . There is a large body of work on Ricci-nonnegative noncompact man-ifolds, and several results show that under some additional assumptions(such as almost Euclidean volume growth), such manifolds must bediffeomorphic to R n [4, 10, 55]. Corollary 1.1 is related to two resultsof this kind by Ledoux and Xia [34, 56], which assert that a complete,Ricci-nonnegative manifold with Euclidean-type Sobolev constant closeto that of Euclidean space must be diffeomorphic to R n . Indeed, Con-dition (3) of Corollary 1.1 above can be seen as a weakening of thisrequirement, since it holds as long as there is some constant whichmakes the Euclidean-type Sobolev inequality valid. This is compen-sated for by Condition (4) on the smallness of the total scale-invariantcurvature.Carron has also pointed out to us that Corollary 1.1 is related to thefollowing statement which can be derived from works of Cheeger [9] andCheeger-Colding [10]: If a complete noncompact manifold with Ric ≥ ´ | Rm | n/ dµ g sufficiently small relative its asymptotic volume ratio(assumed nonzero), then it must be diffeomorphic to R n . Indeed, these Pak-Yeung Chan, Eric Chen and Man-Chun Lee hypotheses ensure by [9, Theorem 4.32] that the manifold must have as-ymptotic volume ratio close to one, from which one can conclude thatthe manifold must be diffeomorphic to R n by [10, Theorem A.1.11.](see also [54, Theorem 5.7] for a recent proof via Ricci flow). Our as-sumptions in Corollary 1.1 differ slightly from this statement’s becauseour required smallness of σ is relative to a lower bound on entropy.Although we will prove below that under bounded curvature, boundedentropy, and small || Rm || L n/ we indeed have almost Euclidean volumegrowth, it is unclear to us whether an entropy lower bound implies anasymptotic volume ratio lower bound in general.We can also apply Theorem 1.2 to obtain a finite diffeomorphism-type result and a Gromov–Hausdorff compactness result in the settingof length spaces. Corollary 1.2.
For all
A > , there is σ ( n, A ) > such that for C , C , the space of compact manifolds ( M, g ) satisfying (a) Ric( g ) ≥ − C ; (b) V ol g ( M ) ≤ C ; (c) inf p ∈ M ν ( B g ( p, , g, ≥ − A ; (d) sup p ∈ M (cid:16) ´ B g ( p, | Rm | n/ dµ g (cid:17) /n ≤ σ contains finitely many diffeomorphism types. Corollary 1.2 follows via an argument analogous to the that in theproof of [31, Theorem 37.1], which was proved by Perelman [42, Re-mark 10.5] using Perelman’s pseudolocality. In our case, the use ofPerelman’s pseudolocality is replaced by Theorem 1.2.
Theorem 1.4.
For any positive integer n ≥ and constant A ≥ n ,there exists constant ε ( n, A ) such that the following holds. Suppose ( M ni , g i , p i ) is a pointed sequence of Riemannian manifolds with thefollowing properties: (a) ( M i , g i ) has bounded curvature; (b) Ric( g i ) ≥ − λ ;(c) (cid:16) ´ B gi ( q, | Rm( g i ) | n/ dµ i (cid:17) /n ≤ ε for all q in M i ; (d) ν ( B g i ( q, , g i , ≥ − A , for all q in M i .Then there exists a smooth manifold M ∞ and a complete distance met-ric d ∞ on M ∞ generating the same topology as M ∞ such that after pass-ing to sub-sequence, ( M i , d g i , p i ) converges in pointed Gromov Haus-dorff sense to ( M ∞ , d g ∞ , p ∞ ) .Remark . The Ricci lower bound assumption on the initial metriccan in fact be weakened to a scalar curvature lower bound and volume comparison control. But we feel it is more natural to state the resultwith the Ricci assumption.
Remark . The initial Ricci curvature lower bound in fact gives aSobolev inequality on the geodesic balls in M , which in turn implies aLog Sobolev inequality and provides a lower bound for the local entropyin terms of its volume. Hence the local entropy ν ( B ( p, , g (0) ,
1) lowerbound condition in the above theorem can be replaced by a uniformvolume lower bound condition for the geodesic balls on M , namely,(1.2) V ( B ( p, ≥ v , for some positive constant v , for all p ∈ M . In that case, the con-stants ε , C and ˆ T also depend on v . In particular, the global entropy ν ( M, g )’s lower bound can be replaced by Ric ≥ k Rm k L n/ bounds have also been obtained for Einstein manifolds as well as forboth compact and noncompact gradient Ricci solitons [1, 6, 25]. Incomparison, Theorem 1.4 does not impose such analytic conditions on( M i , g i ), but does require sufficient smallness of the local scale-invariantcurvature concentration.Theorem 1.4 is also a smoothing result for limit spaces of manifoldswith lower curvature bounds, achieved via distance distortion estimatesand pseudolocality-type estimates of the Ricci flow. There has beenmuch recent work in this direction in many different settings [3, 39, 40,47, 48, 35, 27, 32, 29, 30].The structure of the rest of this paper is as follows. In Section 2,we prove our main smoothing result, Theorem 1.2. In Section 3, weprove our gap result for gradient Ricci soliton, Theorem 1.3. In Section4, we prove our gap result for complete noncompact Ricci nonnegativemanifolds, Corollary 1.1. Finally in Section 5, we prove our Gromov–Hausdorff compactness result, Theorem 1.4. Acknowledgements : The authors would like to thank Peter Toppingfor the interest in this work as well as Gilles Carron for pointing outthe reference [9] and related results. P.-Y. Chan would like to thankBennett Chow, Lei Ni and Jiaping Wang for continuous encouragementand support. E. Chen thanks Guofang Wei and Rugang Ye for theirsupport and was partially supported by an AMS–Simons Travel Grant.M.-C. Lee was partially supported by NSF grant DMS-1709894 andEPSRC grant number P/T019824/1.
Pak-Yeung Chan, Eric Chen and Man-Chun Lee Curvature estimates of Ricci flows
In this section, we will prove the semi-local estimates of the Ricciflow. We begin by fixing some notation below.Suppose ( M n , g ) is an n dimensional complete (not necessarily com-pact) Riemannian manifold and Ω is a connected domain on M withsmooth boundary (boundaryless if M = Ω). Hereinafter, we shall re-serve the positive integer n for the dimension of M . Wang [53] localizedPerelman’s entropy and proved an almost monotonicity in local entropywhen Ω is bounded, generalizing the result in [42]. Using his notation,we have: D g (Ω) := (cid:8) u : u ∈ W , (Ω) , u ≥ k u k = 1 (cid:9) , (2.1) W (Ω , g, u, τ ) := ˆ Ω τ ( Ru + 4 |∇ u | ) − u log udµ (2.2) − n πτ ) − n,ν (Ω , g, τ ) := inf u ∈ D g (Ω) ,s ∈ (0 ,τ ] W (Ω , g, u, s ) , (2.3) ν (Ω , g ) := inf τ ∈ (0 , ∞ ) ν (Ω , g, τ ) . (2.4)In order to prove the curvature estimate of Theorem 1.2, we firstshow that it can be reduced to the preservation of local L n/ control ofRm( g ( t )). Proposition 2.1.
For all
A > , there is c ( n, A ) > such that thefollowing holds. Suppose ( M, g ( t )) , t ∈ [0 , T ] is a complete Ricci flowwith bounded curvature such that for all ( x, t ) ∈ M × [0 , T ] , the followingholds: (a) ν ( B g ( x, n √ T ) , g , T ) ≥ − A ; (b) (cid:16) ´ B g ( t ) ( x, √ t ) | Rm( g ( t )) | n/ dµ g ( t ) (cid:17) n/ ≤ c ε for ε < ,then we have (2.5) sup M | Rm( x, t ) | < εt − for all t ∈ (0 , T ] .Proof. By rescaling, we may assume T = 1. Suppose on the contrarythat the result is not true. Then for some A, ε >
0, we can findsequences of δ i = c i ε i with c i → ε i ∈ (0 ,
1) and { ( M i , g i ( t ) , p i ) } with bounded curvature such that(1) ν ( B g i (0) ( x, n ) , g i (0) , ≥ − A ; (2) (cid:16) ´ B t ( x, √ t ) | Rm( g i ( t )) | n/ dµ i,t (cid:17) /n ≤ δ i → x, t ) ∈ M i × [0 , x i , t i ) ∈ M i × (0 , | Rm i ( x i , t i ) | = ε i t − i . We may choose t i > y, s ) ∈ M i × (0 , t i ),(2.6) | Rm i ( y, s ) | < ε i s − . This can be done since the upper bound of curvature vary continuouslyby boundedness of curvature. Let Q i = t − i ≥
1. Consider the rescaledRicci flow ˜ g i ( t ) = Q i g i ( Q − i t ) for t ∈ [0 ,
1] which satisfies(a) ν ( B g i (0) ( y, n ) , ˜ g i (0) , ≥ − A for all y ∈ M i ;(b) (cid:16) ´ B ˜ gi ( t ) ( y, √ t ) | Rm(˜ g i ( t )) | n/ d ˜ µ i,t (cid:17) /n ≤ δ i → y, t ) ∈ M i × [0 , | Rm ˜ g i ( y, s ) | < s − on M i × (0 , | Rm ˜ g i ( x i , | = ε i .By (a) and [53, Theorem 5.4], we have uniform lower bound of theentropy ν ( B ˜ g i ( t ) ( y, , ˜ g i ( t ) , B ˜ g i ( t ) ( x i , r ) which dependsonly on A and n for any r , t ∈ [1 / , B ˜ g i ( t ) ( x i , t ∈ [1 / , (cid:18) ∂∂t − ∆ (cid:19) | Rm | ≤ | Rm | . Since the curvature is uniformly bounded on [ , ε i = | Rm ˜ g i ( x i , |≤ C ( n, A ) ˆ / B ˜ gi ( t ) ( x i , / | Rm ˜ g i | dµ s ds ≤ C ′ ( n, A ) ˆ / B ˜ gi ( t ) ( x i , / | Rm ˜ g i | n/ dµ s ds ! /n ≤ C ′ ( n, A ) c i ε i → i → ∞ , (2.8)which is impossible if c i is too small. This completes the proof of thelemma. (cid:3) Pak-Yeung Chan, Eric Chen and Man-Chun Lee
Remark . The a-priori curvature boundedness and completeness isin fact unnecessary, see Section 3.Next, we will show that if the initial local L n/ is small enough, thenit is preserved in some semi-local sense. We first begin with the energyevolution of L n/ norm. Lemma 2.1.
Suppose n ≥ and ( M, g ( t )) is a complete solution to theRicci flow, t ∈ [0 , T ] . Then for any α ≥ n , β > and φ ( x, t ) compactlysupported function in spacetime, there exist positive constants C ( α ) and C ′ ( n, α ) such that ddt ˆ M φ ( | Rm | + β ) α dµ t ≤ − C ( α ) ˆ M |∇ ( φ ( | Rm | + β ) α/ ) | dµ t + C ′ ( n, α ) ˆ M φ ( | Rm | + β ) α +1 / dµ t + C ′ ( n, α ) ˆ M |∇ φ | ( | Rm | + β ) α dµ t . + ˆ M φ ✷ φ ( | Rm | + β ) α dµ t , (2.9) where ✷ = ∂∂t − ∆ g ( t ) .Proof. We compute the time derivative of the integral norm as in [16].Using ∂∂t dµ t = − Rdµ t ≤ c ( n ) | Rm | dµ t , we have ddt ˆ M φ ( | Rm | + β ) α dµ t ≤ ˆ M ∂∂t (cid:0) φ ( | Rm | + β ) α (cid:1) dµ t + c ( n ) ˆ M φ ( | Rm | + β ) α +1 / dµ t . (2.10) For the first term on the R.H.S., ˆ M ∂∂t (cid:0) φ ( | Rm | + β ) α (cid:1) dµ t = ˆ M ✷ (cid:0) φ ( | Rm | + β ) α (cid:1) dµ t = ˆ M φ ✷ φ ( | Rm | + β ) α dµ t − ˆ M |∇ φ | ( | Rm | + β ) α dµ t + ˆ M αφ ( | Rm | + β ) α − ✷ | Rm | dµ t − ˆ M α ( α − φ ( | Rm | + β ) α − | Rm | |∇| Rm || dµ t − ˆ M αφ h∇ φ, ∇| Rm |i| Rm | ( | Rm | + β ) α − dµ t , (2.11)where ✷ = ∂∂t − ∆ g ( t ) . To proceed, we apply the evolution equation of | Rm | (see [16] and ref. therein)(2.12) ✷ | Rm | ≤ − |∇ Rm | + 16 | Rm | . It follows from (2.12) that ˆ M αφ ( | Rm | + β ) α − ✷ | Rm | dµ t ≤ − α ˆ M φ ( | Rm | + β ) α − |∇ Rm | dµ t + 16 α ˆ M φ ( | Rm | + β ) α +1 / dµ t . (2.13) Hence by Kato’s inequality and H¨older inequality, ˆ M αφ ( | Rm | + β ) α − ✷ | Rm | dµ t − ˆ M α ( α − φ ( | Rm | + β ) α − | Rm | |∇| Rm || dµ t − ˆ M αφ h∇ φ, ∇| Rm |i| Rm | ( | Rm | + β ) α − dµ t ≤ − C ( α ) ˆ M φ ( | Rm | + β ) α − |∇| Rm || dµ t +16 α ˆ M φ ( | Rm | + β ) α +1 / dµ t + ˆ M αφ |∇ φ ||∇| Rm ||| Rm | ( | Rm | + β ) α − dµ t ≤ − C ′ ( α ) ˆ M φ ( | Rm | + β ) α − | Rm | |∇| Rm || dµ t +16 α ˆ M φ ( | Rm | + β ) α +1 / dµ t + C ′′ ( α ) ˆ M |∇ φ | ( | Rm | + β ) α dµ t . We also have by Cauchy Schwarz inequality C ′ ( α ) ˆ M φ ( | Rm | + β ) α − | Rm | |∇| Rm || dµ t = C ′′′ ( α ) ˆ M φ |∇ ( | Rm | + β ) α | dµ t ≥ C ′′′ ( α )2 ˆ M |∇ ( φ ( | Rm | + β ) α ) | dµ t − C ′′′ ( α ) ˆ M |∇ φ | ( | Rm | + β ) α dµ t . (2.14)All in all, ddt ˆ M φ ( | Rm | + β ) α dµ t ≤ C ( n, α ) ˆ M φ ( | Rm | + β ) α +1 / dµ t + ˆ M φ ✷ φ ( | Rm | + β ) α dµ t + C ( α ) ˆ M |∇ φ | ( | Rm | + β ) α dµ t − C ′′′ ( α )2 ˆ M |∇ ( φ ( | Rm | + β ) α ) | dµ t . (2.15) Our desired inequality (2.9) then follows. (cid:3)
We also need the following lemma showing that the local entropyimplies local Sobolev inequality.
Lemma 2.2.
For all A ≥ n, λ > and δ > , there are positiveconstants C S ( n, A, λ, δ ) and ˆ T ( n, A, λ, δ ) such that the following holds.Suppose ( M, g ( t )) is a complete Ricci flow with bounded curvature on [0 , T ] and for all p ∈ M and all t ∈ (0 , T ] , all of the following conditionsare satisfied (1) R g (0) ≥ − nλ ; (2) ν ( B ( p, , g (0) , ≥ − A ; (3) Ric( p, t ) ≤ δt − .Then we have for any p ∈ M and t ∈ (0 , min { T, ˆ T } ] , (2.16) ν ( B t ( p, , g ( t ) , − ) ≥ − A and for any ϕ ∈ C ∞ ( B t ( p, (cid:18) ˆ B t ( p, | ϕ | nn − dµ t (cid:19) n − n ≤ C S (cid:18) ˆ B t ( p, |∇ ϕ | + ( R + c n λ + 1) ϕ dµ t (cid:19) . Proof.
For (2.16), we apply [53, Theorem 5.4] to get for all small t ≤ min { ˆ T ( n, A, δ ) , T } ν ( B t ( p, , g ( t ) , − ) ≥ ν ( B ( p, , g (0) , t + 32 − ) − t ≥ ν ( B ( p, , g (0) , − A ≥ − A (2.18)This completes the proof of (2.16). Since the Ricci flow has boundedcurvature, it follows from the maximum principle that there exists adimensional constant c n such that for all t ≤ min { ˆ T ( n, A, δ ) , T } (2.19) R ( x, t ) ≥ − c n λ. By the definition of local ν entropy (2.2) and (2.16), we have a uniformLog Sobolev inequality: for any τ ∈ (0 , − ), u ∈ W , ( B t ( p, k u k g ( t ) , = 1, ˆ B t ( p, u log u dµ t ≤ τ ˆ B t ( p, |∇ u | + Ru dµ t − n πτ ) − n + 2 A. (2.20)The uniform Log Sobolev inequality then implies a uniform Sobolevinequality along the Ricci flow as first described in [60] (see also [16, 58]and Theorems 2.1 and 2.2). Indeed, when ∂B t ( p,
2) is nonempty, the same arguments as in [18, 58] will give us Theorems 2.1 and 2.2 belowfor the Dirichlet Sobolev inequality, and these together with (2.20)imply (2.17), finishing the proof. (cid:3)
We shall now state Theorems 2.1 and 2.2 without mentioning theproofs, since they are essentially the same as those found in [18, 58].Let (
N, h ) be a smooth compact Riemannian manifold with metric h and smooth boundary ∂N , H the elliptic operator = − ∆+4 − ( R + c n λ ),where λ and c n are non-negative constants such that R ≥ − c n λ on N ,(2.21) Q ( u, v ) = ˆ N ∇ u · ∇ v + 4 − ( R + c n λ ) u · vdµ h and write Q ( u, u ) as Q ( u ). For t >
0, consider the semigroup e − tH ofthe operator H . For any u ∈ L ( N, h ), the function u ( t ) := e − tH u isthe solution to the Dirichlet evolution equation(2.22) ∂u∂t = − Huu (0) = u u = 0 on ∂N. Theorem 2.1 ([18, 58]) . Let σ ∗ ∈ (0 , ∞ ] . Suppose that for all σ ∈ (0 , σ ∗ ) , (2.23) ˆ N u log u dµ ≤ σ ˆ N |∇ u | + 4 − ( R + c n λ ) u dµ + β ( σ ) is true for any u ∈ W , ( N ) with k u k = 1 , where β is a continuousnonincreasing function and R + c n λ ≥ . If in addition the function, (2.24) τ ( t ) := 12 t ˆ t β ( s ) ds is finite for any t ∈ (0 , σ ∗ ) . Then for each u ∈ L ( N )(2.25) k e − tH u k ∞ ≤ e τ ( t ) k u k for t ∈ (0 , σ ∗ / . Moreover, for all u ∈ L ( N )(2.26) k e − tH u k ∞ ≤ e τ ( t ) k u k for t ∈ (0 , σ ∗ / . Theorem 2.2 ([18, 58]) . Suppose there exist positive constants c and t such that for all t ∈ (0 , t ) and u ∈ L ( N )(2.27) k e − tH u k ∞ ≤ c t − n k u k . Set H = H + 1 . Then for some constant C ( c , t , n ) , (2.28) k H − / u k nn − ≤ C k u k , for any u ∈ L ( N ) . In particular, k u k nn − ≤ C k H / u k ≤ C ( Q ( u ) + k u k ) . (2.29) for all u ∈ W , ( N ) , where H / and H − / denote the fractional op-erator of H and its inverse respectively (see [18, 58] ). With Proposition 2.1 and Lemmas 2.1 and 2.2 in hand, we can nowprove Theorem 1.2 to conclude this section.
Proof of Theorem 1.2.
Let Λ be a constant to be chosen later. Since g ( t ) has bounded curvature, we may let ˆ T to be maximal time suchthat for all ( x, t ) ∈ M × [0 , ˆ T ∧ T ), we have ˆ B g ( t ) ( x, | Rm( g ( t )) | n/ dµ t ! /n ≤ Λ ε. (2.30)By [53, Theorem 3.3], Proposition 2.1 and the injectivity radius esti-mates in [11], it suffices to show that ˆ T is bounded from below uniformlyif ε and Λ are chosen appropriately.We may choose ε small enough so that Λ ε = δ < g ( t ) satisfies(1) (cid:16) ´ B g ( x, | Rm( g (0)) | n/ dµ (cid:17) /n ≤ ε ;(2) (cid:16) ´ B g ( t ) ( x, | Rm( g ( t )) | n/ dµ t (cid:17) /n ≤ δ ;(3) ν ( B g ( x, , g , ≥ − A ;(4) Ric( g ) ≥ − λ for all ( x, t ) ∈ M × [0 , T ∧ ˆ T ). By Proposition 2.1, we may assume(2.31) sup M | Rm( g ( t )) | ≤ c ( n, A ) δt − < t − . Otherwise ˆ T must be bounded from below, then we are done. Now weare ready to estimate ˆ T .For any x ∈ M , we let η ( x, t ) = d t ( x, x )+ c n √ t and define φ ( x, t ) = e − t ϕ ( η ( x, t )) where ϕ ( s ) is a cutoff function on R so that ϕ ≡ −∞ , ], ϕ ≡ −∞ ,
1] and satisfies ϕ ′′ ≥ − ϕ , 0 ≥ ϕ ′ ≥− √ ϕ . By choosing c n large enough, we have from [42, Lemma 8.3]that (cid:18) ∂∂t − ∆ (cid:19) φ ≤ in the sense of barrier and hence in the sense of distribution, see [38,Appendix A].Using Lemma 2.1 with the above choice of φ and α = n/
4, we con-clude that ddt ˆ M φ ( | Rm | + β ) n/ dµ t ≤ − C − n ˆ M |∇ ( φ ( | Rm | + β ) n/ ) | dµ t + C n ˆ M φ ( | Rm | + β ) n/ / dµ t + C n ˆ supp ( φ ) ( | Rm | + β ) n/ dµ t . (2.33)Noted that φ is supported on B t ( x , C − n ˆ M |∇ ( φ ( | Rm | + β ) n/ ) | dµ t ≥ C ( n, A ) (cid:18) ˆ M | ( φ ( | Rm | + β ) n/ ) | nn − dµ t (cid:19) n − n − C ( n, A ) ˆ M φ ( R + c n λ + 1)( | Rm | + β ) n/ dµ t (2.34)while the second term can be estimated by C n ˆ M φ ( | Rm | + β ) n/ / dµ t ≤ C n (cid:18) ˆ supp ( φ ) ( | Rm | + β ) n dµ t (cid:19) n (cid:18) ˆ M (cid:2) φ ( | Rm | + β ) n (cid:3) nn − dµ t (cid:19) n − n ≤ C n δ (cid:18) ˆ M (cid:2) φ ( | Rm | + β ) n (cid:3) nn − dµ t (cid:19) n − n (2.35)as β →
0. We can apply the same argument to ´ M φ R ( | Rm | + β ) n/ dµ t to deduce the same upper bound. Therefore, we conclude that if δ ≤ σ ( n, A ) <<
1, then as β → ddt ˆ M φ ( | Rm | + β ) n/ dµ t ≤ C ( n, A, λ ) ˆ B t ( x , ( | Rm | + β ) n/ dµ t ≤ C ( n, A, λ ) δ n (2.36) By letting β → x , t ) ∈ M × [0 , T ∧ ˆ T ), ˆ B t ( x , ) | Rm | n/ dµ t ≤ e
10 ˜ T ( C ( n, A, λ )Λ n t + 1) ε n . (2.37)Now we claim that for all ( y, t ) ∈ M × [0 , T ∧ ˆ T ], if ˆ T ≤ ˜ T ( n, A ),then we have B t ( y, ⊂ N [ i =1 B t ( x i ,
14 )(2.38)for some N ( n, A, λ ) ∈ N . If the claim is true, then we conclude thatfor all ( y, t ) ∈ M × [0 , T ∧ ˆ T ∧ ˜ T ( n, A )], ˆ B t ( y, | Rm | n/ dµ t ≤ N X i =1 ˆ B t ( x i , ) | Rm | n/ dµ t ≤ N e
10 ˜ T ( C Λ n ˜ T + 1) ε n . (2.39)Therefore if we choose Λ n/ = 4 N and further require ˜ T ≤ (4 n N n C ) − ,then we have contradiction and hence ˆ T ≥ ˜ T ( n, A, λ ). This will com-plete the proof. Hence, it remains to establish the uniform covering.For each ( y, t ) × M × [0 , T ∧ ˆ T ∧ ˜ T ), we let { x i } Ni =1 be a maximal set ofpoints in B t ( y,
1) such that B t ( x i , ) are disjoint from each other and(2.38) holds. By (2.31) and distance distortion estimates [42, Lemma8.3], we have B t ( y, ⊂ B ( y,
2) if ˜ T is small. At the same time, bychoosing δ sufficiently small, we may apply the proof of [28, Lemma2.4] (see also [33, Lemma 2.2]) to show that B t ( x i , ) ⊃ B ( x i , r ) forsome uniformly small r . Therefore, N X i =1 Vol g ( B ( x i , r )) ≤ N X i =1 Vol g (cid:18) B t ( x i ,
18 ) (cid:19) ≤ Vol g ( B t ( y, ≤ Vol g ( B ( y, . (2.40)Since x i ∈ B t ( y, ⊂ B ( y, N then follows fromRic( g ) lower bound and volume comparison. The desired result followsby re-labelling the constants. (cid:3) Gap Theorem of Ricci solitons
In this section we will prove Theorem 1.3, a gap theorem for shrinkingand steady gradient Ricci solitons. We do not assume a-priori boundson the curvature. The novel idea is to obtain local curvature control under the small L n/ curvature and local entropy bound (see also [21]).We first prove the following result, from which Theorem 1.3 shall follow. Theorem 3.1.
For all A ≥ n , there is ε ( n, A ) , C ( n, A ) , ˆ T ( n, A ) > such that the following holds. Suppose ( M, g ( t )) is a Ricci flow on [0 , T ] and p ∈ M be a point such that for all t ∈ (0 , T ] , (1) B t ( p, ⋐ M ; (2) (cid:16) ´ B t ( p, A √ t ) | Rm | n/ dµ t (cid:17) /n ≤ ε for some ε < ε ; (3) ν ( B t ( p, A √ t ) , g ( t ) , t ) ≥ − A ;then we have (3.1) (cid:26) | Rm | ( x, t ) ≤ C ( n, A ) ε t − inj( x, t ) ≥ C ( n, A ) − √ t for all x ∈ B t ( p, A √ t ) , t ≤ T ∧ ˆ T .Proof. We will split the proof into three parts.
Step 1. Rough estimates under stronger assumption.
Wefirst prove the rough curvature estimate: | Rm( x, t ) | ≤ C ( n, A ) t − on B t ( p, A √ t ) under an extra assumption: ⋆ Ric( x, t ) ≤ t − , on B t ( p, √ t ) , t ∈ (0 , T ] . The injectivity radius estimates will follow from the work of [11] by[53, Theorem 3.3]. For x ∈ B t ( p, A √ t ) , t < T , we define(3.2) r ( x, t ) = sup ( < r < A √ t − d t ( x, p ) : sup P ( x,t,r ) | Rm | ≤ r − ) where P ( x, t, r ) = { ( y, s ) : y ∈ B s ( x, r ) , s ∈ [ t − r , t ] ∩ (0 , T ] } . We claimthat if ε is sufficiently small, then we can find c ( n, A ) , T ( n, A ) > x ∈ B t ( p, A √ t ) and t < T ∧ T ,(3.3) F ( x, t ) = r ( x, t ) A √ t − d t ( x, p ) ≥ c . The rough curvature estimate then follows immediately from the claimsince for any x ∈ B t ( p, A √ t ) and t < T ∧ T , | Rm | ( x, t ) ≤ r ( x, t ) ≤ c ( A √ t − d t ( p, x )) ≤ c A t . (3.4) Suppose on the contrary that the claim is not true for some A and n , we can find a sequence of Ricci flow { ( M i , g i ( t ) , p i ) } ∞ i =1 , t i → • Ric i ( x, t ) < t − for all x ∈ B t ( p i , √ t ), t < t i ; • ˆ B gi ( t ) ( p i , A √ t ) | Rm i | n/ dµ i,t ≤ ε for all t < t i ; • ν ( B g i ( t ) ( p, A √ t ) , g i ( t ) , t ) ≥ − A for t < t i ,but for some sequence x i ∈ B t ( p i , A √ t ), we have(3.5) F i ( x i , t i ) = min { F i ( y, s ) : s ∈ (0 , t i ) , y ∈ B s ( p i , A √ s ) } → , where ε is some small positive number to be chosen. Re-scale the flowby ˜ g i ( t ) = Q i g i ( t i + Q − i t ), − Q i t i ≤ t ≤ Q − / i = r i ( x i , t i ) sothat ˜ r i ( x i ,
0) = 1. Then by (3.5) d ˜ g i (0) ( x i , ∂B g i ( t i ) ( p i , A √ t i )) = d g i ( t i ) ( x i , ∂B g i ( t i ) ( p i , A √ t i )) r ( x i , t i ) ≥ A √ t i − d t i ( x i , p i ) r ( x i , t i )= F i ( x i , t i ) − → + ∞ . (3.6)That is to say the pointed Cheeger-Gromov limit of the flow centredat x i is complete provided it exists. Furthermore, we may invoke (3.5)again to see that Q i t i = t i r i ( x i , t i ) > A (cid:18) A √ t i − d t i ( x i , p i ) r i ( x i , t i ) (cid:19) → + ∞ . (3.7)Hence, the limiting solution is ancient if it exists. Next, we would liketo show that after passing to a sub-sequence, the flows converge inCheeger-Gromov sense. The two key ingredients are uniform curvaturebound in i on compact sets in spacetime and the injecivity radius lowerbound at x i w.r.t ˜ g i (0).Let r > y, s ) ∈ ˜ P i ( x i , , r ). Since we have Ric i < t − on B t ( p i , √ t ) and r i ( x i , t i ) = Q − / i << √ t i , by Hamilton-Perelman’s dis-tance estimates ([24, 42]) we have for all large i > N ( n, A, r ),(3.8) d Q − i s + t i ( x i , p i ) ≤ d t i ( x i , p i ) + C n rQ − / i . Hence d Q − i s + t i ( y, p i ) ≤ Q − i r + d Q − i s + t i ( x i , p i ) ≤ C n Q − i r + d t i ( x i , p i )(3.9)It follows from (3.5) and (3.7) that for all large i > N ( n, A, r ), A p Q i t i + s − Q i d t i ( x i , p i ) ≥ A p Q i t i − r − Q i d t i ( x i , p i ) ≥ A p Q i t i − Q i d t i ( x i , p i ) − cAr √ Q i t i = F ( x i , t i ) − − cAr √ Q i t i > C n r. (3.10)Hence by (3.9), y ∈ B Q − i s + t i ( p i , A p Q − i s + t i ). We have by (3.5), (3.9)and (3.10) that˜ r i ( y, s ) = r i ( y, Q − i s + t i ) r i ( x i , t i )= F i ( y, Q − i s + t i ) F i ( x i , t i ) · A p Q − i s + t i − d Q − i s + t i ( y, p i ) A √ t i − d t i ( x i , p i ) ≥ A p Q − i s + t i − d Q − i s + t i ( y, p i ) A √ t i − d t i ( x i , p i ) ≥ A p Q − i s + t i − d Q − i s + t i ( x i , p i ) − Q − / i rA √ t i − d t i ( x i , p i ) ≥ F ( x i , t i ) − − cAr √ Q i t i − C n rF ( x i , t i ) − . (3.11)Thus for all i > N ( n, A, r ),(3.12) ˜ r i ( y, s ) > . This gives the curvature estimates on any compact subset in space-time. By our assumptions, for any r > i > N ( n, A, r ) , theentropy satisfies ν ( e B t ( x i , r ) , ˜ g i ( t ) , Q i t i + t ) ≥ ν ( e B t ( p i , A p Q i t i + t ) , ˜ g i ( t ) , Q i t i + t )= ν ( B t ( p i , A q t i + Q − i t ) , g i ( t i + Q − i t ) , t i + Q − i t ) ≥ − A. (3.13)By virtue of (3.7), (3.12), (3.13) and [53, Theorem 3.3], the vol-ume ratios e V ( e B ( x i ,r )) r n are uniformly bounded from below in i for anyall r ∈ (0 , / x i w.r.t. ˜ g i (0)have a uniform positive lower bound in i . Hence by Hamilton’s com-pactness theorem (see [23], [17]), we can pass ˜ g i ( t ) to a completelimiting Ricci flow ( M ∞ , ˜ g ∞ ( t ) , x ∞ ) which is an ancient complete so-lution with bounded curvature. By the choice of Q − / i and (3.9),we have B t i ( x i , Q − / i ) ⋐ B t i ( p i , A √ t i ). Therefore, for all s <
0, if d Q − i s + t i ( x i , y ) < r , then by (3.8) d Q − i s + t i ( y, p i ) ≤ d Q − i s + t i ( y, x i ) + d Q − i s + t i ( x i , p i ) ≤ r + d t i ( x i , p i ) + C n A Q − / i ≤ r + 2 A √ t i . (3.14)Therefore, if r = √ t i , then we have B Q − i s + t i ( x i , √ t i ) ⋐ B Q − i s + t i ( p i , A q Q − i s + t i )for all i → + ∞ . This together with the assumption implies(3.15) ˆ M ∞ | g Rm | n/ d ˜ µ s ≤ ε for all s ≤
0. Moreover, by the monotonicity of local entropy overdomain, (3.13) and the proof of Lemma 6.28 in [17], we have ν ( M ∞ , ˜ g ∞ ( t )) ≥ − A (3.16)for all t ≤
0. Recall that we have ˜ r i ( x i ,
0) = 1. By applying Propo-sition 2.1 with translation and re-scaling, we see that ˜ g ∞ ( τ ) must beflat for all τ ≤
0. As the entropy is bounded from below for all scales,the manifolds must be of maximum volume growth which implies that˜ g ∞ ( t ) is the static flat Euclidean metric. This contradicts with thecurvature radius at ( x ∞ ,
0) and completes the proof under the assump-tion ⋆ . Now the injectivity radius estimates follows from the curvatureestimate and the work of [11]. Step 2. Removing assumption ⋆ in Step 1. Since B t ( p, ⋐ M for t ≤ T , by smoothness of solution we may find ˜ T ≤ T suchthat | Ric | < t − for x ∈ B t ( p, √ t ) , t ∈ (0 , ˜ T ). W.L.O.G., we mayassume that ˜ T to be small uniformly, otherwise the required estimateon | Rm | follows by Step 1. Hence the result under ⋆ gives the curvatureestimates over a smaller ball, i.e. for some ˆ T ( n, A ),(3.17) | Rm | ( x, t ) ≤ C ( n, A ) t − for all x ∈ B t ( p, A √ t ), t ≤ min { ˜ T , ˆ T ( n, A ) } .We claim that ˜ T ≥ T ∧ ˆ T ( n, A ). Suppose that is not the case,denote s = ˜ T , then by the maximality of ˜ T there is ¯ x ∈ B s ( p, √ s )such that | Ric | (¯ x, s ) = s − . By considering the flow s − g ( st ) , t ∈ [0 , s = 1. By the estimates of inj( x, t ), (3.17),Theorem 3.3 in [53], V s ( A √ s ) is uniformly bounded from below forany s ∈ [1 / , B s (¯ x, A √ s ) for any s ∈ [1 / , B s (¯ x, A √ s ) and theH¨older inequality would imply1 = | Ric | (¯ x, ≤ c ( n ) | Rm | (¯ x, ≤ C ′ ( n, A ) ˆ / B s (¯ x, A √ s ) | Rm | n/ dµ s ds ! /n ≤ C ′′ ( n, A ) ε . (3.18)which is impossible if ε ≤ ε ( n, A ) is sufficiently small. Hence ˜ T ≥ T ∧ ˆ T ( n, A ). This implies the curvature estimate for | Rm | on B t ( p, A √ t )by Step 1. Step 3. Improved curvature estimates.
At this point we havealready obtained a rough curvature estimate on B t ( p, A √ t ) , t ∈ [0 , T ∧ ˆ T ]. For each s ∈ [0 , T ∧ ˆ T ], we may consider ˜ g ( t ) = s − g ( st ) , t ∈ [0 , ,
1] and entropy lower bound, withthe scaling invariant L n/ assumption we can apply iteration [36] againto show that | Rm(˜ g ( x, |≤ C ( n, A ) / B ˜ g ( s ) ( x, A √ s ) | Rm(˜ g ( t )) | n/ dµ s ds ! /n ≤ C ( n, A ) ε . (3.19) This gives an improved coefficient on curvature decay by rescaling itback to g ( t ). (cid:3) We now show how our gap theorem for complete shrinking and steadygradient Ricci solitons with small || Rm || L n/ , Theorem 1.3, follows fromTheorem 3.1. Recall that a complete Riemannian manifold ( M, g ) issaid to be a shrinking (steady) gradient Ricci soliton if there exists asmooth function f such that(3.20) Ric + ∇ f = λg, where the constant λ = (= 0 resp.). Proof of Theorem 1.3.
Let λ = 1 / φ t of the vector field ∇ f − λt with φ being theidentity map. it is known that g ( t ) := (1 − λt ) φ ∗ t g is an ancient solutionto the Ricci flow on M with g (0) = g and t ∈ ( −∞ , λ ) (= R if λ = 0,see [17, 62]). By the reparametrization and the scaling invariance ofConditions 1 and 2 in Theorem 1.3, we have for all t ∈ ( −∞ , λ ) :(1) ν ( M, g ( t )) ≥ − A ;(2) ´ M | Rm | n/ g ( t ) dµ g ( t ) ≤ ε .We are going to show something slightly more general, namely if ( M, g ( t ))is a complete ancient solution to the Ricci flow on ( −∞ ,
0] such that g ( t ) satisfies the above two conditions for each t ∈ ( −∞ , M, g ( t )) is isometric to R n . For any Q > τ ≤
0, we con-sider the rescaled solution h ( t ) := ( Q ˆ T ) − g ( Q ˆ T t − Q + τ ), where t ∈ [0 , ˆ T ]and ˆ T is the constant as in Theorem 3.1. It is not difficult to see that h ( t ) also satisfies the two conditions in Theorem 1.3. Hence we mayapply Theorem 3.1 for all sufficiently small ε to get for any x ∈ MQ | Rm | g ( x, τ ) = ˆ T | Rm | h ( x, ˆ T ) ≤ C ( n, A ) ε. By letting Q → ∞ , we have g ( τ ) is flat. The entropy lower bound atall scales then implies the maximal volume growth of g ( τ ) and thus itis isometric to R n . (cid:3) Gap theorem with small || Rm || L n/ In this section, we will use Ricci flow to discuss Riemannian mani-folds with Ric ≥ || Rm || L n/ which are non-collapsedin term of entropy. We first show that under the assumption of Corol-lary 1.1, we have a long-time solution of the Ricci flow and g hasmaximal volume growth. Theorem 4.1.
For any
A > , there is σ ( n, A ) , C ( n, A ) > suchthat the following holds. Suppose ( M, g ) is a complete non-compactRiemannian manifold with bounded curvature such that (1) Ric( g ) ≥ ; (2) ν ( M, g ) ≥ − A ; (3) (cid:0) ´ M | Rm( g ) | n/ dµ g (cid:1) /n ≤ ε for some ε < σ .Then there is a Ricci flow g ( t ) starting from g on M × [0 , ∞ ) suchthat for all t > , (4.1) (cid:26) sup M | Rm( g ( t )) | ≤ C εt − (cid:0) ´ M | Rm( g ( t )) | n/ dµ t (cid:1) /n ≤ C ε Moreover, g is of maximal volume growth.Remark . The assumption on the global entropy of all scale can alsobe implied by maximal volume growth.
Proof.
For
R >
0, we let g R, = R − g which still satisfies the assump-tions of the Theorem, which are scaling invariant. Therefore we can runShi’s Ricci flow g R ( t ) [46] for a short-time with initial metric g R, . ByTheorem 1.2, if σ is sufficiently small, g R ( t ) exists on M × [0 , T ( n, A )]and satisfies(4.2) ( | Rm( g R ( t )) | ≤ C εt − (cid:16) ´ B gR ( t ) ( x, | Rm( g R ( t )) | n/ dµ R,t (cid:17) /n ≤ C ε for all ( x, t ) ∈ M × [0 , T ]. By re-scaling it back and the uniqueness ofRicci flow [13], we obtain a Ricci flow g ( t ) on [0 , T R ) with | Rm | ≤ C εt − and g (0) = g . Moreover, we have for all R, t > (cid:18) ˆ B t ( x,R ) | Rm( g ( t )) | n/ dµ t (cid:19) /n ≤ C ε. The global integral estimate then follows by letting R → + ∞ .To see that g is of maximal volume growth, thanks to the improvedregularity on curvature and monotonicity of entropy ν , the re-scaledRicci flow g R ( t ) satisfies V ol g R (1) ( B g R (1) ( x, ≥ c. (4.4)Since the lower bound of scalar curvature is preserved along the Ricciflow, together with [47, Corollary 3.3], we have, if σ is sufficiently small, that c ≤ V ol g R (1) ( B g R (1) ( x, ≤ V ol g R (0) ( B g R (0) ( x, V ol g ( B g ( x, R )) R n . (4.5)Since R is arbitrarily large, this completes the proof. (cid:3) Before we prove the Corollary 1.1, we will show that the asymptoticvolume ratio can be improved to be almost Euclidean if we furthershrink the integral curvature and hence is almost Euclidean in thesense of local entropy [54, Lemma 4.10]. This is in spirit similar to thegap theorem proved by Cheeger [9, Theorem 4.32].
Theorem 4.2.
For all
A, λ, δ > , there are σ ( n, A, λ, δ ) , ˆ r ( n, A, λ, δ ) > such that if ( M, g ) is a complete Riemannian manifold of boundedcurvature so that for all p ∈ M , (1) Ric( g ) ≥ − λ ; (2) ν ( B g ( p, , g, ≥ − A ; (3) (cid:16) ´ B g ( p, | Rm( g ) | n/ dµ g (cid:17) /n < σ .Then for all p ∈ M , Vol g ( B g ( p, ˆ r )) ≥ (1 − δ ) ω n ˆ r n . (4.6) Proof.
Let g ( t ) , t ∈ [0 , ˆ T ] be the Ricci flow solution obtained from The-orem 1.2 and Shi’s Ricci flow [46]. We claim that for given δ , thereare constants T ( n, A, λ, δ ) and ˆ σ ( n, A, λ, δ ) such that if σ < ˆ σ , then for( x, t ) ∈ M × [0 , T ],(4.7) Vol g ( t ) (cid:16) B g ( t ) ( x, √ t (cid:17) ≥ (1 − δ ) ω n t n/ . Suppose on the contrary, we can find a sequence of g i ( t ) , t ∈ [0 , ˆ T ]such that g i (0) satisfies the same assumptions as in Theorem 1.2 with ε i → x i ∈ M i and 0 < √ t i → g i ( t i ) (cid:0) B g i ( t i ) (ˆ x i , √ t i ) (cid:1) < (1 − δ ) ω n t n/ i . Consider the rescaled Ricci flow ˜ g i ( t ) = t − i g ( t i t ) on M i × [0 , i and all ( x, t ) ∈ M i × (0 , | Rm(˜ g i ( t )) | ≤ C ( n, A, λ ) ε i t − and inj ˜ g i ( t ) ≥ c ( n, A, λ ) √ t which enable us to pass ( M i , ˜ g i ( t ) , ˆ x i ) to sub-sequential limit ( M ∞ , ˜ g ∞ ( t ) , ˆ x ∞ )in smooth Cheeger-Gromov sense by Hamilton’s compactness [23]. In particular, ˜ g ∞ ( t ) is flat for t ∈ (0 ,
1] since ε i →
0. On the other hand,since t i →
0, we may apply the local monotonicity of entropy in [53,Theorem 5.4] again to show that ν ( M ∞ , ˜ g ∞ ( t )) ≥ − A which implies˜ g ∞ ( t ) is of Euclidean volume growth by Theorem [53, Theorem 3.3]and hence ( M ∞ , ˜ g ∞ (1)) ≡ ( R n , g euc ) which contradicts (4.8).After relabelling the constants, (4.7), together with volume compar-ison implies that for all t ∈ [0 , T ] and r < √ t ,Vol g ( t ) (cid:0) B g ( t ) ( x, r ) (cid:1) ≥ (cid:18) − δ (cid:19) ω n r n . (4.9)By the scalar curvature lower bound of g ( t ) and [47, Corollary 3.3],Vol g ( B g ( x, r )) ≥ e − nλt · Vol g ( t ) ( B g ( x, r )) ≥ e − nλt · Vol g ( t ) (cid:16) B g ( t ) ( x, r − c n √ σt ) (cid:17) ≥ e − nλt (cid:18) − δ (cid:19) ω n (cid:16) r − c n √ σt (cid:17) n ≥ (1 − δ ) ω n r n (4.10)if we choose t, σ small enough and r = √ t . This completes the proof. (cid:3) Proof of Corollary 1.1.
Theorem 4.1 implies that g is of maximal vol-ume growth. By [10, Theorem A.1.11] (see also [54, Theorem 5.7]),it suffices to show that the asymptotic volume growth can be madearbitrarily close to the Euclidean one if we shrink σ . This follows fromTheorem 4.2 and the rescaling argument as in the proof of Theorem 4.1.Alternatively, we can also prove the homeomorphism by showing M = S ∞ i =1 U i where U i is diffeomorphic to a Euclidean ball and U i ⊂ U i +1 for all i using the expansion of injectivity radius, curvature esti-mate | Rm( x, t ) | ≤ εt − from Theorem 4.1. Then the homeomorphismwill follow from the main result of [5], see also [12, Section 3]. Noticethat Gompf’s result says that among the Euclidean spaces only R hasexotic differential structures. So for n >
4, the homeomorphisms canbe made to be diffeomorphisms (see [49]). (cid:3) Regularity of Gromov-Hausdorff limit
In this section, we discuss the compactness of Riemannian mani-folds satisfying small L n/ bound. We remark here that the Gromov-Hausdorff limit follows from Ricci lower bound directly. The key partis to construct the differentiable structure on the limit using the pseu-dolocality of Ricci flows. Proof of Theorem 1.4.
By Shi’s Ricci flow existence [46] and Theo-rem 1.2, by choosing ε small enough we can find a sequence of Ricciflow g i ( t ) on M i × [0 , T ( n, A )] such that(1) Ric( g i (0)) ≥ − λ ;(2) ν ( B g i ( t ) ( x, , g i ( t ) , ) ≥ − A ;(3) | Rm( g i ( t )) | ≤ Cε t for all ( x, t ) ∈ M i × (0 , T ]. By [53, Theorem 3.3] and [11], we can applyHamilton’s compactness to pass ( M i , g i ( t ) , p i ) to ( M ∞ , g ∞ ( t ) , p ∞ ) for t ∈ (0 , T ] in the smooth Cheeger-Gromov sense after passing to sub-sequence. More precisely, there is an exhaustion { Ω i } ∞ i =1 of M ∞ and asequence of diffeomorphism F i : Ω i → M i onto its image such that forany compact subset Ω × [ a, b ] ⋐ M ∞ × (0 , T ], we have F ∗ i g i ( t ) → g ∞ ( t )in C ∞ loc (Ω × [ a, b ]).We now construct the Gromov-Hausdorff limit of g i using F i in amore precise way so that its relation to M ∞ ’s topology is clearer. Thisessentially follows the proof of Gromov’s compactness theorem and thedistance distortion estimates. Since M ∞ is a smooth manifold, we let { x k } ∞ k =1 be a countable dense set with respect to g ∞ (1). Then for each k, l , we have x k , x l ∈ B g ∞ (1) ( p ∞ , R k,l ) and hence by distance distortionestimates [47, Corollary 3.3] using curvature estimates above, we have(5.1) d F ∗ i g i ( x k , x l ) ≤ d F ∗ i g i (1) ( x k , x l ) + C n ≤ C ( k, l )as i → + ∞ . Here we have used the fact that F ∗ i g i (1) converges lo-cally uniformly to g ∞ (1). Therefore, lim i → + ∞ d F ∗ i g i ( x k , x l ) exists afterwe pass it to some sub-sequence which we denote it as d ∞ ( x k , x l ). Re-peating the process for each k, l , we define d ∞ on the dense set. Forgeneral x, y ∈ M ∞ , we define d ∞ ( x, y ) using the density of { x k } . Thisis well defined since if there are two sequences x i , x ′ i → x ∈ M ∞ and y i , y ′ i → y ∈ M ∞ with respect to g ∞ (1), then for i sufficiently large, d ∞ ( x i , y i ) ≤ d ∞ ( x ′ i , y ′ i ) + d ∞ ( x i , x ′ i ) + d ∞ ( y i , y ′ i ) ≤ d ∞ ( x ′ i , y ′ i ) + C (cid:0) d g ∞ (1) ( x i , x ′ i ) (cid:1) / + C (cid:0) d g ∞ (1) ( y i , y ′ i ) (cid:1) / = d ∞ ( x ′ i , y ′ i ) + o (1) . (5.2)by using [28, Lemma 2.4] and [47, Corollary 3.3]. By passing i → + ∞ and switching the sequences, we have the uniqueness of the limit. Inother words, we have(5.3) lim i → + ∞ d F ∗ i g i ( x, y ) = d ∞ ( x, y )for all x, y ∈ M ∞ . Now we claim that d ∞ ( · , · ) is in fact a distance defined on M ∞ × M ∞ .To see this, let y, z ∈ M ∞ be such that d ∞ ( z, y ) = 0. If y = z , thenwe have d g ∞ (1) ( z, y ) > r for some r >
0. For any ε >
0, we can find y ′ , z ′ ∈ { x i } ∞ i =1 such that d g ∞ (1) ( y, y ′ ) + d g ∞ (1) ( z, z ′ ) + d ∞ ( y ′ , z ′ ) < ε andtherefore we can find N ∈ N such that for i > N , d F ∗ i g i ( y ′ , z ′ ) < ε .Applying [28, Lemma 2.4] again, we deduce(5.4) d F ∗ i g i (1) ( y ′ , z ′ ) ≤ C ( n, λ ) ε / . Here we note that although [28, Lemma 2.4] is stated globally, it iseasy to see that the proof holds locally and only require the curvaturebound in form of εt − for ε small enough and an initial Ricci lowerbound which is avaliable in our situation. Therefore, if ε is sufficientlysmall, it will violate the fact that d g ∞ (1) ( y, z ) > r . This shows that d ∞ defines a distance metric on M ∞ .To see that d ∞ generates the same topology as M ∞ , it suffices topoint out that [28, Lemma 2.4] together with a limiting argument im-plies that for d ∞ ( x, y ) <
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Department of Mathematics, University of Cali-fornia, San Diego, La Jolla, CA 92093
Email address : [email protected] (Eric Chen) South Hall, Rm. 6501, Department of Mathematics, Uni-versity of California Santa Barbara, CA 93106-3080
Email address : [email protected] (Man-Chun Lee) Department of Mathematics, Northwestern Univer-sity, 2033 Sheridan Road, Evanston, IL 60208
Current address : Mathematics Institute, Zeeman Building, University of War-wick, Coventry CV4 7AL
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