L 2 Curvature Bounds on Manifolds with Bounded Ricci Curvature
aa r X i v : . [ m a t h . DG ] M a y L CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE
WENSHUAI JIANG AND AARON NABERA
BSTRACT . Consider a Riemannian manifold with bounded Ricci curvature | Ric | ≤ n − B ( p )) > v >
0. The first main result of this paper is to prove that we have the L curvature bound > B ( p ) | Rm | < C ( n , v), which proves the L conjecture of [ChNa15]. In order to prove this,we will need to first show the following structural result for limits. Namely, if ( M nj , d j , p j ) −→ ( X , d , p ) is a GH -limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S ( X ) is n − H n − (cid:0) S ( X ) ∩ B (cid:1) < C ( n , v), which in particular proves the n − n − x ∈ S ( X ) that the tangent cone of X at x is unique and isometric to R n − × C ( S / Γ x ) for some Γ x ⊆ O (4)which acts freely away from the origin. C ONTENTS
1. Introduction 31.1. Main Results on Manifolds 51.2. Main Results on Limit Spaces 61.3. Outline of the proof Theorem 1.6 62. Background and Preliminaries 132.1. Quantitative Stratification 132.2. Volume Monotonicity and Cone Structures 142.3. Cone Splitting 152.4. Harmonic Radius and ǫ -Regularity Theorems 152.5. Heat Kernel Estimates 162.6. δ -Splitting Functions 172.7. Examples 173. ( δ, τ )-Neck Regions 183.1. Basic Properties of Neck Regions 19 Date : May 19, 2016. L -curvature bound on δ -Neck Regions 264.1. Pointwise Riemann curvature estimates 274.2. H -volume on manifold with Ric ≥ − δ ( n −
1) 284.3. H -volume and Local L curvature estimates 324.4. Proof of the L curvature estimate on neck region 345. Splitting Functions on Neck Regions 355.1. Estimates on Standard Green’s Functions on Neck Regions 365.2. Harmonic Function Estimates 415.3. Scale Invariant Hessian Estimates 425.4. Summable Hessian Estimates 455.5. Gradient Estimates 505.6. µ -Splitting Estimates and Proof of Theorem 5.2 526. Completing the Proof of Theorem 3.10 566.1. Proof of Lower Ahlfor’s Regularity bound of Theorem 3.10.1 566.2. Induction Scheme and Proof of Theorem 3.10 on a Manifold 586.3. Proof of Theorem 3.10 607. Neck Decomposition Theorem 617.1. Notation and Ball Decomposition Types 627.2. Cone Splitting and Codimension Four 637.3. Weak c -Ball Covering 657.4. Strong c -Ball Covering and Construction of Maximal Neck Regions 687.5. Refinement of Balls with Less than Maximal Symmetries 717.6. Inductive Covering 747.7. Proof of Neck Decomposition of Theorem 7.1 788. Proof of L -Curvature Estimate 799. Proof of Theorem 1.11 79 CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 3
10. Proof of Theorem 1.14 81Acknowlegments 81References 821. I
NTRODUCTION
The focus of this paper is to understand the regularity of Riemannian manifolds under the bounded Ricciand noncollapsing assumptions | Ric M n | ≤ n − , Vol( B ( p )) > v > . (1.1)A closely related problem, which will also play a central focus in this paper, is the study of Gromov-Hausdorff limit spaces ( M nj , g j , p j ) d GH −→ ( X , d , p ) , (1.2)where the M j satisfy (1.1).There is a good deal of history in studying the regularity of spaces satisfying (1.1) or (1.2). Much ofthe early work focused on closed 4-manifolds under the additional assumptions of bounded topology andbounded diameter. The key use of these assumptions is that by the Chern-Gauss-Bonnet formula one canconclude in the four dimensional case that χ ( M ) < A = ⇒ Z M | Rm | < C ( A , diam( M ) , v) . (1.3)Though a series of paper [A89],[BKN89],[T90] this was used to conclude that a limit space X of fourmanifolds with bounded Ricci, topology, diameter, and uniform lower volume bounds would be a Riemann-ian orbifold with at most isolated singularities. In particular, one gets from this that the singular set is ofcodimension four with bounded n − L p bounds on curvature.In particular, the codimension four conjecture was proved in any dimension under the additional assumptionof a L -bound on the curvature. Though the statement was similar in nature to the four dimensional result, WENSHUAI JIANG AND AARON NABER the techniques to exploit the L -bound in higher dimensions are substantially more involved. This was ex-tended in [Ch2] where under an assumed L curvature bound it was proved that the nonexceptional part ofthe singular set was rectifiable.In [ChNa15] the codimension four conjecture was resolved in full generality, the proof of which requireda variety of new estimates and techniques. In addition, it was shown in [ChNa15] that an improved resultheld in dimension four, where it could be proved under the assumptions of (1.1) the L -bound on curvaturewas in fact automatic, and did not require any apriori assumptions on topology and diameter. As a conse-quence, one could in fact show the topology was itself automatically bounded, which was a conjecture ofAnderson [A94]. In higher dimensions, only weaker curvature estimates were obtained in [ChNa15], whereit was shown that the curvature Rm had apriori L p -bounds for all p <
2. It was conjectured however in[ChNa15],[Na14] that the full L curvature bound should hold in any dimension.After the work of [ChNa15], there were still several open questions left over. First, although the singularset was shown to be of codimension four, the n − L -conjecture of [Na14],[ChNa15], by showing that under the assumptions of (1.1) there exists the L curvaturebound ? B ( p ) | Rm | < C ( n , v) . (1.4)Additionally, in the case of a limit space X of manifolds satisfying (1.1) we will show that not only is thesingular set S ( X ) of codimension four, but we have the finiteness estimate H n − ( S ( X ) ∩ B ) < C ( n , v) , (1.5)which proves the n − S ( X ) is n − M into two types of pieces: neck regions, which look like the singular CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 5 space R n − × C ( S / Γ ) on many scales, and ǫ -regularity regions which have scale invariant uniform curvaturebounds. The proof of the L estimate will then rely on both our ability to give effective estimates for thenumber of pieces in this decomposition, and our ability to do more refined analysis on the neck regionsthemselves. The challenge of doing analysis on the neck region is that there are an uncontrollable number ofscales involved in a neck region, and to get global information one needs estimates which are summable andsmall over all these scales. These estimates will depend heavily on a new type of superconvexity estimatewhich we will prove for the L hessian of a harmonic function on these neck regions. We refer the reader toSection 1.3, where we give a much more detailed outline of the proof.1.1. Main Results on Manifolds.
Let us now discuss in precision our main results concerning pointed Rie-mannian manifolds ( M n , g , p ) with bounded Ricci curvature | Ric | ≤ n − B ( p )) > v >
0. Our main regularity result in this context is the following:
Theorem 1.6 ( L -Estimate) . Let ( M n , g , p ) be a pointed Riemannian manifold such that | Ric M n | ≤ n − and Vol( B ( p )) > v > , then there exists C ( n , v) > such that ? B ( p ) | Rm | ≤ C . (1.7)Let us remark that the above is certainly sharp in that one cannot expect L p estimates on the curvaturefor any p > L estimate to hold without the noncollapsing assumption, see Example 2.23 andExample 2.24. An application of the above is to prove the following weak L estimate on the injectivityradius, which follows immediately from [ChNa13] once the L curvature bound has been established: Theorem 1.8 (Injectivity Radius Estimate) . Let ( M n , g , p ) be a pointed Riemannian manifold such that | Ric M n | ≤ n − and Vol( B ( p )) > v > , then there exists C ( n , v) > such that we have the weak L estimateon the injectivity radius given by Vol (cid:0) { x ∈ B ( p ) : inj x < r } (cid:1) < C ( n , v) r . (1.9) Remark . One could replace the injectivity radius inj by the harmonic radius r h in the above theorem ifone wishes.Let us end by remarking on one more result, which tells us that if we additionally have a bound on thegradient of the Ricci curvature, in particular an Einstein manifold, then we also have a sharp apriori L p estimate on the gradient of the curvature. Precisely: Theorem 1.11 (Gradient L / -Estimate) . Let ( M n , g , p ) satisfy | Ric M n | ≤ n − , |∇ Ric | ≤
A and the noncol-lapsing assumption
Vol( B ( p )) > v > , then there exists C ( n , v , A ) > such that ? B ( p ) |∇ Rm | / ≤ C . (1.12) WENSHUAI JIANG AND AARON NABER
Main Results on Limit Spaces.
We now turn our attention to our main results on pointed Gromov-Hausdorff limits ( M nj , g j , p j ) d GH −→ ( X , d , p ) (1.13)of sequences of manifolds satisfying the Ricci curvature bounds and noncollapsing of (1.1). Our main resultin this direction is the following, which is concerned the structure of the singular sets of such limits: Theorem 1.14.
Let ( M nj , g j , p j ) d GH −→ ( X , d , p ) be a Gromov-Hausdorff limit of manifolds with | Ric M nj | ≤ n − and Vol( B ( p j )) > v > . Then the following hold(1) The singular set S ( X ) is n − rectifiable.(2) In particular, we have the hausdorff measure estimate H n − (cid:0) S ( X ) ∩ B (cid:1) < C ( n , v) .(3) For n − a.e. x ∈ S ( X ) the tangent cone of X is unique and isometric to the conespace R n − × C ( S / Γ x ) , where Γ x ≤ O (4) acts freely away from the origin.Remark . In fact, the proof gives stronger minkowski and packing estimates on the singular set. Pre-cisely, if { B r i ( x i ) } is any disjoint collection of balls with x i ∈ S ( X ) ∩ B then P r n − i < C ( n , v). See [NV15].In particular, the above proves the n − Outline of the proof Theorem 1.6.
In this subsection we given an outline of the proof’s of the maintheorems of this paper. Primarily, we will focus on the L -curvature estimate of Theorem 1.6, howeverthe same technical ingredients will go into the proofs of the other main results of the paper, including thestructure results for limit spaces given by Theorem 1.14. This section is just an outline, and many of thecomputations are rough in nature, however the morals are the correct ones which will be applied throughoutthe paper. For simplicity we will assume in the outline that Ric ≡
0, which will essentially save on havingto discuss error terms that arise in the general case, which are of little consequence but often times quiteinvolved (especially in the proof of the superconvexity estimate below).1.3.1. ( δ, τ ) -Neck Regions. There are several new types of estimates as well as a new decomposition typetheorem that the proofs rely on. The new estimates and decompositions all center around the notion of whatwe call a ( δ, τ )-neck region N ⊆ B . The precise definition of a neck region is a bit technical, and we referthe reader to Section 3 for this, but roughly a ( δ, τ )-neck is an open set N = B ( p ) \ [ x ∈ C B r x ( x ) ≡ B \ B r x ( C ) , (1.16)where C is a closed set of ball centers and r x < δ is a radius function. To qualify as a neck region, we willhave for each r x < r < B δ − r ( x ) is δ r -Gromov-Hausdorff close to a ball in R n − × C ( S / Γ ), and thatthe ball centers look roughly like a covering of the singular set R n − × { } at every scale. Thus we can view CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 7 C as a discrete approximation to the singular set, and it is natural and convenient to associate to the neckregion the n − µ ≡ X r n − x δ x . (1.17)Before continuing let us discuss a simple example. Analyzing this example will help solidify what it iswe hope to hold for a general neck region: Example . Let E be the standard Ricci flat four manifold given by the Eguchi-Hanson metric, seeExample 2.23, and let E η ≡ η − E be the rescaled Ricci flat metric so that the central 2-sphere has radius η .Let us pick a point y c ∈ E which is an element of this central sphere. Note then that (cid:0) E η , y c (cid:1) η → −→ (cid:0) R / Z , M n η ≡ R n − × E η , and let r x : R n − → R + be a positive function with |∇ r x | < δ .Then if we consider any discrete subset C = { x i } ⊆ B (0 n − ) × { y c } with { B τ r i ( x i ) } a maximal disjoint subset,where r i = r x i , then for all δ > τ < τ ( n ) if η ≤ η ( n , δ ) is sufficiently small then N ≡ B (0 , y ) \ S B r x ( C )is a ( δ, τ )-neck. (cid:3) There are two important pieces of information to take away from Example 1.18. The first is that if weconsider the packing measure µ = P r n − i δ x i associated to the covering, then µ is uniformly Ahlfor’s regular.More precisely, for all r i < r < A ( n , τ ) − r n − ≤ µ ( B r ( x i )) ≤ A ( n , τ ) r n − . (1.19)This holds because in the context of Example 1.18 we have that { B r i ( x i ) } is a Vitali covering of B (0 n − ).The second piece of information we get from Example 1.18 is that we have curvature control on the neckregion N ≡ B \ B r x ( C ). In particular, regardless of the ( δ, τ )-neck of the Example, there is a uniform L bound on the curvature R N | Rm | ≤ δ ′ . In fact, it is not hard to check that as δ → δ ′ → η << δ a neck region is cutting out the central 2-spheres, which is whereall the L is concentrating.Our main theorem on the structure of general neck regions is Theorem 3.10, which tell us that the pack-ing measure and L -curvature control which held in the previous easy example continue to hold on arbitrary( δ, τ )-neck regions. The proofs of these points will take several new ingredients, which we will outlineshortly, however let us first describe in detail what the results say.1.3.2. Structural Theorems on Neck Regions: Ahlfor’s Regularity.
Our first main structural result on neckregions given in Theorem 3.10 is that the packing measure of a neck region has uniform n − δ sufficiently small Theorem 3.10 tells us that:For each ball center x ∈ C and r x < r with B r ( x ) ⊆ B we have A − r n − ≤ µ (cid:16) B r ( x ) (cid:17) ≤ A ( n , τ ) r n − . (1.20) WENSHUAI JIANG AND AARON NABER
The proof of this uniform Ahlfor’s regularity is quite involved, and we will see that the Ahlfors bound itselfis tied into essentially every result of this paper. The lower and upper bounds in the estimate are provedseparately. Let us mention just a few words about the lower bound now, and we will come back to the upperbound near the end of the outline.The moral of the lower bound estimate is the following. Roughly, the restriction of the ball centers C ∩ B r ( p ) to any ball look discretely homeomorphic to a ball B r (0 n − ) ⊆ R n − . This will be made pre-cise by proving a discrete version of a Reifenberg theorem in Theorem 3.24, which will find a collec-tion C ′ ⊆ B r (0 n − ) and a bih¨older mapping φ : C ′ → C which satisfies a variety of properties. Letus now also choose a 1-lipschitz Gromov-Hausdorff map u : B r ( p ) → B r (0 n − ). Then the composition u ◦ φ : C ′ ⊆ B r (0 n − ) → B r (0 n − ) is a bilholder map which looks close to the identity. If we ignore thediscrete nature of the problem then we could pretend that u ◦ φ is a degree 1 homeomorphism from B r toitself. In particular, that would prove A ( n ) − r n − = Vol( u ◦ φ ( B r )) ≤ Vol n − ( φ ( B r )) ≈ Vol n − ( C ) ≈ µ ( C ∩ B r ),where we have used that u is 1-lipschitz and that µ is approximating the n − C . Witha little work this argument will be made precise in Section 6.1 . We will come back to the outline of theupper Ahlfor’s bound, which is much more anlaytic in nature, at the end of the outline.1.3.3. Structural Theorems on Neck Regions: L -Estimate. Let us now discuss our second structural resultabout ( δ, τ )-neck regions from Theorem 3.10. Specifically, we have on a neck region N ⊆ B that we canprove the desired L -curvature bound Z N ∩ B | Rm | ≤ δ ′ . (1.21)The proof of the L curvature estimate (1.21) will, in fact, require that we have already proven the Ahlfor’sregularity (1.20). In order to explain the main technical lemma involved in the proof, let us recall thedefinition of the H -volume given by H t ( x ) ≡ Z M (4 π t ) − n e − d x , y )4 t dv g ( y ) . (1.22)We will see in Section 4 that the H -volume is monotone nonincreasing. Our main technical result toward theproof of the L -estimate (1.21) on neck regions is Proposition 4.3, which will prove, under the assumptionof the Ahlfor’s condition (1.20), that for each x ∈ N with 2 r = d ( x , C ) we have the estimate Z B r ( x ) | Rm | ≤ C ( n ) Z B r ( x ) | H r − H − r | ( y ) d µ ( y ) . (1.23)That is, if the measure µ satisfies the n − L energyof a ball B r ( x ) in the neck region by the H -volume drop on that scale over µ in a slightly bigger ball. Let ussee how the L estimate (1.21) follows from the local estimate (1.23). Indeed, it is not so difficult to build aVitali covering N ∩ B ⊆ [ α N α [ i = B s α, i ( x α, i ) , (1.24) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 9 where d ( x α, i , C ) = s α, i , s α, i ∈ (2 − α − , − α ], s α = − α and { B − s α, i ( x α, i ) } are disjoint. Then we can roughlyestimate Z N ∩ B | Rm | ≤ X α X i Z B s α, i ( x α, i ) | Rm | ≤ C ( n ) X α X i Z B s α, i ( x α, i ) | H s α, i − H − s α, i | ( y ) d µ ( y ) ≤ C ( n ) X α Z B / (cid:12)(cid:12)(cid:12) H s α − H − s α (cid:12)(cid:12)(cid:12) ( y ) d µ ( y ) ≤ C ( n ) Z B / (cid:12)(cid:12)(cid:12) X α ( H s α − H − s α ) (cid:12)(cid:12)(cid:12) ( y ) d µ ( y ) ≤ C ( n ) Z B / (cid:12)(cid:12)(cid:12) H − H − r y (cid:12)(cid:12)(cid:12) ( y ) d µ ( y ) , (1.25)where we have used the monotonicity of H to bring the sum inside the absolute value sign. However, since N is a neck region we have that B δ − ( y ) and B δ − r y ( y ) are both Gromov-Hausdorff close to R n − × C ( S / Γ ),and therefore we have that | H − H − r y | ( y ) → δ →
0. Using our Ahlfor’s condition againwe see for δ sufficiently small that the L estimate (1.21) of Theorem 3.10 on the neck region is proved.1.3.4. Neck Decomposition Theorem.
Thus, we have now discussed several structural results from the paperwhich tell us that a general ( δ, τ )-neck region analytically behaves much like we might hope from Example1.18. One is still in the position of understanding the relevance of these results, in particular in the contextof proving the L curvature estimate of Theorem 1.6.In more detail, in order to exploit the structural results of Theorem 3.10 about ( δ, τ )-neck regions, weneed to see that there are lots of such neck regions. Otherwise, we are proving theorems about a set whichmay not really appear in practice. Indeed, the next primary result of the paper which we wish to discuss isthe neck decomposition of Theorem 7.1, which will show that every point either lies in a neck region or inan ǫ -regularity ball. More precisely, we can cover our space B ( p ) ⊆ [ a N a ∪ [ b B r b ( x b ) , (1.26)where N a ⊆ B r a ( x a ) are ( δ, τ )-neck regions, and each ball B r b ( x b ) is a uniformly smooth ball in that theharmonic radius satisfies r h ( x b ) > r b , see Section 2.4 for a review of the harmonic radius. The key aspectof the decomposition of Theorem 7.1 is that we will prove the n − X a r n − a + X b r n − b ≤ C ( n , v , τ, δ ) . (1.27)We will mostly avoid discussing the proof of the above estimate in this outline, as it involves numeroustechnical covering arguments which first decomposes B into five types of balls with the help of four pa-rameters, and then proceeds to recover these balls until only the neck and ǫ -regularity regions are left. It isworth mentioning however that this decomposition, and in particular the content bound, relies heavily on theAhlfor’s bound of (1.20). Without (1.20) the techniques of [ChNa13] could be used build this decompositionunder the weaker content estimate P a r n − − δ a + P b r n − − δ b ≤ C ( n , v , δ ), but as we will see shortly it is crucialfor the applications in this paper, and in particular the L curvature estimate of Theorem 1.6, to have the sharp n − Proving the L Curvature Estimate.
Though we have not yet outlined the proof of the Ahlfor’s regu-larity result of Theorem 3.10, let us now take a moment to see how the neck decomposition of Theorem 7.1combined with the structural results of Theorem 3.10 on neck regions we have discussed lead to a proof ofthe L -estimate of Theorem 1.6. Indeed, using the neck decomposition we can write Z B ( p ) | Rm | ≤ X a Z N a | Rm | + X b Z B rb | Rm | . (1.28)Let us now observe that on the regularity balls B r b ( x b ) that we can use the harmonic radius bound r h ( x b ) > r b and standard elliptic estimates in order to prove the scale invariant curvature bound r − nb Z B rb | Rm | ≤ C ( n ) . (1.29)On the other hand, applying the L curvature estimate of Theorem 3.10 on ( δ, τ )-neck regions explainedin (1.21) leads to the scale invariant estimate on neck regions given by r − na Z N a | Rm | ≤ C . (1.30)If we now combine the content estimate (1.27), the regularity ball estimate (1.29), and the ( δ, τ )-neckestimate (1.30), then we see we can prove the desired L -curvature estimate by computing Z B ( p ) | Rm | ≤ X a Z N a | Rm | + X b Z B rb | Rm | , ≤ C ( n ) X a r n − a + C ( n ) X b r n − b ≤ C ( n , v , τ, δ ) , (1.31)which would indeed finish the proof of Theorem 1.6.1.3.6. Harmonic Splitting Functions on Neck Regions.
Therefore, what is left in our outline is to understandhow to prove the upper bound of the Ahlfor’s regularity estimate (1.20), which is one of the main technicalchallenges of this paper. This estimate itself is based heavily on a key new technical estimate for harmonicsplitting functions.More precisely, let N ⊆ B be a ( δ, τ )-neck, so that in particular B δ − is Gromov-Hausdorff close to R n − × C ( S / Γ ). Let u : B → R n − be a harmonic δ -splitting map associated to this geometry, so that wehave the estimates ? B |∇ u | , ? B (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) < δ , sup B |∇ u | ≤ . (1.32) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 11
Our main new technical achievement is to show that if we restrict u to the ball centers C , which recall actas a discretization of the singular set, then u is a bilipschitz map over most of C . To accomplish this we needto address what is apriori a logical loop. Namely, we will use this bilipschitz bound in order to prove theAhlfor’s regularity, however we need the Ahlfor’s regularity in order to prove the bilipschitz bound in thefirst place. We will address this issue in the next subsection of the outline, for now let us simply assume thatfor some B > x ∈ C with B r ( x ) ⊆ B we have that B − r n − ≤ µ (cid:16) B r ( x ) (cid:17) ≤ Br n − . (1.33)Mentally, one should view B >> A , where A is the Alhfor’s regularity constant we aim to prove. Thus,our goal is effectively to show if one has a bad Alhfor’s regularity constant B , then with δ small one in facthas a better bound of A for free. With a little technical footwork we will see how this can be used to dealwith the apriori logical loop.With this in mind, our main result for splitting functions on ( δ, τ )-neck regions is Theorem 5.2, whichtells us that for each ǫ > δ ≤ δ ( n , ǫ, τ, B ), then there exists a subset C ǫ ⊆ C ∩ B such that µ ( C ǫ ) > (1 − ǫ ) µ ( C ∩ B ) , ∀ x , y ∈ C ǫ we have (cid:12)(cid:12)(cid:12)(cid:12) | u ( x ) − u ( y ) | − d ( x , y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ d ( x , y ) . (1.34)Let us spend a few moments on outlining the proof of the above. To begin with, let us not try and hitevery detail, and instead just focus on what is the main new technical estimate needed in the proof. That is,in Theorem 5.55 we prove the estimate Z N ∩ B r − h |∇ u | ≈ Z B (cid:16) X r a = − a ? N ∩ A ra + , ra ( x ) r a |∇ u | (cid:17) d µ [ x ] < ǫ , (1.35)where ǫ may be taken arbitrarily small so long as δ < δ ( n , τ, ǫ, B ) is taken sufficiently small. When combinedwith the Ahflors condition (1.33) and a telescoping argument, one can conclude directly the bilipschitzestimate (1.34) from this, see Section 5 for details.Therefore, we will focus on the estimate (1.35). Let us begin by considering the Green’s function associ-ated to the packing measure µ given by − ∆ G µ = µ . (1.36)Because µ has the Ahlfor’s regularity bounds (1.33) and approximates the singular set of R n − × C ( S / Γ ), onecan imagine G µ being well approximated by r − h , where r h is the harmonic radius, which is itself roughly thesame as the distance to C . Indeed, in Lemma 5.6 we will see that if we define the Green’s distance functionby G µ = b − , then we will have the estimates on the neck region N given by C ( n , B ) − r h ≤ b ≤ C ( n , B ) r h , C ( n , B ) − ≤ |∇ b | ≤ C ( n , B ) , Vol (cid:0) { b = r } ∩ B / (cid:1) ≤ Cr . (1.37) Now if φ is a cutoff function for which φ ≡ N , see Lemma 3.15 for a precise construc-tion, then we can define the quantities F ( r ) ≡ r − Z b = r |∇ u | |∇ b | φ , H ( r ) = rF ( r ) . (1.38)Let us observe that an estimate on H ( r ) represents a scale invariant hessian estimate along the b = r slice.A key computation takes place in Proposition 5.57, where we will see that H ( r ) satisfies the superconvexitytype estimate r ¨ H + r ˙ H − (1 − ǫ ) H ≥ − ǫ X r n − x δ [ C − r x , Cr x ] − ǫ r δ [0 , C ] , (1.39)where C = C ( n ) and ǫ may be taken arbitrarily small so long as δ is sufficiently small (indeed, the aboveis actually simpler than Proposition 5.57 due to the assumption Rc ≡ Z ∞ r H ( r ) ≤ ǫ . (1.40)Combining with the Green’s function estimates (1.37) and a coarea formula one immediately concludesfrom this the estimate Z N r − h |∇ u | ≤ C Z M b − |∇ u | φ , ≤ C Z ∞ r − Z b = r |∇ u ||∇ b | φ , = C Z ∞ r H ( r ) < ǫ , (1.41)which is the claimed estimate.1.3.7. Ahlfor’s Regularity on Neck Regions.
Now we end our outline by sketching how the bilipschitz es-timate (1.34) implies the Ahlfors regularity estimate (1.20). As mentioned previously, there is a seeminglylogical loop in that we needed that Ahlfors condition in order to prove the bilipschitz estimate itself.In order to circumnavigate this issue in Section 6, we will use an inductive procedure which is motivatedfrom [NV15]. Precisely, let us consider the radii r α ≡ − α , and let our goal be to inductively prove the upperbound in (1.20) for all r ≤ r α , recalling that we have already outlined the lower bound independently. First,since M is a smooth manifold we have that r x > r > x ∈ C .In particular, if r α is the largest radius for which r x ≥ r α for all x ∈ C , then the result is trivial for this α bythe simple definition of µ . This is the base step of our induction.Therefore, our focus is to prove (1.20) for some r α , assuming we have proved it for r α + . Let us beginwith a weaker estimate, which will be useful to establish. Namely, assume B r ( x ) ⊆ B with r ≤ r α . CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 13
Then by a standard covering argument we can cover B r ( x ) by at most C ( n ) balls of radius 10 − r ≤ r α , andtherefore by applying our inductive assumption we have for some B = C ( n ) A the strictly weaker estimate µ (cid:0) B r ( x ) (cid:1) ≤ Br n − . (1.42)While this estimate is not good enough for the inductive step, indeed if one were to iterate it in α theconstant would blow up horribly, it is enough for us to apply Theorem 5.2 and obtain the ǫ -bilipschitz resultof (1.34). More precisely, if u : B r ( x ) → R n − is a δ -splitting function, which exists because we are in a( δ, τ )-neck, then there exists C ǫ ⊆ C ∩ B r ( x ) such that µ ( C ǫ ) > (1 − ǫ ) µ ( C ∩ B r ) , ∀ x , y ∈ C ǫ we have (cid:12)(cid:12)(cid:12)(cid:12) | u ( x ) − u ( y ) | − d ( x , y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ d ( x , y ) . (1.43)However, the set { B r i ( x i ) } C is a Vitali covering, which tells us that for ǫ sufficiently small that { B r i ( u ( x i )) } C ∩ B r is itself a Vitali covering of B r (0 n − ). This immediately implies the improved Ahlfor’s upper bound µ ( C ∩ B r ) ≤ − ǫ µ ( C ǫ ) = X C ǫ r n − i ≤ C Vol( B r (0 n )) ≤ A r n − . (1.44)This proves the inductive step, and therefore finishes the outline of the proof.2. B ACKGROUND AND P RELIMINARIES
In this section we review several standard constructions and techniques which will be used throughoutthe paper.2.1.
Quantitative Stratification.
Let us briefly review the notion of symmetry and stratification, as theseideas are commonplace throughout this paper. We begin by recalling the standard notion of symmetry:
Definition 2.1.
Given a metric space Y we define the following:(1) We say Y is k-symmetric at y ∈ Y if there exists a pointed isometry ι : R k × C ( Z ) → Y with ι (0 k , z c ) = y , where Z is compact and z c is a cone point.(2) We say Y is k -symmetric with respect to L k ⊆ Y if L k = ι (cid:0) R k × { z c } (cid:1) .Associated to the notion of symmetry is that of stratification: Definition 2.2.
Given a metric space Y we define the closed k th -stratum by S k ( X ) = : { x ∈ X : no tangent cone at x is ( k + } (2.3)This notion of symmetry leads to a natural quantitative generalization, first introduced in [ChNa13]. It isthe notion of quantitative symmetry which will play the most important role for us in this paper, in particular in the discussion of neck regions. Let us begin with a discussion of quantitative symmetry: Definition 2.4.
Given a metric space Y with y ∈ Y , r > ǫ >
0, we say(1) B r ( y ) is ( k , ǫ )-symmetric if there exists a pointed ǫ r -GH map ι : B r (0 k , z c ) ⊆ R k × C ( Z ) → B r ( y ) ⊆ Y .(2) B r ( y ) is ( k , ǫ )-symmetric with respect to L k ǫ ⊆ B r ( y ) if L k ǫ ≡ ι (cid:0) R k × { z c } (cid:1) To state the definition in words, we say that B r ( y ) ⊆ Y is ( k , ǫ )-symmetric if the ball B r ( y ) looks veryclose to having k -symmetries. The quantitative stratification is then defined as follows: Definition 2.5.
For each ǫ >
0, 0 < r < k ∈ N , define the closed quantitative k -stratum, S k ǫ, r ( X ), by S k ǫ, r ( X ) ≡ { x ∈ X : for no r ≤ s ≤ B s ( x ) a ( k + , ǫ )-symmetric ball } . (2.6)Thus, the closed stratum S k ǫ, r ( X ) is the collection of points such that no ball of size at least r is almost( k + S k ǫ, r ( X ) is small in a very strong sense. To saythis a little more carefully, if one pretends that the k -stratum is a well behaved k -dimensional submanifold,then one would expect the volume of the r -tube around the set to behave like Cr n − k . In [ChNa13] thefollowing slightly weaker result was shown: Theorem 2.7 (Quantitative Stratification, [ChNa13]) . Let M n satisfy Ric ≥ − ( n − with Vol( B ( p )) > v > .Then for every ǫ, η > there exists C = C ( n , v , ǫ, η ) such that Vol (cid:16) B r (cid:16) S k ǫ, r ( M ) ∩ B ( p ) (cid:17)(cid:17) ≤ Cr n − k − η . (2.8)One of the consequences of the main Theorems of this paper is that for spaces with bounded Ricci cur-vature one can improve the above to Vol (cid:16) B r (cid:16) S n − ǫ, r ( M ) ∩ B ( p ) (cid:17)(cid:17) ≤ Cr for the top stratum of the singular set.2.2. Volume Monotonicity and Cone Structures.
When considering a Riemannian manifold with a lowerRicci bound Ric ≥ − ( n − κ g , the key tool which separates the study of collapsed versus noncollapsedspaces is that of a monotone quantity. Throughout this paper we will consider several, all of which areessentially equivalent, however it will be more convenient to work with one or another depending on thecontext. Let us begin by recalling the classical volume ratio V r ( x ) = V κ r ( x ) ≡ Vol (cid:0) B r ( x ) (cid:1) Vol − κ ( B r ) , (2.9)where Vol − κ ( B r ) is the volume of the ball of radius r is the space form M n − κ of constant curvature − κ . Itis a classic consequence of the Bishop-Gromov monotonicity that for each x ∈ M that V r ( x ) is monotone CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 15 nonincreasing in r > V r ( x ) → r →
0. Note that the space being noncollapsed is now equivalentto there being a lower bound on V , so that we have a bounded monotone quantity. The importance of thiscomes into play because there is a rigidity, which tells us that when this quantity is very pinched we musthave symmetry. Precisely: Theorem 2.10 ([ChCo1]) . Let ( M n , g , p ) with ǫ > , then there exists δ ( n , ǫ ) > such that if Ric ≥ − δ and V ( p ) ≥ (1 − δ ) V ( p ) , then B ( p ) is (0 , ǫ ) -symmetric. In Section 4 we will discuss some generalizations of this point using some distinct monotone quantities.The advantage of the approach of Section 4 will be that we will be able to obtain some sharp estimates onthe cone structures, which will be required in the proof of the L estimates.2.3. Cone Splitting.
We saw in the previous subsection how to force 0-symmetries by using a monotonequantity. In this subsection we want to review one method of forcing higher orders of symmetries. Therelevant concept for this paper is one introduced in [ChNa13] called cone splitting. The main result in thisdirection from [ChNa13] is the following:
Theorem 2.11 (Cone-Splitting) . For every ǫ, τ > there exists δ ( n , ǫ, τ ) > such that if(1) Ric ≥ − δ .(2) B ( p ) is ( k , δ ) -symmetric with respect to L k δ ⊆ B ( p ) .(3) There exists z ∈ B ( p ) \ B τ L k δ such that B ( z ) is (0 , δ ) -symmetric.Then B ( p ) is ( k + , ǫ ) -symmetric. Therefore, the above is telling us that nearby 0-symmetries interact to force higher order symmetries.2.4.
Harmonic Radius and ǫ -Regularity Theorems. In this subsection we review two ǫ -regularity theo-rems which will play a prominent role in this paper. To make these results precise let us recall the notion ofthe harmonic radius: Definition 2.12.
For x ∈ M n we define its harmonic radius r h ( x ) > r > φ : B r ( x ) → R n with the following properties:(1) φ is a harmonic mapping.(2) φ is a diffeomorphism onto its image with B r (0 n ) ⊆ φ ( B r ( x )), and hence defines a coordinate chart.(3) The coordinate metric g i j = h∇ φ i , ∇ φ j i on B r (0 n ) satisfies || g i j − δ i j || C ( B r ) < − n . Remark . Note that by a standard implicit function type argument one has that r h ( x ) > Remark . The C -norm above is taken with respect to the scale invariant Euclidean norm. That is, || g i j − δ i j || C ( B r ) ≡ sup B r | g i j − δ i j | + r sup B r | ∂ k g i j | . Remark . Note that if we have the Ricci bound | Ric | ≤ A , then for every α < p < ∞ we have theapriori estimate || g i j − δ i j || C ,α ( B r / ) , || g i j − δ i j || W , p ( B r / ) < C ( n , A , α, p ). This follows from elliptic estimateswhich exploit the harmonic nature of the coordinate system.Now the ǫ -regularity theorems of this subsection are meant to find weak geometric conditions underwhich we can be sure there exists a definite lower bound on the harmonic radius. The first which we willdiscuss goes back to Anderson and tells us that if the volume of a ball is sufficiently close to that of a Eu-clidean ball, then we must have smooth estimates. Precisely: Theorem 2.16 ([A90]) . There exists ǫ ( n ) > such that if ( M n , g , p ) satisfies | Rc | ≤ ǫ and V ( p ) > (1 − ǫ ) ,then we have that r h ( p ) > .Remark . Recall that V r ( p ) = V ǫ r ( p ) ≡ Vol( B r ( x )Vol − κ ( B r ) , and thus V > − ǫ gives us that the volume of B ( p )is close to the volume of B (0 n ).We end with the following ǫ -regularity result proved in [ChNa15]. This result can be viewed as a con-sequence of the proof of the codimension four conjecture, and tells us that if a ball has a sufficient amountof symmmetry, then the ball must be smooth. Results like the following are why the notion of quantitativesymmetry play such a crucial role in the analysis and estimates of this paper: Theorem 2.18 ([ChNa15]) . Let ( M n , g , p ) satisfy | Rc | ≤ n − and Vol( B ( p )) > v > . Then there exists ǫ ( n , v) > such that if B ( p ) is ( n − , ǫ ) -symmetric then r h ( p ) > . Heat Kernel Estimates.
In this subsection we record estimates on heat kernels on Riemannian mani-folds with lower Ricci curvature bounds. The estimates of this subsection are either classical or very minormodifications of classical estimates which are better suited to our purposes. Let us summarize the basicestimates on heat kernels, which follow from the results in [LiYau86], [SY], [Ha93], [Kot07]:
Theorem 2.19 (Heat Kernel Estimates) . Let ( M n , g , x ) be a pointed Riemannian manifold with Vol( B ( x )) ≥ v > and Rc ≥ − ( n − . Then for any < t ≤ and ǫ > , the heat kernel ρ t ( x , y ) = (4 π t ) − n / e − f t satisfies(1) C ( n , ǫ ) t − n / e − d ( x , y )2(4 − ǫ ) t ≤ ρ t ( x , y ) .(2) ρ t ( x , y ) ≤ C ( n , v , ǫ ) t − n / e − d ( x , y )2(4 + ǫ ) t .(3) t |∇ f t | ≤ C ( n , v , ǫ ) + d ( x , y )(4 − ǫ ) t .(4) − C ( n , v , ǫ ) + d ( x , y )(4 + ǫ ) t ≤ f t ≤ C ( n , v , ǫ ) + d ( x , y )(4 − ǫ ) t CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 17 δ -Splitting Functions. In this subsection we recall the basics about splitting functions, which act as abridge between geometric notions and notions in analysis. Let us begin with a definition, which is similar tothe one introduced in [ChNa15]:
Definition 2.20.
We say u : B r ( p ) → R k is a harmonic δ -splitting function if the following holds:(1) ∆ u = > B r ( p ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) < δ .(3) r > B r ( p ) (cid:12)(cid:12)(cid:12) ∇ u (cid:12)(cid:12)(cid:12) < δ .(4) sup B r ( p ) |∇ u | ≤ B r splitting offan R k -factor. Precisely, we have the following: Theorem 2.21.
For every ǫ > there exists δ ( n , ǫ ) > such that if Ric ≥ − δ then the following hold:(1) If B δ − ( p ) is δ -GH close to R k × Y, then there exists an ǫ -splitting u : B ( p ) → R k .(2) If there exists a δ -splitting u : B ( p ) → R k , then B ( p ) is ǫ -GH close to R k × Y.Remark . The above theorem is a mild extension of one of the main accomplishments of [ChCo1], withthe sharp gradient bound of condition (4) having been proved in [ChNa15].2.7.
Examples.
In this subsection, we recall some standard examples which play an important role in guid-ing the results of this paper. We discuss these in only minimal detail, and refer the reader to the appropriatereferences for more.
Example . The Eguchi-Hanson space E = ( T ∗ S , g ) is a complete Ricciflat metric which is diffeomorphic to the cotangent bundle of S . The asymptotic cone of E convergesrapidly to R / Z , where Z acts on R by x → − x . That is, if one considers the spaces E η ≡ ( T ∗ S , η g ),then we have the Gromov Hausdorff limit E η η → −→ C ( RP (3)) = R / Z . This is the simplest example whichshows that even under the assumption of Ricci flatness and noncollapsing, Gromov-Hausdorff limit spacescan contain codimension 4 singularities.It is an interesting exercise, using the Chern-Gauss-Bonnet formula, that one has R E η | Rm | = π in-dependent of η , however for q > R E η | Rm | q → ∞ as η →
0, which shows us that an apriori L bound on the curvature is the most one can expect. Example L Blow up under collapsing) . We now briefly study an example of Anderson [A93], whichwill tell us that the noncollapsing assumption is necessary in that there exists a sequence of manifolds M j such that | Ric j | → , diam( M j ) = , Vol( M j ) → , Z M j | Rm j | → ∞ . (2.25)Indeed, the rough construction is as follows. Anderson built a complete simply connected Ricci flatfour manifold ( S , g ) such that outside a compact set we have that S is diffeomorphic, and in fact quicklybecoming isometric, to the metric product R × S . The space S has the property that R S | Rm | = C > M j let us begin with the flat four torus T × S j − , where T = ( S ) is the standard threetorus and S j − is the circle of radius j − . One may then glue copies of S j − ≡ j S into T × S j − . As j → ∞ one may glue an arbitrarily large number of copies while perturbing the Ricci flat condition arbitrarily smallamount. Each glued copy of S j − contributes roughly C to the total L norm of | Rm | , and therefore it is easyto check that (2.25) is satisfied. 3. ( δ, τ )-N ECK R EGIONS
A central theme of this paper will be that of a ( δ, τ )-neck region. In this section we will define this, andthen state our main results on such regions. The proofs themselves will take place over the next severalsections, as they will be a bit involved. We begin with a formal definition:
Definition 3.1.
We call N ⊆ B ( p ) a ( δ, τ )-neck region if there exists a closed subset C = C ∪ C + = C ∪ { x i } and a radius function r : C → R + with 0 < r x ≤ δ on C + and r x = C such that N ≡ B \ B r x ( C ) satisfies(n1) { B τ r x ( x ) } ⊆ B ( p ) are pairwise disjoint.(n2) For each r x ≤ r ≤ δ r -GH map ι x , r : B δ − r (0 n − , y c ) ⊆ R n − × C ( S / Γ ) → B δ − r ( x ),where Γ ⊆ O (4) is nontrivial.(n3) For each r x ≤ r with B r ( x ) ⊆ B ( x ) we have that L x , r ≡ ι x , r (cid:0) B r (0 n − ) × { y c } (cid:1) ⊆ B τ r ( C ).(n4) | Lip r x | ≤ δ .For each τ ≤ s ≤ N s ≡ B \ B s · r x ( C ). Remark . If A ⊆ M is a closed subset with a x : A → R + a nonnegative continuous function, then theclosed tube B a x ( A ) is by definition the set S x ∈ A B a x ( x ). Remark . Recall if f : A ⊆ M → R is a function defined on some subset A ⊆ M , then the lipschitzconstant of f is defined | Lip f | ≡ sup x , y ∈ A | f ( x ) − f ( y ) | d ( x , y ) . Remark . The notation L x , r is based on the comparible notation for the quantitative stratification givenin Definition 2.4, since B r ( x ) is ( n − , δ )-symmetric. Remark . In the above we may take B ( p ) to be either in a manifold M n or a limit space M nj → X . In thecase when B ( p ) is in a manifold then necessarily C = ∅ and C < ∞ , since C ⊆ Sing( X ) and inf r x > CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 19
Remark . We also say N ⊆ B R ( p ) is a neck region for some R > B ( ˜ p ) = R − B R ( p ) we have that N satisfies the above. Remark . Note that if N ⊆ B ( p ) is a ( δ, τ )-neck region and B s ( q ) ⊆ B ( p ), then N ∩ B s ( q ) defines a( δ, τ )-neck on B s ( q ).An important concept associated to any neck region is the induced packing measure. In the same mannerthat the set C is a potentially discrete approximation of the singular set, the packing measure is a potentiallydiscrete approximation of the n − Definition 3.8.
Let N ≡ B \ B r x ( C ) be a neck region, then we define the associated packing measure µ = µ N ≡ X x ∈ C + r n − x δ x + λ n − | C , (3.9)where λ n − | C is the n − C .Let us now state what is our main result on neck regions. The proof of this result will take place over thecourse of the next several sections, as it will be one of the main technical accomplishments of the paper: Theorem 3.10.
Let ( M nj , g j , p j ) → ( X , d , p ) be a Gromov-Hausdorff limit with Vol( B ( p j )) > v > , τ < τ ( n ) and ǫ > . Then for δ ≤ δ ( n , v , τ, ǫ ) if | Rc j | < δ and N = B ( p ) \ B r x ( C ) is a ( δ, τ ) -neck region, then thefollowing hold:(1) For each x ∈ C and r > r x such that B r ( x ) ⊆ B ( p ) the induced packing measure µ satifies theAlhfors regularity condition A ( n , τ ) − r n − < µ ( B r ( x )) < A ( n , τ ) r n − .(2) C is n − rectifiable.(3) X is a manifold on N and we have the L -curvature bound R N ∩ B | Rm | < ǫ . Let us briefly outline this section. We begin in Section 3.1 by discussing some basic properties of neckregions. This includes the definition of wedge regions and the construction of a canonical cutoff function,which will be used frequently in the analysis. In Section 3.2 we will show that every neck region on asingular limit space M j → X may be approximated by neck regions on the smooth spaces M nj themselves.This will allow us to primarily work on manifolds and then pass our results off to the limit automatically.Finally in Section 3.3 we will prove a discrete Reifenberg type theorem for the effective singular set C . Thiswill end up playing an important role in the proof of the lower bound in our Ahlfor’s regularity.3.1. Basic Properties of Neck Regions.
In this subsection we record some simple properties of neck re-gions which will play a role in the analysis.
Regularity in the Neck Region.
Let us begin with the following, which tells us that the harmonicradius at a point in the neck region is roughly the distance of that point to the effective critical set. The proofis immediate using ( n
3) and ( n
4) of a neck region, but it is worth mentioning the result explicitly:
Lemma 3.11.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > with | Rc | < δ and N ⊆ B ( p ) a ( δ, τ ) -neck region.Then for each ǫ > we have for δ < δ ( n , v , ǫ ) and τ < τ ( n , ǫ ) that for each y ∈ N − it holds that (1 − ǫ ) d ( y , C ) ≤ r h ( y ) ≤ (1 + ǫ ) d ( y , C ) . (3.12)3.1.2. Wedge Regions in ( δ, τ ) -Necks. A region which will play a useful technical role for us is that of awedge region. Precisely:
Definition 3.13.
Let N = B ( p ) \ B r x ( C ) be a ( δ, τ )-neck region with τ, δ < − n . Then for each center point x ∈ C we define the wedge region W x associated to x by W x ≡ n y ∈ A − r x , ( x ) : d ( y , x ) < · d ( y , C ) o . (3.14)Wedge regions are the correct scale invariant regions that many of our estimates will live on. Let us pointout that although the lipschitz condition ( n
4) of the radius function is quite useful for many of the technicalresults, the only point where it plays a role of any consequential importance is in controlling the wedgeregions. Indeed, note that W x ⊆ N − , however if a neck region consisted of an arbitrary covering which didnot necessarily satisfy ( n Cutoff Functions on Neck Regions.
In this subsection we build a natural cutoff function associatedwith a neck region. We will record some of the basic properties and estimates associated to it, which will beuseful throughout the paper. Precisely:
Lemma 3.15.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > with | Rc | < δ and N ⊆ B ( p ) a ( δ, τ ) -neck region.Then for τ < τ ( n ) and δ < δ ( n , v , τ ) there exists a cutoff function φ = φ · φ ≡ φ N : B → [0 , such that(1) φ ( y ) = if y ∈ B / ( p ) with supp ( φ ) ⊆ B / ( p ) .(2) φ ( y ) ≡ if y ∈ N − with supp ( φ ) ∩ B ⊆ N − .(3) |∇ φ | , | ∆ φ | ≤ C ( n ) .(4) supp ( |∇ φ | ) ∩ B / ⊆ A − r x , − r x (cid:0) C (cid:1) with r x |∇ φ | , r x |∇ φ | < C ( n ) in each B r x ( x ) . CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 21
Proof.
Recall from [ChCo1] that for each annulus A s , r ( x ) we can build a cutoff function φ such that φ ≡ B s ( x ) , φ ≡ B r ( x ) , |∇ φ | ≤ C ( n ) | r − s | , | ∆ φ | ≤ C ( n ) | r − s | . (3.16)It will be important to recall that φ is built as a composition φ = c ◦ f , where c is a smooth cutoff on R with c = , s ] and c = , r ], and f satisfies ∆ f = n on B r ( x ) and is uniformly equivalent tothe square distance d x .Thus we can let φ be the cutoff associated to the annulus A , ( p ). To build φ let us begin by definingfor each x ∈ C the cutoff φ x = c x ◦ f x as in (3.16) associated to the annulus A − r x , − r x ( x ). Using ellipticestimates we can get the pointwise estimate |∇ f x | < C ( n ) r − x on A − r x , − r x ( x ) ∩ N − . In particular wecan get the estimate r x |∇ φ x | , r x |∇ φ x | < C ( n ) on A − r x , − r x ( x ) ∩ N − . (3.17)Now with τ < − let us define the cutoff function φ ≡ Y (cid:16) − φ x (cid:17) . (3.18)We have that supp φ ∩ B ⊆ N − and for each point y ∈ B ( p ) we have by ( n
4) that there are at most C ( n )of the cutoff functions φ x which are nonvanishing at y . In particular, the estimates (3.17) then easily implythe required estimates on φ . (cid:3) Approximating Neck Regions on Limit Spaces.
Most of our theorems on neck regions for limitspaces will be proved by first showing the corresponding results on smooth manifolds and then passing to alimit. The reason for this is two fold. First, it is simply more convenient to work on a manifold. However,the more important reason is that there is a subtle point in the inductive proof of the Ahlfor’s regularityestimates which force one to first prove all results in the case C = ∅ . Essentially, this is because there isno base step for the induction argument otherwise. The following approximation results will allow us toapproximate the general case by such a scenario: Theorem 3.19.
Let ( M nj , g j , p j ) GH → ( X , d , p ) with Vol( B ( p j )) > v > and N ≡ B ( p ) \ B r x ( C ) a ( δ, τ ) -neckregion. Then there exists a sequence of ( δ i , τ i ) -neck regions N i ≡ B ( p i ) \ B r x , i ( C i ) such that N i → N in thefollowing sense:(1) δ i → δ , τ i → τ .(2) If φ i : B ( p i ) → B ( p ) are the Gromov-Hausdorff maps then φ i ( C i ) → C in the Hausdorff sense.(3) r x , i → r x : C → R + uniformly.(4) If µ i , µ are the packing measures of N i and N , respectively, then if we limit µ i → µ ∞ we get theuniform comparison µ ≤ C ( n , τ ) µ ∞ .(5) Conversely, we have the estimate µ ∞ ≤ C ( n , τ ) µ . Remark . We will prove (1)-(4) in full generality now. The estimate (5) we will prove under the as-sumption that C is rectifiable, which will be a consequence of Theorem 3.10. We will only sketch (5) asthe main results in the paper will not rely on it. Remark . More generally, we will see without prior knowledge of rectifiability that µ ∞ is uniformlyequivalent to µ + = µ ∩ C + plus the upper minkowski n − C . The point then is that when C is k -rectifiable we have that the n − n − N ⊆ B ( p ) by a se-quence of neck regions ˜ N a ⊆ B ( p ) which also live in the singular space, but which are discrete. After this isaccomplished we will approximate much more directly the discrete neck regions ˜ N a by neck regions livingin the smooth approximating spaces. Let us begin by constructing the discrete neck regions ˜ N a : Lemma 3.22.
Let ( M nj , g j , p j ) GH → ( X , d , p ) with Vol( B ( p j )) > v > and N ≡ B ( p ) \ B r x ( C ) a ( δ, τ ) -neckregion. Then there exists a sequence of ( δ a , τ a ) -neck regions ˜ N a ≡ B ( p ) \ B r x ( ˜ C a ) with r x , a > r a > uniformly bounded from below and ˜ N a → N in the following sense:(1) δ a → δ , τ a → τ .(2) ˜ C a → C in the Hausdorff sense.(3) r x , a → r x : C → R + uniformly.(4) If ˜ µ a , µ are the packing measures of ˜ N a and N , respectively, then if we limit ˜ µ a → ˜ µ ∞ we get theuniform comparison µ ≤ C ( n , τ ) ˜ µ ∞ .(5) We have the estimate ˜ µ ∞ ≤ C ( n , τ ) µ .Proof. For any r > r x on C given by ˜ r x ≡ max { r x , r } . Note that | Lip ˜ r x | < δ ,and that all the properties of a neck region are satisfied with C and ˜ r x except potentially the Vitali condition.Thus let us choose a maximal subset ˜ C r ≡ { x ri } ⊆ C such that the balls (cid:8) B τ r i ( x ri ) (cid:9) are disjoint. It is easy tocheck that ˜ N r ≡ B \ B ˜ r x ( ˜ C r ) is then a ( δ, τ )-neck region itself for each r >
0. We need to understand theconvergence of these neck regions to N . Conditions (1) → (3) are clear. In order to see (4) and (5) we canfocus on the set C , as on C + the limit of ˜ µ r is exactly µ + = µ ∩ C + . Thus let us observe the following. If y ∈ C then for all r << s we have by the definition of ˜ µ r the uniform minkowski estimates r − Vol (cid:0) B s ( y ) ∩ B r ( C ) (cid:1) ≤ C ( n , τ ) ˜ µ r ( B s + r ( y ))˜ µ r ( B s − r ( y ) ∩ B r ( C )) ≤ C ( n , τ ) r − Vol (cid:0) B s ( y ) ∩ B r ( C ) (cid:1) . (3.23)Limiting r → µ ∞ ∩ C is equivalent to the minkowski n − C is rectifiable, then a standard geometric measure theory argument tells us that theminkowski content is itself uniformly equivalent to the hausdorff measure, which thus proves (5). (cid:3) Let us now finish the proof of Theorem 3.19: CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 23
Proof of Theorem 3.19.
Let us begin by remarking that because of Lemma 3.22 we need only consider thecase when our neck region N satisfies inf r x >
0. Indeed, in the general case we may approximate N by asequence ˜ N a of such neck regions with ˜ N a → N . If we can therefore find for each of these neck regionssmooth approximations N ai ⊆ B ( p i ) with N ai → ˜ N a , then by taking a diagonal subsequence we have ap-proximated N itself in the sense of Theorem 3.19.Thus let us assume N satisfies inf r x >
0. In particular notice that C = C + is a finite set in this case. Let φ i : B ( p ) → B ( p i ) be the ǫ i -Gromov Hausdorff maps. For i sufficiently large with ǫ i << inf r x note thatwe can consider the center points C i ≡ { φ i ( x ) } x ∈ C with the radius function r x , i ≡ r φ − i ( x ) . With C finite it isthen an easy exercise to check that N i ≡ B \ B r x , i ( C i ) are ( δ i , τ i )-neck regions which satisfy the criteria ofthe Theorem, as claimed. (cid:3) Reifenberg and Lower Ahlfor’s Regularity.
In this section we build a Reifenberg type map for thecenter points of our neck regions. The constructions of this section generalize the usual Reifenberg construc-tion of [Rei60], [ChCo2] in that the mapping is built with respect to a general covering. The constructionof this section is related to the constructions of [NV15]. Our main application of this will be in Section 6 toprove the lower Ahlfor’s regularity bound of Theorem 3.10. Let us begin with the precise construction:
Theorem 3.24.
Let ( M nj , g j , p j ) GH → ( X , d , p ) satisfy Vol( B ( p j )) > v > with Ric j ≥ − δ and let N ≡ B ( p ) \ B r x ( C ) a ( δ, τ ) -neck region. For each ǫ > if τ < τ ( n ) and δ ≤ δ ( n , v , ǫ ) , then there exists a map Φ : C → R n − such that(1) (1 − ǫ ) d ( x , y ) ǫ ≤ | Φ ( x ) − Φ ( y ) | d ( x , y ) ≤ (1 + ǫ ) d ( x , y ) − ǫ .(2) For each x ∈ C and r x ≤ r with B r ( x ) ⊆ B there exists s x , r = s > such that (1 − ǫ ) (cid:16) d ( x , y ) r (cid:17) ǫ ≤ s − | Φ ( x ) − Φ ( y ) | r − d ( x , y ) ≤ (1 + ǫ ) (cid:16) d ( x , y ) r (cid:17) − ǫ .(3) If s x ≡ s x , r x then { B τ s x ( Φ ( x )) } are disjoint while we have the covering B − ǫ (0 n − ) ⊆ S B s x (cid:0) Φ ( x ) (cid:1) .(4) | Lip s x | < ǫ s x r x .Remark . Condition (1) tells us that the mapping Φ is uniformly bih¨older, while condition (2) tells us thatup to rescaling Φ continues to be uniformly bih¨older on all scales. It follows from (1) that s ∈ ( r + α , r − α ). Remark . Note that in particular condition (2) tells us that if we consider the rescaled map Φ : B ( x ) ⊆ r − M → s − R n − then Φ is an ǫ -GH map. Remark . The only manner in which the neck region structure is used is to see that C is a reifenberg set.Indeed, it is not hard to check that the constructions of this theorem work for a general reifenberg type set,with the exception of (4), and need not arise as a neck region in a space with lower Ricci bounds. Proof.
Let us choose the constants η ≡ − n ǫ and δ ≤ η , and let us consider the scales t β ≡ η β . We willessentially prove the Theorem inductively on β . More precisely, for each x ∈ C let us define r β x ≡ max { t β , r x } and consider the tubular neighborhood C β ≡ B r β x ( C ). Our goal is to build maps Φ β : C β → R n − such thatthe following hold:( β. Φ : C = B ( C ) → R n − is a η − GH map , ( β.
1) for each x ∈ C ∃ s β x > Φ β x ≡ r β x ( s β x ) − Φ β : B r β x ( x ) → R n − is a η r β x − GH map , ( β.
2) for each x ∈ C we have (cid:12)(cid:12)(cid:12) Φ β − Φ β − (cid:12)(cid:12)(cid:12) < η s β x on B r β x ( x ) . (3.28)Before building the maps Φ β let us check to see that we can finish the Theorem from their construction.Indeed, notice from ( β.
1) that we get(1 − η ) s β y r β y < s β x r β x < (1 + η ) s β y r β y for any y ∈ B r β − x ( x ) . (3.29)and from ( β.
2) we get (1 − η ) s β − x r β − x < s β x r β x < (1 + η ) s β − x r β − x . (3.30)Using the above with ( β.
2) again we also get for each x ∈ C the estimate (cid:12)(cid:12)(cid:12) Φ β x − Φ β − x (cid:12)(cid:12)(cid:12) < η r β x on B r β x ( x ) . (3.31)From (3.30) and ( β.
0) we can compute ( r β x ) η < s β x r β x < ( r β x ) − η , (cid:16) r β x r α x (cid:17) η s α x r α x < s β x r β x < (cid:16) r β x r α x (cid:17) − η s α x r α x . (3.32)Now let us iterate ( β.
2) in order to compare Φ α and Φ β to compute (with α < β say) (cid:12)(cid:12)(cid:12) Φ α − Φ β (cid:12)(cid:12)(cid:12) < η s α x on B r β x ( x ) . (3.33)In particular, this gives us uniform bounds on Φ β on C which are independent of β , so after possibly passingto a subsequence we can limit Φ β → Φ : C → R n − which satisfies (cid:12)(cid:12)(cid:12) Φ α − Φ (cid:12)(cid:12)(cid:12) ( x ) < η s α x for any x ∈ C . (3.34)Let us focus on first proving the H ¨older continuous estimate of (1) for the mapping Φ , the proof of (2) iscompletely analogous. So indeed, let x , y ∈ C with β such that r β x ≥ r β y is the smallest radius such that r β x > d ( x , y ). Then we can compute | Φ ( x ) − Φ ( y ) | ≤ | Φ ( x ) − Φ β ( x ) | + | Φ β ( y ) − Φ ( y ) | + | Φ β ( x ) − Φ β ( y ) |≤ η s β x + η s β y + ( d ( x , y ) + η r β x ) ( r β x ) − s β x ≤ (( r β x ) − d ( x , y ) + η ) · s β x ≤ (1 + η ) d ( x , y ) − η , (3.35) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 25 where in the last lines we have used (3.30), (3.31) and that r β + x = η r β x ≤ d ( x , y ). A similar computation and(3.32) give us the lower bound: | Φ ( x ) − Φ ( y ) | ≥ | Φ β ( x ) − Φ β ( y ) | − | Φ ( x ) − Φ β ( x ) | − | Φ β ( y ) − Φ ( y ) |≥ ( d ( x , y ) − η r β x ) ( r β x ) − s β x − η s β y ≥ (1 − η ) d ( x , y ) · ( r β x ) η − η ( r β x ) + η ≥ (1 − η ) d ( x , y ) + η , (3.36)which finishes the proof of (1) and hence (2) by a verbatim argument which uses the second equation in(3.32). The proof of (4) then follows from (3.30). Therefore the main point is to check that Φ satisfies thecovering estimate of (3). Indeed, it follows from (2) and ( n
3) that for any x ∈ C and r ≥ r x that Φ (cid:0) C ∩ B r ( x ) (cid:1) is ( τ + η ) s x , r -dense in B s x , r (cid:0) Φ ( x ) (cid:1) . (3.37)So now let us assume the covering statement of (3) fails, and try and find a contradiction. Therefore let¯ y ∈ B − ǫ (0 n − ) be a point such that ¯ y < S B s x ( Φ ( x )). Let y ∈ C be such that d (¯ y , Φ ( y )) = min x d (¯ y , Φ ( x )) . (3.38)Note that d (¯ y , Φ ( y )) > s y by our covering assumption. Now let us choose r > r y to be the largest radius suchthat d (¯ y , Φ ( y )) ≤ s x , r . Note by the continuity condition of (3.32) we have that d (¯ y , Φ ( y )) > (2 − η ) s x , r .However by (3.37) we then have that there must exist z ∈ C for which d (¯ y , Φ ( z )) < d (¯ y , Φ ( y )), which is ourdesired contradiction, and thus proves (3).Therefore we have shown that if the inductive condition ( β ) holds for all β ∈ N then our main Theoremfollows. Let us now focus on proving this inductive statement. To accomplish this we must first construct Φ , and then show the inductive step which will build Φ β + from Φ β . Let us begin with the construction of Φ . Indeed, by assumption there exists a δ -GH map φ : B ( p ) ∩ C → R n − . For δ ≤ η let us simply define Φ to be this φ restricted to C .Assume now that Φ β has been constructed to satisfy ( β.
1) and ( β. Φ β + . Thusfor each x ∈ C let φ x : B r β x ( x ) ∩ C → R n − be a η r β x -GH map. In order to normalize our mappings let uschoose s β + x > v x ∈ R n − and R x ∈ O ( n −
4) so that( s β + x , v x , R x ) ≡ arg inf ( s , v , R ) (cid:16) inf B r β x ( x ) ∩ C β (cid:12)(cid:12)(cid:12)(cid:12) Φ β − (cid:0) ( r β + x ) − s R ◦ φ x + v (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:17) , (3.39)and let us define the mapping φ ′ x ≡ ( r β + x ) − s β + x R x ◦ φ x + v x . Observe by the properties of Φ β and φ x wenecessarily have ( s β + x ) − r β + x φ ′ x : B r β + x ( x ) → R n − is a η r β + x -GH map. , | φ ′ x − Φ β | < η s β + x on B r β x ( x ) . (3.40) Now let us choose a maximal Vitali subcovering of C ⊆ S B r β x ( x ) such that { B − r β j ( x j ) } are disjoint. Thenwe can build a partition of unity { p j } with X j p j ( y ) = y ∈ [ x B r β x ( x ) = C β , supp( p j ) ⊆ B r β j ( x j ) . (3.41)We now build our mapping Φ β + : C β + → R n − to be defined by Φ β + ( x ) ≡ X j p j ( x ) φ ′ x j ( x ) . (3.42)It is now a short exercise using (3.40) to see that Φ β + satisfies ( β.
1) and ( β. (cid:3)
4. P
ROVING THE L - CURVATURE BOUND ON δ -N ECK R EGIONS
Recall that the proof of Theorem 3.10 will be done by a rather involved induction scheme, so that inthe end each of the conclusions of Theorem 3.10 will be proved simultaneously. This section is thereforededicated to proving the L curvature bound on neck regions under the assumption that we have alreadyproved some Ahlfor’s regularity bound. Precisely, our main result in this section is the following: Theorem 4.1.
Let ( M n , g , p ) satisfy Vol( B ( p j )) > v > with δ ′ , B > fixed. Then if τ < τ ( n ) and δ < δ ( n , v , τ, δ ′ , B ) are such that N = B ( p ) \ S x ∈ C B r x ( x ) is a ( δ, τ ) -neck region for which(1) | Rc | < δ ,(2) For each x ∈ C and r x < r with B r ( x ) ⊆ B we have B − r n − < µ (cid:0) B r ( x ) (cid:1) < Br n − ,then for each B r ( x ) ⊆ B we have the curvature estimate r − n R N − ∩ B r ( x ) | Rm | ≤ δ ′ .Remark . In particular, we have that R N − ∩ B | Rm | ≤ δ ′ .The above is the key estimate required for the proof of the L curvature bound of Theorem 3.10.3. Thekey result toward the proof of Theorem 4.1 is the following local L curvature estimate: Proposition 4.3.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > . Then if τ < τ ( n ) and δ < δ ( n , v , τ, B ) are suchthat N = B ( p ) \ S x ∈ C B r x ( x ) is a ( δ, τ ) -neck region for which(1) | Rc | < δ ,(2) For each x ∈ C and r x < r with B r ( x ) ⊆ B we have B − r n − < µ (cid:0) B r ( x ) (cid:1) < Br n − ,then for any y ∈ N − with d ( y , C ) = r, we have Z B r ( y ) | Rm | ≤ C ( n , v , B , τ ) δ r n − + C ( n , v , B , τ ) Z B r ( y ) (cid:12)(cid:12)(cid:12) H r ( x ) − H r / ( x ) (cid:12)(cid:12)(cid:12) d µ ( x ) . (4.4) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 27
We will give a proof of Proposition 4.3 and Theorem 4.1 in the end of this section. The proof of Propo-sition 4.3 relies on a few new and sharp estimates involving about the H -volume and a pointwise curvatureestimate in Lemma 4.5.4.1. Pointwise Riemann curvature estimates.
In this subsection, we get a pointwise bound on the curva-ture based on bounds on its Ricci curvature and control over some splitting functions. This estimate willprovide us a manner in which to prove the local L curvature estimate if we can find enough linear indepen-dently functions with good estimates. Our main result of this subsection is the following: Lemma 4.5 (Curvature estimate of Level sets) . Let ( M n , g , p ) be a Riemannian manifold with a map Φ = ( h , f , · · · , f n − ) : B r ( p ) → R + × R n − . Assume f = h − P n − i = f i ≥ c r > and | f i | ≤ c − r onB r ( y ) ⊂ B r ( p ) . Let A = ( a i j ) be a ( n − × ( n − symmetric matrix with a i j ( x ) = h∇ f i ( x ) , ∇ f j ( x ) i . Assumefurther | det A | ( x ) ≥ c > and |∇ f i | ≤ c on B r ( y ) . Then for any x ∈ B r ( y ) , we have the following scaleinvariant estimate r | Rm | ( x ) ≤ C ( n , c , c ) r | Rc | + r |∇ h | + n − X i = r |∇ f i | + F + F , where F = |∇ h − g | + P n − i = r |∇ f i | .Remark . The picture here is that B r ( p ) ≈ R n − × C ( Z ) with B r ( y ) away from the cone point, where f , ..., f n − represent linear splitting functions and f is the distance function coming from the cone factor.Therefore h represents the square distance coming from the cone point itself. Proof.
Let
Ψ = ( f , f , · · · , f n − ) : B r ( p ) → R n − . For any x ∈ B r ( y ), since | det A | ≥ c >
0, byconstant rank theorem, we have N = Ψ − (cid:16) Ψ ( x ) (cid:17) ∩ B r ( y ) is a smooth 3 dimensional submanifold of M . Let { e , e , e } be horizontal vector fields on M which form a (local) orthonormal basis of the level set N . Then {∇ f , ∇ f , · · · , ∇ f n − , e , e , e } form a basis of M at x . Note that to control the curvature | Rm | ( x ) we onlyneed to control the curvature tensor with the { e j } factors in the basis. Indeed, sinceRm( X , Y , ∇ f , Z ) = ∇ f ( X , Y , Z ) − ∇ f ( Y , X , Z ) , then if the curvature Rm involves one normal direction ∇ f i for some 1 ≤ i ≤ n − | Rm( · , · , ∇ f i , · ) | ≤ |∇ f i | . Similarly for i =
0, we have that | Rm( · , · , ∇ f , · ) | = (2 f ) − | Rm( · , · , ∇ f , · ) | ≤ f − |∇ f | . Hence, toprove the lemma, it suffices to estimate Rm( e i , e j , e k , e l ) with i , j , k , l = , ,
3. We will use Gauss-Codazziequation to estimate these terms. Let us consider the second fundamental form Π . Denote by g N therestricted metric on N . By definition, we know at x that Π = a ∇ f + n − X i = a i ∇ f i , (4.7)where a i depend on the matrix A − and hence | a j | ≤ C ( n , c , c ). Since ∇ f = (2 f ) − (cid:0) ∇ h − P n − i = ∇ f i ⊗∇ f i − P n − i = f i ∇ f i (cid:1) and Π only takes values in the horizontal vector fields, the term 2 P n − i = ∇ f i ⊗ ∇ f i givesno contribution to Π . Therefore Π = a f − g N + E with | E | ≤ C ( n , c , c ) (cid:16) r − |∇ h − g | + P n − i = |∇ f i | (cid:17) . By Gauss-Codazzi equation Rm( X , Y , Z , W ) = Rm N ( X , Y , Z , W ) + h Π ( Y , W ) , Π ( X , Z ) i − h Π ( X , W ) , Π ( Y , Z ) i , wehave Rc N = Rc + a f − g N + E , (4.8)with | E | ≤ C ( n . c , c ) (cid:16) r − |∇ h | + P n − i = |∇ f i | + r − | E | + | E | (cid:17) . Thus we have the scalar curvature estimate R N = a f − + E with | E | ≤ C ( n , c , c ) ( | Rc | + | E | ). On the other hand, since dim N =
3, we haveRm N = − R N g N ◦ g N − Rc N − R N g N ! ◦ g N , (4.9)where g N ◦ g N is the Kulkarni-Nomizu product, see [P]. ThenRm N = − a f g N ◦ g N + E , (4.10)with | E | ≤ C ( n , c , c ) | E | . Using the Gauss-Codazzi equation again, we finally arrive at | Rm( e i , e j , e k , e l ) | ≤ C ( n , c , c ) (cid:16) | E | + | E | + | E | (cid:17) . (4.11)Therefore, combining with the estimates on normal direction, we have | Rm | ( x ) ≤ C ( n , c , c ) r − |∇ h | + n − X i = |∇ f i | + | E | + r − | E | + | E | ( x ) (4.12) ≤ C ( n , c , c ) | Rc | + r − |∇ h | + n − X i = |∇ f i | + r − | E | + | E | ( x ) . (4.13) (cid:3) By noting the curvature estimate above, to prove Proposition 4.3, we only need to find n − H -volume on manifold with Ric ≥ − δ ( n − . Let ( M , g , p ) be pointed manifold with Rc ≥ − ( n − δ and Vol( B ( p )) ≥ v. We define the H -volume as H δ t ( p ) = Z M (4 π t ) − n / e − d x , p )4 t dx − Z t s Z M ( L δ ( x ) − n )(4 π s ) − n / e − d x , p )4 s dxds , (4.14)where L δ ( x ) = + n − d ( p , x ) √ δ coth (cid:0) √ δ d ( p , x ) (cid:1) . Note that L ≡ n for spaces with nonnegativeRicci curvature, so that this second term is purely a correction term. On a first reading of this sectionwe recommend the reader sets δ =
0, most of the formulas simplify quite a bit in this case. By directcomputation, we have ∂ t H δ t ( p ) = Z M (cid:0) d ( p , x )4 t − L δ ( x )4 t (cid:1) (4 π t ) − n / e − d x , p )4 t dx . (4.15) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 29
Noting that ∆ e − d x , p )4 t = (cid:18) − t ∆ d ( p , x ) + d ( p , x )4 t (cid:19) e − d x , p )4 t , we have ∂ t H δ t ( p ) = t Z M (cid:0) ∆ d ( p , x ) − L δ ( x ) (cid:1) (4 π t ) − n / e − d x , p )4 t dx ≤ , (4.16)where we use the Laplacian comparison in the last inequality. For simplicity of notation, we will drop the δ of H δ t when there is no confusion, but one should keep in mind the dependence of H t on the lower Riccicurvature bound − ( n − δ .Let us consider the following heat flow ∂ t f t = ∆ f t − n (4.17) f ( x ) = nU ( d ( p , x )) , (4.18)where U ( r ) = R r sinh − ( n − ( √ δ t ) R t sinh n − ( √ δ s ) dsdt . We begin by recording some basic points about f t which will be useful: Lemma 4.19. If ( M , g ) satisfies Rc ≥ − ( n − δ and f t is as in (4.17) then we have the following:(1) ∆ f t ≤ n for all t ≥ .(2) − nt ≤ f t ≤ f = nU ( d ( p , x )) .(3) If Q t ≡ e − n − δ t ( |∇ f | − f − nt ) − n − δ ( t f + nt ) , then Q t ≤ for all t ≥ .(4) If P t ( x ) = e − n − δ t ( |∇ f | − f + t ( ∆ f − n )) − n − δ ( t f + nt ) + t δ ( n − ∆ f − n ) then wehave P t ≤ for all t ≥ .Remark . The last two expressions will be used to give sharp estimates on the gradient and hessian ofour smooth approximation.
Proof.
By Laplacian comparison, we have ∆ U ( d ( p , x )) ≤
1. On the other hand, since ( ∂ t − ∆ )( ∆ f − n ) = ∂ t f ≤ ∂ t − ∆ )( f + nt ) = f t ( x ) ≥ − nt .For the gradient estimate, we have by direct computation that( ∂ t − ∆ )( |∇ f | − f ) = − |∇ f | − ∇ f , ∇ f ) + n ≤ n − δ |∇ f | + n . (4.21)By noting our lower bound f t + nt ≥ ∂ t − ∆ ) Q t ( x ) ≤ . (4.22)Combined with Q ≤ ∂ t − ∆ ) P t ≤ − e − n − δ t |∇ f − g | . (4.23)Combined with P ≤ (cid:3) Let us observe that a consequence of (3) above is the gradient estimate |∇ f | ≤ (cid:0) + n − δ t e n − δ t (cid:1)(cid:0) f + nt (cid:1) ≤ (cid:0) + n − δ t e n − δ t (cid:1)(cid:0) f + nt (cid:1) . (4.24)Let us now prove a couple more refined estimates on f t that depend on the pinching of our H -volume.Precisely, we have the following: Lemma 4.25.
Let ( M , g , p ) satisfy Vol( B ( p )) ≥ v > and Ric ≥ − ( n − δ . Denote by η = | H r / ( p ) − H r ( p ) | the H -entropy pinching at scale r, then we have the estimates:(1) > r > B r ( p ) |∇ f t − g | dydt ≤ C ( n , v)( δ r + η ) . (2) For any t ≤ r , we have sup y ∈ B r ( p ) | f t − f | ( y ) ≤ C ( n , v) ǫ ( η ) r , where ǫ ( η ) → if η → .(3) For any t ≤ r , we have > B r ( p ) | ∆ f t − n | + r − > B r ( p ) |∇ f − ∇ f t | ≤ C ( n , v) η. Moreover, if we assume further the Ricci curvature upper bound | Ric | ≤ ( n − δ and harmonic radiusr h ( x ) ≥ r for some x ∈ B r ( p ) , then for any r / ≤ t ≤ r , we have sup B r / ( x ) |∇ f t − g | + r ? B r / ( x ) |∇ f t | ≤ C ( n , v)( δ r + η ) . (4.26) Proof.
Consider the heat flow (4.17), then we have ( ∂ t − ∆ )(2 n − ∆ f ) =
0. Thus we have(2 n − ∆ f t )( x ) = Z (2 n − ∆ f )( y ) ρ t ( x , dy ) . Then by the heat kernel estimate of Theorem 2.19, for any t ≤ r , one can compute ? B r ( p ) | n − ∆ f t | ≤ C ( n , v) η . Hence, we have ? B r ( p ) | f − f t | ≤ Z t ? B r ( p ) (2 n − ∆ f s ) ds ≤ C ( n , v) η t . (4.27)By the gradient estimate of f t in (4.24), we therefore have sup B r ( p ) | f − f t | ≤ C ( n , v) ǫ ( η ) r . The gradient L in (3) follows from integrating by part and the L estimate of (2 n − ∆ f ). In fact, let φ be a cutoff functionas in [ChCo1] with support in B r ( p ) and φ ≡ B r ( p )) and r |∇ φ | + r | ∆ φ | ≤ C ( n ). By integrating byparts, we have Z M φ |∇ f − ∇ f t | ≤ Z M |∇ φ | · | f − f t | · φ |∇ f − ∇ f t | + Z M φ | ∆ f − ∆ f t | · | f − f t | (4.28) ≤ Z M φ |∇ f − ∇ f t | + C ( n ) r − Z B r ( p ) | f − f t | + sup B r ( p ) | f − f t | Z B r ( p ) | ∆ f − ∆ f t | . Combining with (4.27) and sup B r ( p ) | f − f t | ≤ C ( n , v) ǫ ( η ) r and the L estimates of | ∆ f t − n | , we have R M φ |∇ f − ∇ f t | ≤ C ( n , v) η r + n . Hence, we prove (2) and (3). CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 31
To prove (1), recall that P t ( x ) = e − n − δ t ( |∇ f | − f + t ( ∆ f − n )) − n − δ ( t f + nt ) + t δ ( n − ∆ f − n )satisfies P t ≤ ϕ be a cutoff function as in [ChCo1] with support in B r ( p ) and ϕ ≡ B r ( p )) and r |∇ ϕ | + r | ∆ ϕ | ≤ C ( n ). Multiplying such ϕ to (4.23) and integrating by parts, we have Z r ? B r ( p ) |∇ f − g | ≤ C ( n , v) ? B r ( p ) | P r | + C ( n , v) ? r ? B r ( p ) | P t | . (4.29)To estimate > B r ( p ) | P t | , one only needs to estimate > B r ( p ) (cid:12)(cid:12)(cid:12)(cid:12) |∇ f | − f (cid:12)(cid:12)(cid:12)(cid:12) . This can be controlled by consideringthe evolution of f t − f . In fact, we have( ∂ t − ∆ )( f − f ) = (4 n + f − f ) − f ( ∆ f − n ) − |∇ f | − f ) + |∇ f | − f ) . (4.30)Let ψ be a cutoff function as in [ChCo1] with support in B r ( p ) and ψ ≡ B r ( p )) and r |∇ ψ | + r | ∆ ψ | ≤ C ( n ). By noting 0 ≤ (4 f − |∇ f | )( y ) ≤ C ( n ) δ d ( p , y ) for d ( p , y ) ≤
10, we can show Z r / Z M e − n − δ t ( |∇ f | − f ) ψ ≥ − C ( n , v)( δ r + η ) r + n . (4.31)Using the above, the L estimate on the laplacian of f t and | f t | ≤ C ( n ) r on B r ( p ) for t ≤ r /
2, we have Z r / Z M P t ψ ≥ − C ( n , v)( δ r + η ) r + n . Noting that P t ≤
0, we have ? r / ? B r ( p ) | P t | ≤ C ( n , v)( δ r + η ) r , (4.32)which finishes the proof of (1).Now we wish to prove the estimates of (4.26). Indeed, under the assumption r h ( x ) ≥ r we have that ? B r / ( x ) | Rm | q < C ( n , q ) , (4.33)for all q < ∞ . Using this, the estimates of (4.26) are fairly standard, so we will only sketch the argument.Denote H f = ∇ f − g , then we can compute( ∂ t − ∆ ) | H f | = − |∇ f | + Rm ∗ H f ∗ H f + Ric ∗ H f + ∇ (Ric( ∇ f , · )) ∗ H f , (4.34)where ∗ means tensorial linear combinations and the exact expression can be computed as in Lemma 5.28.Then we can apply a standard parabolic moser iteration using (4.33) and (1) in order to conclude the point-wise estimate in (4.26). In order to conclude the L estimate on ∇ f , let us simply multiply (4.34) by acutoff function and integrate using (4.33) and the pointwise estimate on | H f | , which finishes the sketch. (cid:3) H -volume and Local L curvature estimates. The main purpose of this subsection is to prove thelocal L curvature estimate in Proposition 4.3. The key ingredient is the parabolic estimate of H -volumein Lemma 4.25 and pointwise curvature estimate in Lemma 4.5. First, let us introduce the concept ofindependent points Definition 4.35 (( δ, ρ )-independent points) . Assume B r ( x ) is δ r -GH close to B r (0 n − , y ) ⊂ R n − × C ( S / Γ )for some δ ≤ − with δ r -GH map ι : B r (0 n − , y ) → B r ( x ) and ι (0 n − , y ) = x . We say n − { x , x , · · · , x n − } ⊂ B δ r (cid:0) ι ( B r (0 n − ) ×{ y } ) (cid:1) are ( δ, ρ )-independent points on B r ( x ), if { x i } i ≥ ⊂ B r ( x ) \ B ρ r ( x )and { x i } i ≥ k + ⊂ B r ( x ) \ B ρ r (cid:16) ι (cid:0) B r (0 k ) × (0 n − − k , y ) (cid:1)(cid:17) with { x , · · · , x k } ⊂ B ρ r (cid:16) ι (cid:0) B r (0 k ) × (0 n − − k , y ) (cid:1)(cid:17) for1 ≤ k ≤ n −
4. Here we denote by (0 k , n − − k ) = n − the origin of R k × R n − − k = R n − .Now we are ready to prove Proposition 4.3. Proof of Proposition 4.3.
The main idea for the proof is to use the pointwise curvature estimate in Lemma4.5. The key ingredient is to find n − h , u , . . . , u n − which satisfy the conditions of this Lemma.Intuitively, such h should be a square distance function and ( u , . . . , u n − ) splitting functions which form acone map to R n − × C ( S / Γ ). Based on this observation, we will construct such functions in detail in thefollowing paragraphs. Claim 1:
Let η ≡ > B r ( y ) (cid:12)(cid:12)(cid:12) H r ( x ) − H r / ( x ) (cid:12)(cid:12)(cid:12) d µ ( x ). For τ ≤ τ ( n ) and δ ≤ δ ( n , B , τ ), there exists ρ ( n , B , τ )and L ( n , τ, B ) such that there exists ( δ, ρ )-independent points { x , · · · , x n − } ⊂ ˜ C ∩ B r ( y ), where ˜ C = { x ∈ C ∩ B r ( y ) : (cid:12)(cid:12)(cid:12) H r ( x ) − H r / ( x ) (cid:12)(cid:12)(cid:12) ≤ L ( n , τ, B ) η } . Proof of Claim 1:
We divide the proof into two cases, namely when r ≥ C ( n , B ) max x ∈ C ∩ B r ( y ) r x is largecompared to the singular ball radii or the opposite case of r < C ( n , B ) max x ∈ C ∩ B r ( y ) r x .Case 1: If r ≥ C ( n , B ) max x ∈ C ∩ B r ( y ) r x , then we prove the claim by constructing the points induc-tively. First, we consider L ( n , τ, B ) ≥ in the definition of ˜ C , and then µ ( ˜ C ) ≥ (1 − − ) µ ( B r ( y )).By the Ahlfors assumption of µ , we have µ ( ˜ C ) ≥ ¯ C ( n , B ) r n − >
0. Since ˜ C ∩ B r ( y ) , ∅ , fix x ∈ ˜ C ∩ B r ( y ). Then µ ( B r ( x ) ∩ ˜ C ) ≥ ¯ C ( n , B ) r n − for some ¯ C ( n , B ). By the definition of ( δ, τ )-neck region, B r ( x ) is δ r -GH close to B r (0 n − , y ) ⊂ R n − × C ( S / Γ ) with δ r -GH map ι : B r (0 n − , y ) → B r ( x )such that ι (0 n − , y ) = x . Moreover, by the definition of ( δ, τ )-neck region and cone-splitting Theo-rem 2.11, by choosing δ small, ˜ C ∩ B r ( x ) ⊂ B δ r (cid:0) ι ( B r (0 n − ) × { y } ) (cid:1) . Let ρ > ρ r ≥ r x , then µ ( B ρ r ( x )) ≤ C ( n , B ) ρ n − r n − . If ρ is sufficiently small, then µ ( B r ( x ) ∩ ˜ C \ B ρ r ( x )) ≥ n − n ¯ C ( n , B ) r n − . Then there exists x ∈ B r ( x ) ∩ ˜ C \ B ρ r ( x ). Without loss of generality, we assume x ⊂ B ρ r (cid:16) ι (cid:0) B r (0 ) × (0 n − , y ) (cid:1)(cid:17) . By the δ r -GH approximation, there exists C ( n ) ρ − many balls with ra-dius 2 ρ r covering B ρ r (cid:16) ι (cid:0) B r (0 ) × (0 n − , y ) (cid:1)(cid:17) . In particular, by the Ahlfor’s regularity assumption and ρ r ≥ max x ∈ B r ( y ) r x , we have that µ (cid:16) B ρ r (cid:16) ι (cid:0) B r (0 ) × (0 n − , y ) (cid:1)(cid:17)(cid:17) ≤ C ( n , B ) ρ n − r n − . For ρ small, we have µ (cid:16) B r ( x ) ∩ ˜ C \ (cid:16) B ρ r (cid:16) ι (cid:0) B r (0 ) × (0 n − , y ) (cid:1)(cid:17)(cid:17)(cid:17) ≥ n − n ¯ C ( n , B ) r n − . (4.36)Thus we can choose x ∈ (cid:16) B r ( x ) ∩ ˜ C \ (cid:16) B ρ r (cid:16) ι (cid:0) B r (0 ) × (0 n − , y ) (cid:1)(cid:17)(cid:17)(cid:17) . Without loss of generality, weassume { x , x , x } ⊂ (cid:16) B ρ r (cid:16) ι (cid:0) B r (0 ) × (0 n − , y ) (cid:1)(cid:17)(cid:17) . By choosing ρ ( n , B ) small and δ ( n , B ) small, continuing CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 33 the construction, we can find x , · · · , x n − ∈ ˜ C . By the construction, such { x , · · · , x n − } satisfy Definition4.35. Hence, we complete the first part of the proof.Case 2: Let us consider the case r ≤ C ( n , B ) max x ∈ C ∩ B r ( y ) r x . Denote by r ¯ x = max x ∈ C ∩ B r ( y ) r x . By thelipschitz condition ( n
4) in the definition of neck region, we have for any x ∈ C ∩ B r ( y ) and δ ≤ δ ( n , B ) that1 / r ¯ x ≤ r x ≤ r ¯ x . Therefore by the Vitali condition of our covering we have C ∩ B r ( y ) < C ( n , B , τ ), andby the definition of µ we then have µ ( B r ( y )) < L ( n , B , τ ) µ ( x ) for each x ∈ C ∩ B r ( y ). In particular, for L ( n , B , τ ) we have that our integral η estimate becomes a pointwise estimate and ˜ C ∩ B r ( y ) = C ∩ B r ( y )must be the whole set.Therefore, by the definition of neck region and choosing ρ ( n , B , τ ) small, it’s not hard to find n − { x , · · · , x n − } such that there are ( δ, ρ )-independent by choosing δ small. Indeed, this just follows fromthe (n3) condition in the definition of neck region. More carefully, let us first choose x ∈ C ∩ B r ( y ).Then we have a δ r -GH map ι : B r (0 n − , y c ) ⊂ R n − × C ( S / Γ ) → B r ( x ) with ι (0 n − , y c ) = x . By thecondition (n3), we choose x i ∈ C ∩ B r ( x ) such that ι ((0 i , r ) , y c ) ∈ B τ r ( x i ) for 1 ≤ i ≤ n −
4, where(0 i , r ) = (0 , , · · · , r , , · · · , ∈ R n − with the i th factor equals r . By choosing δ small and τ ≤ − , thesepoints are ( δ, − )-independent. Hence we finish the whole proof of the claim. (cid:3) Now recall that η = > B r ( y ) (cid:12)(cid:12)(cid:12) H r ( x ) − H r / ( x ) (cid:12)(cid:12)(cid:12) d µ ( x ). Applying Lemma 4.25 to each x i , we have n − f i , t such that > r > B r ( x i ) |∇ f i , t − g | dxdt ≤ C ( n , v , B , τ )( δ r + η ) . For 1 ≤ i ≤ n −
4, let w i , t = (cid:16) f i , t − f , t − d ( x , x i ) (cid:17) / d ( x , x i ). Using the estimates for f i , t in Lemma 4.19, we have r ? r ? B r ( y ) |∇ w i , t | dxdt ≤ C ( n , v , B , τ )( δ r + η ) . Claim 2:
There exists a ( n − , n − D with | D | ≤ C ( n , B ) such that if ( v i , t ) ≡ ( w i , t ) D then ¯ f , t ≡ f , t − P n − i = v i , t satisfies ¯ f , r ≥ C ( n , B ) r > B r ( y ). Further, if we denote v , r = q ¯ f , r on B r ( y ), then wehave min x ∈ B r ( y ) | det A | ( x ) ≥ /
2, where A ( x ) = h∇ v i , r , ∇ v j , r i ( x ) for i , j = , · · · , n − Proof of Claim 2:
We prove this claim by contradiction, therefore let us assume this is not true. Thenfor δ a → δ a , τ )-neck region N a ⊂ B ( p a ) ⊂ M a with y a ∈ N a , d ( y a , C a ) = r a and ( δ a , ρ )-independent points { x , a , · · · , x n − , a } ⊂ C a , but there is no matrix D a with | D a | ≤ C ( n , B ) satisfying the claimfor N a , where C ( n , B ) will be determined later. Let us rescale each metric g a to ˜ g a such that d ( y a , C a ) = M a → R n − × C ( S / Γ ) with y a → y ∞ and x i , a → x i , ∞ , where { x i , ∞ } is a (0 , ρ )-independentset. By the C estimate of Lemma 4.25 we have that | ˜ f i , , a − d x i , a | →
0, and hence ˜ f i , , a → ˜ f i , , ∞ ≡ d x i , ∞ . Onthe one hand, it is then clear that ˜ w i , , ∞ ≡ (cid:16) ˜ f i , , ∞ − ˜ f , , ∞ − d ( x , ∞ , x i , ∞ ) (cid:17) / d ( x , ∞ , x i , ∞ ) is a linear function.As { x i , ∞ } is a (0 , ρ )-independent set, one can choose a matrix D ∞ with | D ∞ | ≤ C ( n , ρ ) ≡ C ( n , B ) such that(˜ v i , , ∞ ) ≡ ( ˜ w i , , ∞ ) D represents the standard coordinate functions of R n − × { y c } ⊂ R n − × C ( S / Γ ) with(0 n − , y c ) = x , ∞ . Then ¯ f , , ∞ ≡ ˜ f , , ∞ − P n − i = ˜ v i , , ∞ is the distance square function d R n − ×{ y c } . Thus the( n − , n −
3) matrix A ∞ defined in B ( y ∞ ) ⊂ R n − × C ( S / Γ ) satisfies A ∞ = ( δ i j ). However, for the rescaledmetric ˜ g a we have by the harmonic radius lower bound of y a in Lemma 3.11 and the hessian estimate inLemma 4.25 that ˜ f i , , a → ˜ f i , , ∞ = d x i , ∞ in C sense on B ( y ∞ ). By choosing D a = D ∞ for a sufficientlylarge, this derives our contradiction and proves the result. (cid:3) Now we plan to use such functions to prove our expected curvature estimates by using Lemma 4.5. ByClaim 2, let us consider the map
Φ = ( h , u , · · · , u n − ) ≡ ( f , r , v , r , · · · , v n − , r ) on B r ( y ). Using theharmonic radius lower bound r h ( y ) ≥ r / f i , r in Lemma 4.25, and notingthat u i are linear combinations of the f i , r with uniformly bounded constants, we have the following scaleinvariant estimatessup B r / ( y ) |∇ h − g | + r ? B r / ( y ) |∇ h | + n − X i = sup B r / ( y ) r |∇ u i | + r ? B r / ( y ) |∇ u i | ≤ C ( n , v , B , τ )( η + δ r ) . (4.37)Moreover, by the pointwise nondegeneration of A ( x ) in Claim 2, we can now use Lemma 4.5 to deduce thecurvature estimates. In fact, for any z ∈ B r ( y ), we have scale invariant estimates r | Rm | ( z ) ≤ C ( n , v , B ) | Rc | r + r |∇ h | + n − X i = r |∇ u i | + F + F ( z ) , (4.38)where F = |∇ h − g | + P n − i = r |∇ u i | . By the pointwise hessian estimate for u i , we have r | Rm | ( z ) ≤ C ( n , v , B , τ ) | Rc | r + r |∇ h | + n − X i = r |∇ u i | + |∇ h − g | + n − X i = r |∇ u i | ( z ) . (4.39)Integrating over B r ( y ), we get r ? B r ( y ) | Rm | ≤ C ( n , v , B , τ )( η + δ r ) ≤ C ( n , v , B , τ ) (cid:16) δ r + ? B r ( y ) (cid:12)(cid:12)(cid:12) H r ( x ) − H r / ( x ) (cid:12)(cid:12)(cid:12) d µ ( x ) (cid:17) . (4.40)This completes the proof. (cid:3) Proof of the L curvature estimate on neck region. In this subsection, based on the local L curvatureestimate in Proposition 4.3, we prove Theorem 4.1. Proof of Theorem 4.1.
Since the estimates are scale invariant, without loss of generality we will assume r =
1. By the local L curvature estimate of Proposition 4.3, for any y ∈ N − with 2 s = d ( y , C ), we havethe estimate Z B s ( y ) | Rm | ≤ C ( n , v , B , τ ) δ s n − + C ( n , v , B , τ ) Z B s ( y ) | H s − H − s | ( x ) d µ ( x ) . (4.41)In order to use such an estimate, we first construct a Vitali covering. For any x ∈ N − with d ( x , C ) = s x ,consider the covering { B s x / ( x ) } of N − . We can choose a Vitali covering { B s a ( x a ) } such that { B s a / ( x a ) } are disjoint and N − ∩ B ⊆ [ a B s a ( x a ) . (4.42) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 35
Rearranging x a such that d ( x α, i , C ) = s α, i with s α, i ∈ (2 − α − , − α ], then we have N − ∩ B ⊆ [ α N α [ i = B s α, i ( x α, i ) , (4.43)and { B − s α, i ( x α, i ) } are disjoint. Moreover, by Ahlfors assumption, for any fixed α we have that ♯ i { B s α, i ( x α, i ) } ≤ C ( n , B , v) s − n α with s α = − α . Then we have Z N − ∩ B | Rm | ≤ X α X i Z B s α, i ( x α, i ) | Rm | (4.44) ≤ C ( n , v , B , τ ) X α X i δ s n − α, i + Z B s α, i ( x α, i ) | H s α, i − H − s α, i | ( x ) d µ ( x ) ≤ C ( n , v , B , τ ) X α δ s α + C ( n , v , B , τ ) X α Z B / (cid:12)(cid:12)(cid:12) H s α − H − s α (cid:12)(cid:12)(cid:12) ( x ) d µ ( x )By the monotonicity of H -volume, we have Z N − ∩ B | Rm | ≤ C ( n , v , B , τ ) δ + C ( n , v , B , τ ) Z B / (cid:12)(cid:12)(cid:12) X α ( H s α − H − s α ) (cid:12)(cid:12)(cid:12) ( x ) d µ ( x ) ≤ C ( n , v , B , τ ) δ + C ( n , v , B , τ ) Z B / (cid:12)(cid:12)(cid:12) H − H − r x (cid:12)(cid:12)(cid:12) ( x ) d µ ( x ) . (4.45)However, since N is a neck region we have that both B δ − ( x ) and B δ − r x ( x ) are Gromov-Hausdorff close to R n − × C ( S / Γ ), and therefore by volume convergence we have that | H − H − r x | ( x ) → δ → δ ≤ δ ( n , v , δ ′ , τ ), we have proved the estimates. (cid:3)
5. S
PLITTING F UNCTIONS ON N ECK R EGIONS
The main goal of this section is to do some analysis on neck regions, and more specifically to study thebehavior of splitting functions on neck regions. We will show that splitting functions are in fact better be-haved than the standard basic estimates would lead one to believe. This will be the key technical ingredientin the proof of Theorem 3.10.More precisely, with δ, δ ′ , τ, B > | Rc | < δ . (S2) N = B ( p ) \ B r x ( C ) is a ( δ, τ )-neck region . (S3) For each x ∈ C and r x < r with B r ( x ) ⊆ B we have B − r n − < µ ( B r ( x )) < Br n − . (S4) u : B ( p ) → R k is a δ ′ -splitting function . (5.1)Note that it will eventually be a consequence of Theorem 3.10 that we can take B = A ( n , τ ), and therefore( S
3) will be a redundant assumption. However, at this stage it is important to not make this assumption, as the results of this section will factor heavily in the proof of this point. The main goal of this section is toprove the following:
Theorem 5.2.
Let ( M nj , g j , p j ) → ( X , d , p ) be a limit space with Vol( B ( p j )) > v > . Then for every ǫ, B > and τ < τ ( n ) if δ, δ ′ ≤ δ ( n , ǫ, τ, B , v) is such that assumptions (S1)-(S4) of (5.1) hold, then thereexists a subset C ǫ ⊆ C ∩ B such that(b1) µ ( (cid:0) C ∩ B (cid:1) \ C ǫ ) < ǫ .(b2) For each x , y ∈ C ǫ we have that − ǫ < | u ( x ) − u ( y ) | d ( x , y ) < + ǫ .(b3) For each x ∈ C ǫ and r x ≤ r ≤ we have that u : B r ( x ) → R n − is an ǫ -splitting.Remark . Recall that we say a limit M nj → X satisfies | Ric | < δ if | Ric j | < δ j where δ j → δ . Remark . Essentially the entire section will focus on the case when X is a manifold, since by applyingthe neck approximation of Theorem 3.19 we will immediately conclude the general statement.Before continuing let us make some observations about the above result. The difficulty in proving theresult is due to the smallness of the set C . If a δ -splitting function were to have pointwise estimates on thehessian, as it does for instance in the bounded curvature case, then the result would be trivial. However, withonly L bounds on the hessian apriori, one can only use the hessian information to prove ( b
3) away from aset of codimension 2 + ǫ , which is far away from the requirements of ( b L bound on the curvature, then the most one can proveis L estimates on the hessian, see [Ch2], which are still not strong enough to prove ( b → ( b Estimates on Standard Green’s Functions on Neck Regions.
In this section we discussion standardGreen’s functions on neck regions. We will use these estimates in the next section to discuss Green’sfunctions with respect to the packing measure µ . Definition 5.5.
Given ( M n , g , p ) with N ⊆ B ( p ) a ( δ, τ )-neck region with packing measure µ = µ N , wedefine the following:(1) We denote by G x ( y ) a Green’s function at x . That is, − ∆ y G x ( y ) = δ x , where δ x is the dirac delta at x .(2) We denote by G µ ( y ) the function G µ ( y ) = R G x ( y ) d µ ( x ) a Green’s function which solves − ∆ G µ = µ .(3) We denote by b ( y ) ≡ G − µ the µ -Green’s distance function to the center points C .The intuition is that b should behave in a manner which is comparable to the Green’s function from thesingular set in R n − × C ( S / Γ ), which itself is a multiple of d ( S n − , · ) − ≈ r − h . The main result of this CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 37 subsection will be to prove just that. Precisely:
Lemma 5.6.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > . Then ∀ B > there exists δ ( n , B , v) , τ ( n , B ) ,C ( n , B , v) > and a Green function G µ such that if the assumptions (S1)-(S3) of (5.1) hold, then(1) For x ∈ N − , we have C − d ( x , C ) ≤ b ( x ) ≤ C d ( x , C ) .(2) For x ∈ N − , we have C − ≤ |∇ b | ≤ C. This subsection is dedicated to proving the above, though we will need to work through several prelimi-naries first. Let us begin by collecting together a list of useful computations:
Lemma 5.7.
Given ( M n , g , p ) with N ⊆ B ( p ) a ( δ, τ ) -neck region and b ( y ) an associated Green’s distancefunction, then for a smooth function f : M → R the following hold:(1) µ [ f ] = − R b ≤ r G µ ∆ f + r − R b = r f |∇ b | + r − R b ≤ r ∆ f .(2) µ [ f ] = − R b ≤ r G µ ∆ f + r − R b = r f |∇ b | + r ddr (cid:16) r − R b = r f |∇ b | (cid:17) .Proof. Let us observe that (cid:0) b − − r − (cid:1) vanishes on b = r , is smooth in a neighborhood of this set, andsupp n ∆ (cid:0) b − − r − (cid:1)o ⊆ { b < r } . Thus we can use standard properties of the distributional laplacian to compute Z b ≤ r ∆ f (cid:16) b − − r − (cid:17) = − µ (cid:2) f (cid:3) − Z b = r f h ∇ b |∇ b | , ∇ b − i = − µ (cid:2) f (cid:3) + r − Z b = r f |∇ b | , (5.8)which proves the first formula. To compute the second formula let us first compute ∆ b = b − |∇ b | , (5.9)and recall that the mean curvature of the b = r level set is given by div (cid:0) ∇ b |∇ b | (cid:1) . Therefore we can compute ddr r − Z b = r f |∇ b | ! = − r − Z b = r f |∇ b | + r − Z b = r h∇ f , ∇ b |∇ b | i + r − Z b = r f h∇|∇ b | , ∇ b |∇ b | i + r − Z b = r f |∇ b | div (cid:0) ∇ b |∇ b | (cid:1) = r − Z b = r ∆ f − r − Z b = r f |∇ b | + r − Z b = r f h∇|∇ b | , ∇ b |∇ b | i + r − Z b = r f |∇ b | (cid:16) ∆ b |∇ b | − h∇ b , ∇|∇ b ||∇ b | i (cid:17) = r − Z b = r ∆ f . (5.10) (cid:3) With the help of the above we can now compute the following, which will be the key use of the Green’sdistance function:
Lemma 5.11.
Given ( M n , g , p ) with N ⊆ B ( p ) a ( δ, τ ) -neck and b ( y ) = b N ( y ) the associated Green’sdistance function, then the following hold:(1) For every compactly supported f : B ( p ) → R we haveddr (cid:16) r − Z b = r f |∇ b | (cid:17) = r − Z b ≤ r ∆ f . (2) r d dr (cid:16) r − Z b = r f |∇ b | (cid:17) + ddr (cid:16) r − Z b = r f |∇ b | (cid:17) = r − Z b = r ∆ f |∇ b | − . Proof.
The proof of (1) is just (5.10) in the previous Lemma. Taking derivative of (2) in Lemma 5.7 withrespect to r , we have r d dr (cid:16) r − Z b = r f |∇ b | (cid:17) + ddr (cid:16) r − Z b = r f |∇ b | (cid:17) = ddr Z b ≤ r b − ∆ f ! (5.12) = ddr Z r ds Z b = s s − ∆ f |∇ b | − ! = r − Z b = r ∆ f |∇ b | − . (cid:3) Green function estimates on manifolds with Ricci curvature lower bound.
In order to prove Lemma5.6, we first consider the Green function G x for any center point x ∈ C in a neck region. Besides the basicexpected estimates, we need to see that at every point y ∈ N in the neck region itself there is a fixed direction v y ∈ T y M for which G x has positive gradient in the direction of v y . This will be used heavily when we inte-grate to construct G N in order to see that the gradient of G µ has a definite lower bound in the neck region.Precisely, we have the following: Lemma 5.13.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > and | Rc | < δ with N = B ( p ) \ S x ∈ C B r x ( x ) a ( δ, τ ) -neck region. For each R > if δ ≤ δ ( n , R , v) and < τ ≤ τ ( n , v) then there exists C ( n , v) > such that ∀ x ∈ C there exists a Green’s function G x such that:(1) For all y ∈ B δ − / ( p ) we have C − d − nx ( y ) ≤ G x ( y ) ≤ Cd − nx ( y ) .(2) For all y ∈ B δ − / ( p ) we have |∇ y G x | ( y ) ≤ Cd − nx ( y ) (3) For all y ∈ N − if r = d ( y , C ) and x ∈ B Rr ( y ) ∩ C then there exists a unit vector v y ∈ T y M with h∇ y G x ( y ) , v y i > c ( R , n ) r − n > , (5.14) (4) In particular, if y ∈ N − with r = d ( y , C ) and x ∈ B r ( y ) ∩ C then h∇ y G x ( y ) , v y i > c ( n ) r − n . (5.15) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 39
Proof.
For each x ∈ C , since the estimates in (1) and (2) are scale invariant, in order to estimate the Green’sfunction on the ball B δ − / ( p ) with | Rc | < δ , it suffices to construct and estimate the Green function on theball B ( p ) with | Rc | <
1. In this case we construct the Green function G x in the following manner. Let G x , ( y ) = Z ρ t ( x , y ) dt . Then by heat kernel estimate of Theorem 2.19, we can compute C − d − nx ( y ) ≤ G x , ( y ) ≤ Cd − nx ( y ). Ad-ditionally, we can compute that ∆ G x , ( y ) = R ∂ t ρ t ( x , y ) dt = ρ ( x , y ) − δ x ( y ) . Therefore let us solve G x , by ∆ G x , ( y ) = ρ ( x , y ) , (5.16)on B ( x ) with G x , ( y ) = G x , ( y ) on ∂ B ( x ). We define our Green’s function G x = G x , − G x , . We claim that G x satisfies the Lemma. Indeed, by noting the uniform bound on the heat kernel ρ ( x , y ) wemay use a standard maximal principle and Cheng-Yau gradient estimates on (5.16) as in [Ch01], in order toshow that | G x , | + |∇ G x , | < C ( n ) in B / ( p ) is uniformly bounded. Coupling with the estimates of G x , , weprove the estimates on G x . Thus we have prove (1) and (2).Now we only give a proof of (5.14), the argument is the same for (5.15). We argue by contradiction.Therefore assume for some R > δ i , τ )-neck regions N i with δ i → y i ∈ N i , − and r i = d ( y i , C i ) such that sup v ∈ T yi M , | v | = inf x i ∈ B Rri ( y i ) ∩ C i h∇ y G x i ( y i ) , v i < i − r − ni . (5.17)Scaling B Rr i ( y i ) to ball ˜ B R ( y i ) and denoting the corresponding Green function to be ˜ G x i , thensup v ∈ T yi M , | v | = inf x i ∈ ˜ B R ( y i ) ∩ C i h∇ y ˜ G x i ( y i ) , v i < i − . (5.18)To deduce a contradiction, we will show that the Green function ˜ G x i converges to a function D d − nx on R n − × C ( S / Γ ) with constant C − ( n , v) < D < C ( n , v). Since the convergence is C on the neck region dueto the harmonic radius control, we can take v y to be any vector which approximates the radial direction onthe C ( S / Γ ) factor in order to conclude the result.Thus, we only need to show the Green function ˜ G x i → D d − nx . On one hand, we notice that ˜ B δ − / i ( x i )converges to the same limit R n − × C ( S / Γ ) for any sequence x i ∈ ˜ B R ( y i ) ∩ C i . On the other hand, by (1) and(2) of the Lemma we have on ˜ B δ − / i ( x i ), we have C − d − nx i ≤ ˜ G x i ≤ Cd − nx i and |∇ ˜ G x i | ≤ Cd − nx i . By standardAscoli we have that ˜ G x i converges to a function G x on the limit space R n − × C ( S / Γ ) which satisfies theestimates C − d − nx ≤ G x ≤ Cd − nx , and |∇ G x | ≤ Cd − nx . (5.19)In fact, if we use some RCD theory it is not hard to see that G x is itself the Green’s function at x , howeverwe will prove something slightly weaker in order to avoid such techniques. Indeed, since ˜ G x i convergessmoothly on the regular part of R n − × C ( S / Γ ) we at least have that G x is harmonic away from the singular set. If we lift G x to a function G ˜ x on R n , we get away from ˜ x that G ˜ x is locally lipschitz and harmonic awayfrom a set of zero capacity. Hence, G x is harmonic away from ˜ x with the bounds (5.19). Now the onlyharmonic functions on R n \ ˜ x with estimates (5.19) are multiples of the Green’s function. Hence we have G x = D d − nx for some constant C − ( n , v) < D < C ( n , v) as claimed, which finishes the proof. (cid:3) Proof of Lemma 5.6.
Noting that G µ ( y ) = R G x ( y ) d µ ( x ) we will use the pointwise Green functionestimates in Lemma 5.13 and the Ahlfor’s regularity assumption in order to conclude the proof of Lemma5.6. Indeed, for any y ∈ N − let r ≡ d ( y , C ), and let us estimate the upper bound of G N ( y ) as follows: G µ ( y ) ≤ C Z d − nx ( y ) d µ ( x ) = C Z B r ( y ) d − nx ( y ) d µ ( x ) + C ∞ X i = Z A ir , i + r ( y ) d − nx ( y ) d µ ( x ) (5.20) ≤ Cr − n Z B r ( y ) d µ ( x ) + Cr − n ∞ X i = − i ( n − Z A ir , i + r ( y ) d µ ( x ) , ≤ C · Br − (cid:16) + X − i (cid:17) ≡ C ( n , v , B ) r − , (5.21)as claimed. To prove the lower bound of G µ we can similarly compute G µ ( y ) ≥ C − Z d − nx ( y ) d µ N ( x ) ≥ C − Z B r ( y ) d − nx ( y ) d µ ( x ) ≥ C − − n r − n µ (cid:16) B r ( x ) (cid:17) ≥ C − ( n , v , B ) r − . (5.22)Since b − ( y ) = G µ ( y ), we have C − r ≤ b ( y ) ≤ Cr , or that C − d ( y , C ) ≤ b ( y ) ≤ C d ( y , C ). Hence we haveproven (1) of Lemma 5.6. For the gradient estimate, using the same computational strategy as above wehave |∇ G µ ( y ) | = Z |∇ G x ( y ) | d µ ( x ) ≤ C Z d − nx ( y ) d µ ( x ) ≤ Cr − . (5.23)By noting that b − = G µ , then 2 b − |∇ b | = |∇ b − | = |∇ G µ | ≤ Cr − . By the upper bound estimate of b , wehave |∇ b | ≤ C . For the gradient lower bound, for any fixed unit vector v ∈ T y M , we have h∇ G µ ( y ) , v i = Z h∇ G x ( y ) , v i d µ ( x ) = Z B rR ( y ) h∇ G x ( y ) , v i d µ ( x ) + Z M \ B rR ( y ) h∇ G x ( y ) , v i d µ ( x ) . (5.24)By the gradient upper bound estimate |∇ G x ( y ) | ≤ Cd − nx ( y ), we have | Z M \ B rR ( y ) h∇ G x ( y ) , v i d µ ( x ) | ≤ Z M \ B rR ( y ) |∇ G x ( y ) | d µ ( x ) ≤ CR − r − . (5.25)On the other hand, for fixed R we have by the Green function estimates of Lemma 5.13 that if δ ≤ δ ( R , n , v),then there is a unit vector v y ∈ T y M such that Z B rR ( y ) h∇ G x ( y ) , v y i d µ ( x ) ≥ Z B r ( y ) r − n c ( n ) d µ ( x ) ≥ C ( n , B ) r − . (5.26) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 41
Therefore, combining the estimates above, we have h∇ G µ ( y ) , v y i ≥ Z B rR ( y ) h∇ G x ( y ) , v i d µ ( x ) − (cid:12)(cid:12)(cid:12)(cid:12) Z M \ B rR ( y ) h∇ G x ( y ) , v i d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ C ( B , n ) r − − C ( B , n ) R − r − . (5.27)Choosing R = R (v , n , B ) large enough we conclude h∇ G µ ( y ) , v i ≥ C ( B , n )2 r − . In particular, this gives us theestimate |∇ G µ | ( y ) ≥ Cr − and hence the desired estimate |∇ b | ( y ) ≥ C for y ∈ N − . This finishes the proofof the Lemma. (cid:3) Harmonic Function Estimates.
In this short subsection we record several estimates about harmonicfunctions. We begin with some basic computations, mainly for the convenience of the reader. Our list is thefollowing:
Lemma 5.28.
Let ∆ u = be a harmonic function on an open set. Then the following hold:(1) ∆ ∇ i u = R ia ∇ a u.(2) ∆ |∇ u | = |∇ u | + Rc ( ∇ u , ∇ u ) .(3) ∆ ∇ i ∇ j u = (cid:0) ∇ j R ia − ∇ a R i j + ∇ i R ja (cid:1) ∇ a u − R a bi j ∇ a ∇ b u + R ia ∇ a ∇ j u + R a j ∇ i ∇ a u.(4) ∆ |∇ u | = |∇ u | + (cid:10)(cid:0) ∇ j R ia − ∇ a R i j + ∇ i R ja (cid:1) ∇ a u , ∇ i ∇ j u (cid:11) − Rm ( ∇ u , ∇ u ) + Rc ( ∇ a ∇ u , ∇ a ∇ u ) . Let us now record the following hessian estimates which will play a role in subsequent subsections:
Lemma 5.29.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > with Ric ≥ − δ . Then if u : B → R n − is a δ ′ -splitting function, then the following hessian estimate holds: Z B ( x ) d − nx |∇ u | ≤ C ( n , v) , (5.30) where d x ( y ) = d ( x , y ) is the distance function.Remark . Let us make the important observation that the constant in the above estimate cannot be takento be small, simply bounded.
Proof.
Using Lemma 5.28 we have the formula12 ∆ |∇ u | + δ |∇ u | ≥ |∇ u | . (5.32)Now for x ∈ B let G x ( y ) be the Green’s function. Recall from Section 5.1 that for y ∈ B we have theestimate C ( n , v) − d − nx ( y ) ≤ G x ( y ) ≤ C ( n , v) d − nx . (5.33) Now recall as in [ChCo1] that we may build a cutoff function φ such that φ ≡ B ( x ), φ ≡ B / ( x ) and such that |∇ φ | , | ∆ φ | ≤ C ( n ). Therefore let us multiply both sides of (5.32) by φ G x and integratein order to compute Z B ( x ) d − nx |∇ u | ≤ C ( n , v) Z φ G x |∇ u | ≤ C ( n , v) Z φ G x ∆ ( |∇ u | − + C ( n , v) δ Z φ G x |∇ u | , ≤ C ( n , v) (cid:16) − |∇ u | ( x ) + δ + Z (cid:16)(cid:0) ∆ φ G x + h∇ φ, ∇ G x i (cid:1)(cid:0) |∇ u | − (cid:1)(cid:17) , ≤ C ( n , v) , (5.34)as claimed. (cid:3) By using the Green’s function estimates of Lemma 5.6 we can argue as above to prove the followingestimate:
Lemma 5.35.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > . If the assumptions (S1)-(S4) of (5.1) hold with δ, δ ′ < δ ( n ) , then Z b ≤ b − |∇ u | ≤ C ( n , B ) . Scale Invariant Hessian Estimates.
In this subsection we prove that a harmonic splitting function ona neck region continues to have scale invariantly small hessian on all scales within the neck region. This isnot generally true for a harmonic function which is not living on a neck region, and the estimate will play animportant role in our analysis in subsequent sections (particularly on making certain errors small, instead ofjust bounded, which will be crucial). In the next subsection we will prove a much stronger estimate whichbasically says this error is not just small but summably small, at least when averaged over C , which is whatis required to prove the gradient estimate in (5.105).The main result of this subsection is the following: Theorem 5.36 (Pointwise Scale Invariant Hessian Estimate) . Let ( M n , g , p ) satisfy Vol( B ( p )) > v > .Then for every ǫ > and τ < τ ( n ) if δ, δ ′ ≤ δ ( n , ǫ, τ, v) is such that assumptions (S1)-(S4) of (5.1) hold, thenwe have for every x ∈ C and r x < r < that r ? B r ( x ) |∇ u | < ǫ. (5.37) Remark . Let us observe a corollary: If y ∈ N − with 2 r = d ( y , C ) ≈ r h ≈ b , then for δ, δ ′ ≤ δ ( n , ǫ, τ, v)we can use elliptic estimates to obtain r |∇ u | < ǫ on B r ( y ). We can rephrase this as the pointwise hessianestimate |∇ u | ( y ) < ǫ r − h ( y ) on N − . CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 43
The strategy for the proof of the above will be by contradiction. Assuming the result is false, we willtake a sequence of ( δ, τ )-neck regions with δ ′ -splitting functions such that δ, δ ′ →
0, but for which the con-clusions of the theorem presumably fail. We will prove in subsection 5.3.2 that the limit harmonic functionis actually linear, and therefore has vanishing hessian. One must be able to conclude from this the almostvanishing of the hessian for the original sequence of harmonic functions. Thus a key technical result will beshown in subsection 5.3.1, which will prove, among other things, a form of H -convergence for the sequenceof harmonic functions.5.3.1. Weak Sobolev Convergence of Limiting Harmonic Functions.
The main goal of this subsection is toprove the following convergence result, which will play an important role in subsequent sections.
Theorem 5.39.
Let ( M nj , g j , p j ) → ( X , d , p ) satisfy | Ric j | ≤ n − and Vol( B ( p j )) > v > . Assume forsome R , C > that u j : B R ( p j ) → R is a sequence of harmonic functions with | u j | ≤ C. Then there existsharmonic u : B R ( p ) → R such that after possibly passing to a subsequence(1) u j → u uniformly on compact subsets of B R .(2) For each x j → x ∈ X with B r ( x j ) ⊆ B R ( p j ) we have |∇ u | ( x ) ≤ lim inf j ||∇ u j || L ∞ ( B r ( x j )) .(3) For each < p < we have that we have |∇ u j | p → |∇ u | p in measure. In fact, for each x j → x ∈ Xwith B r ( x j ) ⊆ B R ( p j ) we have that R B r ( x j ) |∇ u j | p → R B r ( x ) |∇ u | p .(4) For each < p < we have that we have |∇ u j | p → |∇ u | p in measure. In fact, for each x j → x ∈ Xwith B r ( x j ) ⊆ B R ( p j ) we have that R B r ( x j ) |∇ u j | p → R B r ( x ) |∇ u | p .Remark . Recall that a harmonic function on X is in the RCD sense, which is to say R h∇ u , ∇ v i = v . Remark . For clarity, we mean in the above that R B r ( x ) |∇ u | p ≡ R R ( X ) ∩ B r ( x ) |∇ u | p , where R ( X ) is theregular set of X . Though we will not explicitly say it, we will see as a consequence of the proof that for p < |∇ u | p cannot have a distributional term on S ( X ), so that this is reasonable. Proof.
Points (1) and (2) are standard, see for instance [ChCo2], and only require a lower bound on theRicci curvature. In a little detail, if the u j are harmonic with the uniform bounds | u j | ≤ C , then by usingstandard Cheng-Yau estimates (or one of several others), we can conclude that for every r < R there exists C r > |∇ u j | ≤ C r , (5.42)uniformly. Using a standard Ascoli argument one can conclude the existence of u : B R → R such that u j → u uniformly such that (2) holds. Using [ChCo2] one can conclude u is itself harmonic.The proofs of (3) and (4) are very similar, and both require heavily the two sided bound on the Riccicurvature and the codimension four nature of the singular set as in [ChNa15]. We will focus on (3), as the proof of (4) is essentially the same. Thus, using Theorem 7.20 of [ChNa15] one can conclude that for every p < r < R there exists C pr such that Z B R ( x j ) |∇ u j | p < C pR . (5.43)Let us define the effective regular and singular sets R r ≡ { x : r h ( x ) > r } , S r ≡ { x : r h ( x ) ≤ r } . (5.44)Recall that the singular set S ( X ) ≡ S ( X ) is a set of codimension four, and in fact we have the much strongervolume estimate Vol (cid:0) S r ∩ B R (cid:1) ≡ Vol (cid:0) B r (cid:8) r h < r (cid:9) ∩ B R (cid:1) ≤ C ǫ ( n , v , R ) r − ǫ . (5.45)Now on the regular set R r ( X ) we have, due to our lower bound on the harmonic radius, for every α < u j → u in C ,α uniformly on compact subsets. Now let x j ∈ M j → x ∈ X such that B r ( x j ) ⊆ B R , andlet us fix p <
4. For every s > C ,α convergence we just commented on that Z R s ∩ B r ( x j ) |∇ u j | p → Z R s ( X ) ∩ B r ( x ) |∇ u | p . (5.46)To finish the proof, we need to see that R S s ∩ B r ( x j ) |∇ u j | p → s →
0, uniformly in the u j . Therefore letus choose p < p ′ < R B r ( x j ) |∇ u j | p ′ < C ′ uniformly as in (5.43). Then a Cauchy inequality gives usthat Z S s ∩ B r ( x j ) |∇ u j | p ≤ Vol( S s ∩ B r ) p ′ − p (cid:0) Z S s ∩ B r ( x j ) |∇ u j | p ′ (cid:1) pp ′ ≤ C ( n , R , p , p ′ ) s p ′ − p ) , (5.47)which shows in particular that R S s ∩ B r ( x j ) |∇ u j | p → s →
0, as claimed. (cid:3)
Blow ups and the Proof of Theorem 5.36.
Let us now use the tools of the previous subsection in orderto finish the proof of Theorem 5.36. Thus, let us assume the result is false, so that for some ǫ > δ j , δ ′ j → M nj with Vol( B ( p j )) > v > N j ⊆ B ( p j ) ( δ j , τ )-neck regions with u j : B → R n − δ ′ j -splitting maps such that for some x j ∈ C j and r x j < r j < r j ? B rj ( x j ) |∇ u j | ≥ ǫ . (5.48)Let us begin by observing that r j →
0, since for any r > r ? B r ( x j ) |∇ u j | ≤ r − n Z B ( x j ) |∇ u j | ≤ r − n δ ′ j → . (5.49) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 45
Now let us consider the rescaled spaces ˜ M j = r − j M j with ˜ u j = r − j (cid:0) u j − u j ( x j ) (cid:1) : B r − j ( x j ) → R n − therenormalized maps. Notice that we have the equality r j ? B rj ( x j ) |∇ u j | = ? ˜ B ( ˜ x j ) |∇ ˜ u j | . (5.50)Now by the definition of neck regions we have that˜ M j → R n − × C ( S / Γ ) , (5.51)and by Theorem 5.39 we have that ˜ u j → ˜ u : R n − × C ( S / Γ ) → R n − , (5.52)where ˜ u is a harmonic function for which |∇ ˜ u | ≤ lim inf |∇ ˜ u j | =
1. However, we can lift ˜ u to a Γ -invariantharmonic function ˜ u : R n → R n − , and since we have a global gradient bound we can apply Liouville’stheorem to see we have that ˜ u is a linear function. In particular, |∇ ˜ u | ≡ . (5.53)By again applying Theorem 5.39 we have that r j ? B rj ( x j ) |∇ u j | = ? ˜ B ( ˜ x j ) |∇ ˜ u j | → ? ˜ B ( ˜ x j ) |∇ ˜ u | = , (5.54)which is our desired contradiction. (cid:3) Summable Hessian Estimates.
In this section we prove a vastly refined version of Theorem 5.36which tells us that the L -norm of the hessian is scale invariantly summable. As we will see, this is the keyestimate in the proof of Theorem 3.10. Theorem 5.55 ( L Summable Hessian Estimate) . Let ( M n , g , p ) satisfy Vol( B ( p )) > v > . Then for every ǫ, B > and τ < τ ( n ) if δ, δ ′ < δ ( n , ǫ, B , τ, v) is such that assumptions (S1)-(S4) of (5.1) hold, then we havethe estimate Z N − r − h |∇ u | < ǫ , (5.56) where N − is the enlarged neck region as in Definition 3.1. To prove the above we will first discuss a new superconvexity estimate in Section 5.4.1. Analysis of thederived ode together with the µ -Green’s estimates of Section 5.1 will then be applied in order to concludeTheorem 5.55 itself. The Superconvexity Equation in Einstein Case.
In this subsection we prove a new superconvexityestimate on the L Hessian with respect to the µ -Green’s distance. The estimates of this section are the cor-nerstone of Theorem 5.55 and hence Theorem 5.2. Let us begin by stating our main result for this subsection: Proposition 5.57.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > . Then for every ǫ, B > and τ < τ ( n ) ifassumptions (S1)-(S4) of (5.1) hold with δ, δ ′ ≤ δ ( n , ǫ, B , τ, v) , then for F ( r ) = r − R b = r p |∇ u | + ǫ φ N |∇ b | and H ( r ) = rF ( r ) , we have the following evolution equationr H ′′ ( r ) + rH ′ ( r ) − H ( r ) ≥ e ( r ) = Z b = r − q |∇ u | + ǫ ′ | ∆ φ N | − c ( n ) | Rm | · |∇ u | φ N + U + U ! |∇ b | − , where U = h∇ φ N , ∇ p |∇ u | + ǫ i and U = (cid:16) ∇ j ( R ia ∇ a u ) − ∇ a ( R i j ∇ a u ) + ∇ i ( R ja ∇ a u ) (cid:17) ∇ i ∇ j u (cid:17) φ N √ |∇ u | + ǫ .Remark . The function φ N is the neck cutoff function from Lemma 3.15. Remark . Let us remark that H ( r ) represents the scale invariant L norm of the hessian over the surface b = r . Our main goal will then be to get a Dini integral estimate on H ( r ), see Theorem 5.66. Proof.
Using Lemma 5.28 we can compute12 ∆ ( |∇ u | + ǫ ) = |∇ u | − h Rm ∗ ∇ u , ∇ u i + ∇ ( Rc ( ∇ u , · )) ∗ ∇ u , (5.60)where h Rm ∗ ∇ u , ∇ u i = R a bi j ∇ a ∇ b u · ∇ i ∇ j u and ∇ ( Rc ( ∇ u , · )) ∗ ∇ u = (cid:16) ∇ j ( R ia ∇ a u ) − ∇ a ( R i j ∇ a u ) + ∇ i ( R ja ∇ a u ) (cid:17) ∇ i ∇ j u . Let f = φ N p |∇ u | + ǫ . Then ∆ f = ∆ φ N p |∇ u | + ǫ + ∆ p |∇ u | + ǫ φ N + h∇ φ N , ∇ p |∇ u | + ǫ i (5.61) ≥ − p |∇ u | + ǫ | ∆ φ N | − c ( n ) | Rm | · |∇ u | φ N + U + U , where U and U are defined as in the lemma. Thus if H ( r ) = rF ( r ) = r − R b = r f |∇ b | with b − = G µ , then bythe Green formula and the computation of Lemma 5.11, we have H ′′ ( r ) + r H ′ − r H = r − Z b = r ∆ f |∇ b | − (5.62) ≥ r − Z b = r (cid:16) − p |∇ u | + ǫ | ∆ φ N | − c ( n ) | Rm | · |∇ u | φ N + U + U (cid:17) |∇ b | − . Thus we finish the proof. (cid:3)
Of course, in order to exploit the above superconvexity we will want to apply a maximum principle inorder to obtain bounds for H . To accomplish this we first need to find special solutions of the above ode tocompare to: Lemma 5.63 (Special solution) . Consider on the interval (0 , R ) the odeh ′′ ( r ) + r h ′ ( r ) − r h ( r ) = g ( r ) , CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 47 for some bounded g ( r ) which may change signs. Then we have a solution h ( r ) such that Z R h ( r ) r dr = − Z R sg ( s ) ds + R Z R s g ( s ) ds , (5.64) with h ( R ) = − R R R s g ( s ) ds and h (0) = .Proof. Let us define h ( r ) = (cid:18) − r R Rr gds − r − R r s gds (cid:19) , which one can easily check is a solution. We willsee that h ( r ) satisfies the desired estimates. Indeed, − Z R r − h ( r ) dr = Z R Z Rr g ( s ) dsdr + Z R r − Z r s gdsdr (5.65) = Z R g ( s ) Z s drds + Z R s g ( s ) Z Rs r − drds = Z R sg ( s ) ds − R Z R s g ( s ) ds Hence we finish the proof. (cid:3)
We will now use the above to provide estimates on the L norm of the δ ′ -splitting functions on a neckregion: Theorem 5.66.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > . Then for every ǫ, B > and τ < τ ( n ) if δ, δ ′ ≤ δ ( n , ǫ, B , τ, v) > is such that assumptions (S1)-(S4) of (5.1) hold and if F ( r ) = r − R b = r p |∇ u | + ǫ φ N |∇ b | with H ( r ) ≡ rF ( r ) , then we have the Dini estimate: Z ∞ r H ( r ) ≤ C ( n , v , B ) ǫ . (5.67) Proof.
By the definition of φ N and Lemma 5.6, there exists R = R ( n , v , B ) such that φ N ≡ { b ≥ R } .Therefore we have Z ∞ r H ( r ) = Z R r H ( r ) . Now let us begin by using Lemma 5.63 to choose a special solution h ( r ) and h ( r ) of our ode such that h ′′ + r h ′ − r h = r − Z b = r U |∇ b | − = r − Z b = r h∇ φ N , ∇ p |∇ u | + ǫ i|∇ b | − , (5.68)with h ( R ) = − R R b ≤ R h∇ φ N , ∇ p |∇ u | + ǫ i , h (0) = Z R h ( r ) r dr = − Z b ≤ R b − h∇ φ N , ∇ p |∇ u | + ǫ i + R Z b ≤ R h∇ φ N , ∇ p |∇ u | + ǫ i . (5.69)On the other hand, we have special solution h ( r ) of h ′′ + r h ′ − r h = r − Z b = r U |∇ b | − , (5.70) such that h ( R ) = − R R b ≤ R U , h (0) = Z R h ( r ) r dr = − Z b ≤ R b − U + R Z b ≤ R U , (5.71)where U = (cid:16) ∇ j ( R ia ∇ a u ) − ∇ a ( R i j ∇ a u ) + ∇ i ( R ja ∇ a u ) (cid:17) ∇ i ∇ j u (cid:17) φ N √ |∇ u | + ǫ as in Proposition 5.57.Now let us use Lemma 5.63 one last time to produce a special solution h ( r ) of h ′′ + r h ′ − r h = r − Z b = r (cid:16) − p |∇ u | + ǫ | ∆ φ N | − c ( n ) | Rm | · |∇ u | φ N (cid:17) |∇ b | − , (5.72)such that h ( R ) = − R R b ≤ (cid:16) − p |∇ u | + ǫ | ∆ φ N | − c ( n ) | Rm | · |∇ u | φ N (cid:17) , h (0) = Z R h ( r ) r dr = Z b ≤ R − b − + R ! (cid:16) − p |∇ u | + ǫ | ∆ φ N | − c ( n ) | Rm ||∇ u | φ N (cid:17) . (5.73)Finally let us choose h ( r ) such that h ′′ + r h ′ − r h = h (0) = h ( R ) = | h ( R ) | + | h ( R ) | + | h ( R ) | + | h ( R ) | ≡ A . Note that we can explicitly solve h ( r ) = Ar / R . On the other hand, we now have that H − X i = h i ′′ ( r ) + H − X i = h i ′ ( r ) − r H − X i = h i ( r ) ≥ , (5.75)with (cid:16) H − P i = h i (cid:17) (0) = (cid:16) H − P i = h i (cid:17) ( R ) ≤
0. Therefore, using maximal principal to equation (5.75),we have H − P i = h i ≤
0. Let us point out that the negative sign on the zero order term in (5.75) is importantfor the maximum principle to hold. Observe also that H ≥ h i may change signs. We may nowestimate Z R H ( r ) r dr ≤ X i = Z R h i ( r ) r dr . (5.76)Therefore, to estimate the Dini integral for H ( r ), we only need to control the Dini integral of each h i .Beginning with h , we have by (5.69) and integrating by parts that Z R h ( r ) r dr = − Z b ≤ R b − h∇ φ N , ∇ p |∇ u | + ǫ i + R Z b ≤ R h∇ φ N , ∇ p |∇ u | + ǫ i (5.77) ≤ C ( n , v , B ) Z b ≤ R (cid:16) b − |∇ φ N | + b − | ∆ φ N | (cid:17) p |∇ u | + ǫ ! , where we have used that φ N ≡ { b = R } and |∇ b | ≤ C ( n , v , B ) on supp φ N . Now using Theorem 5.36and Lemma 5.6 with δ, δ ′ ≤ δ ( ǫ, n , v) we can obtain the apriori estimate |∇ u | ≤ ǫ b − on supp φ N . Pluggingthis in gives us Z R h ( r ) r dr ≤ C ( n ) ǫ Z b ≤ R (cid:0) b − |∇ φ N | + b − | ∆ φ N | (cid:1)! . (5.78) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 49
To estimate h let us see by (5.71), integration by parts, and our Ricci curvature bound that we have Z R h ( r ) r dr = − Z b ≤ R b − U + R Z b ≤ U (5.79) ≤ C ( n ) δ Z b ≤ R b − |∇ ∇ u p |∇ u | + ǫ | φ N + Z b ≤ R b − |∇ b | φ N + Z b ≤ R b − |∇ φ N | ≤ C ( n ) δǫ Z b ≤ R b − |∇ u | φ N + δ Z b ≤ R b − |∇ b | φ N + δ Z b ≤ R b − |∇ φ N | ! , where δ comes from the Ricci bound | Ric | ≤ δ . Finally for h , by (5.73) and H ¨older inequality we have Z R h ( r ) r dr = Z b ≤ R − b − + ! (cid:16) − p |∇ u | + ǫ | ∆ φ N | − c ( n ) | Rm | · |∇ u | φ N (cid:17) (5.80) ≤ C ( n ) ǫ Z b ≤ R b − | ∆ φ N | + C Z b ≤ R | Rm | φ N ! / Z b ≤ R b − |∇ u | φ N ! / . By applying the L curvature estimate on neck region in Theorem 4.1 and the hessian L estimate in Lemma5.35 with ǫ we therefore get Z R h ( r ) r dr ≤ C ( n ) ǫ Z b ≤ R b − | ∆ φ N | + C ( n , B ) ǫ . (5.81)Combining these with (5.76) we get Z R H ( r ) r dr ≤ C ( n , B , v) ǫ + Z b ≤ R b − | ∆ φ N | + Z b ≤ R b − |∇ φ N | + Z b ≤ R b − |∇ u | φ N ! . (5.82)Hence, to get the Dini estimate it suffices to estimate each term of (5.82). In fact, from the definition of φ N in Lemma 3.15, the definition of neck region, and by the comparison of b and d C in Lemma 5.6, one caneasily get Z b ≤ R b − | ∆ φ N | + Z b ≤ R b − |∇ φ N | ≤ C ( n , B , v) Z d C ≤ d − C | ∆ φ N | + Z d C ≤ d − C |∇ φ N | ! (5.83) ≤ C ( n , B , v) X x ∈ C ∩ B ( p ) Z B rx ( x ) (cid:16) d − C | ∆ φ N | + d − C |∇ φ N | (cid:17) + Z supp φ N d − C ≤ C ( n , B , v) X x ∈∩ B ( p ) r n − x + Z { d C ≤ }∩ supp φ N d − C ≤ C ( n , B , v) µ N ( B ( p )) + C ( n , B , v) ≤ C ( n , B , v) , where the term R supp φ N d − C comes from the derivative of φ in Lemma 3.15. On the other hand, by usingthe L p estimates arising from the codimension four theorem of [ChNa15], we know that R B ( p ) |∇ u | / ≤ C ( n , v). Therefore, by the H ¨older inequality we have Z b ≤ R b − |∇ u | φ N ≤ Z b ≤ R b − φ N ! / Z b ≤ R |∇ u | / φ N ! / ≤ C ( n , B , v) . (5.84)Thus we have proved the Lemma. (cid:3) Proof of Theorem 5.55.
Let us first observe that Z N − r − h |∇ u | < Z r − h |∇ u | φ N , (5.85)where φ N is the cutoff function from Lemma 3.15. In order to finish the proof let us apply Theorem 5.66with ǫ ′ in order to conclude Z ∞ r − Z b = r |∇ u | φ N < Z ∞ r − Z b = r q |∇ u | + ǫ ′ φ N < C ( n , B , v) ǫ ′ . (5.86)By using the coarea formula we can rewrite this as Z b − |∇ b | |∇ u | φ N < ǫ ′ . (5.87)Finally, using the Green’s function estimates |∇ b | > C − ( n , B ) and b ≤ C ( n , B ) r h from Lemma 5.6 this givesus the estimate Z r − h |∇ u | φ N < C ( n , B , v) ǫ ′ < ǫ , (5.88)where in the last line we have chosen ǫ ′ < C ( n , B , v) − ǫ in order to finish the proof. (cid:3) Gradient Estimates.
In this subsection we provide the main gradient estimate required in the proof ofTheorem 5.2. Precisely, we have the following:
Theorem 5.89.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > . For every ǫ, B > and τ < τ ( n ) , if (S1)-(S4) of (5.1) hold with δ, δ ′ < δ ( n , ǫ, B , τ, v) , then for every x ∈ C ∩ B ( p ) and r x < r < we have that ? B r ( x ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) < ǫ + C ( n , v) Z W x r − nh |∇ u | . (5.90)The proof of the above will rely on first proving a telescoping type estimate in Lemma 5.91. This estimatewill be applied iteratively along a wedge region to then conclude the above.5.5.1. Telescoping Estimate.
Our main result of this subsection is a telecoping type estimate which allowsus to compare the average of a function along different nearby balls. Precisely, we have the following:
Lemma 5.91.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > and Ric ≥ − ( n − , and let f : B ( p ) → R be a H -function. Then if B r ( x ) , B r ( x ) , B r ( x ) ⊆ B satisfy r , r > − n and are such that B r ( x ) , B r ( x ) ⊆ B r ( x ) , then we have the estimate (cid:12)(cid:12)(cid:12)(cid:12) ? B r ( x ) f − ? B r ( x ) f (cid:12)(cid:12)(cid:12)(cid:12) < C ( n ) ? B r ( x ) |∇ f | (5.92) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 51
Remark . In practice, we will be applying this estimate to f = |h∇ u a , ∇ u b i − δ ab | , where u : B → R n − is one of our harmonic splitting functions. Proof.
The proof is an application of the Poincare inequality for spaces with lower Ricci curvature bounds.To see this, let us choose r ′ ≡ r − max { r , r } >
0, so that B r ( x ) , B r ( x ) ⊆ B r ′ ( x ) ⊆ B r ( x ). Now we canestimate (cid:12)(cid:12)(cid:12)(cid:12) ? B r ( x ) f − ? B r ′ ( x ) f (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ? B r ( x ) (cid:12)(cid:12)(cid:12) f − ? B r ′ ( x ) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ≤ Vol( B r ( x )) − (cid:12)(cid:12)(cid:12)(cid:12) Z B r ′ ( x ) (cid:12)(cid:12)(cid:12) f − ? B r ′ ( x ) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ≤ C ( n ) (cid:12)(cid:12)(cid:12)(cid:12) ? B r ′ ( x ) (cid:12)(cid:12)(cid:12) f − ? B r ′ ( x ) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n ) ? B r ( x ) |∇ f | , (5.94)where in the last line we have used volume monotonicity and the Poincare inequality [SY] using that r − r ′ > − n . Applying the same argument to B r ( x ) gives the estimate (cid:12)(cid:12)(cid:12)(cid:12) ? B r ( x ) f − ? B r ′ ( x ) f (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n ) ? B r ( x ) |∇ f | , (5.95)and by combining these we obtain the desired result. (cid:3) Proof of Gradient Estimate of Theorem 5.89.
The proof will come in two steps. To begin with, letus fix γ = − and consider the sequence of scales s α ≡ γ − α , and for x ∈ C let us consider a sequence ofpoints x , x , . . . such that x α ∈ ∂ B s α ( x ) is the maximizer of the quantity D α = min y ∈ ∂ B s α ( x ) d ( y , x ) d ( y , C ) . Note that by our Gromov-Hausdorff condition we have that 1 + δ > D α > − δ , at least for s α > r x /
10. Letus observe the ball inclusions given by B − s α ( x α ) , B − s α + ( x α + ) ⊆ B − s α ( x α ) ⊆ W x . (5.96)In particular, if we consider the function f = |h∇ u a , ∇ u b i − δ ab | then we can apply Lemma 5.91 in orderto conclude (cid:12)(cid:12)(cid:12)(cid:12) ? B − s α ( x α ) f − ? B − s α + ( x α + ) f (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n ) s α ? B − s α ( x α ) |∇ f | , ≤ C ( n ) s α ? B − s α ( x α ) |∇ u | , ≤ C ( n ) s − n α Z W x ∩ A s α/ , s α ( x ) |∇ u | . (5.97) In particular, for every s α > r x /
10 we can sum in order to obtain the estimate (cid:12)(cid:12)(cid:12)(cid:12) ? B − s α ( x α ) f − ? B − ( x ) f (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( n ) X s α > r x / s − n α Z W x ∩ A s α/ , s α ( x ) |∇ u |≤ C ( n ) Z W x r − nh |∇ u | . (5.98)Since u : B ( p ) → R n − is a δ -splitting we have for δ < δ ( ǫ ) that (cid:12)(cid:12)(cid:12)(cid:12) ? B − ( x ) f (cid:12)(cid:12)(cid:12)(cid:12) ≤ − ǫ , (5.99)which by combining with our previous estimates therefore gives us ? B − s α ( x α ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) ≤ − ǫ + C ( n ) Z W x r − nh |∇ u | , (5.100)for all s α > r x / r > r x and pick s α such that | s α − r | is minimized. Then by applying Lemma 5.91 we have the estimate (cid:12)(cid:12)(cid:12)(cid:12) ? B − s α ( x α ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) − ? B r ( x ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) , ≤ C ( n ) r ? B r ( x ) |∇ f | ≤ C ( n ) r ? B r ( x ) |∇ u | . (5.101)But if we apply the scale invariant estimate of Theorem 5.36 with ǫ ′ = C ( n ) − − ǫ then we obtain theestimate (cid:12)(cid:12)(cid:12)(cid:12) ? B − s α ( x α ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) − ? B r ( x ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ≤ − ǫ . (5.102)Combining this with our previous estimates gives us our desired estimate ? B r ( x ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) ≤ ǫ + C ( n ) Z W x r − nh |∇ u | , (5.103)as claimed. (cid:3) µ -Splitting Estimates and Proof of Theorem 5.2. In this subsection we now build the ingredientsrequired to finish the proof of Theorem 5.2.5.6.1. µ -Maximal Function Estimates. We begin in this subsection by studying a maximal function definedwith respect to the µ -measure. This will be used in the next subsection to define for us our bilipschitz set ofcenter points. Our main result for this subsection is the following: CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 53
Proposition 5.104.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > . Then for every ǫ, B > and τ < τ ( n ) if δ, δ ′ ≤ δ ( n , ǫ, B , τ, v) is such that assumptions (S1)-(S4) of (5.1) hold, then if we define the µ -supportedmaximal function m u : C → R + bym ( x ) ≡ sup r x ≤ r ≤ ? B r ( x ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) + r ? B r ( x ) (cid:12)(cid:12)(cid:12) ∇ u (cid:12)(cid:12)(cid:12) ! , then we have the estimate Z B m ( x ) d µ ( x ) < ǫ . (5.105) Remark . The hessian part of the estimate is a consequence of Theorem 5.36. Therefore it is thegradient aspect of the above estimate which is the challenging part of the Theorem.In order to prove the Proposition we need a lemma which relates integrals over wedge regions to integralsover neck regions. Precisely:
Lemma 5.107.
Let ( M n , g , p ) satisfy Vol( B ( p )) > v > with τ < τ ( n ) and δ, δ ′ ≤ δ ( n , B , τ, v) such thatassumptions (S1)-(S4) of (5.1) hold. Then for f : B → R + a nonnegative function then we can estimate Z B (cid:16) Z W x r − nh f (cid:17) d µ ≤ C ( n , B ) Z N − f . (5.108) Remark . This is essentially an effective version of a Fubini type theorem.
Proof.
We have by Lemma 3.11 that r h ( y ) ≥ C ( n ) d C ( y ). Hence, we only need to estimate R B (cid:16) R W x d − n C f (cid:17) d µ .Moreover, since W x ⊂ N − we can assume without loss that supp f ⊂ N − . Let s i = − i . We rewritethe integral in the following manner. Z B ( p ) Z W x d − n C ( y ) f ( y ) d µ ( x ) dy = Z B ( p ) ∞ X i = Z W x ∩{ s i + ≤ d C ( y ) ≤ s i } d − n C ( y ) f ( y ) d µ ( x ) dy (5.110) ≤ ∞ X i = s − ni + Z B ( p ) Z W x ∩{ s i + ≤ d C ( y ) ≤ s i } f d µ ( x ) dy . Let χ i ( x , y ) be a function on M × M such that χ i ( x , y ) = y ∈ W x ∩ { s i + ≤ d C ( y ) ≤ s i } , otherwise χ i ( x , y ) =
0. By Fubini theorem and Ahlfor’s assumption on the measure µ , we have Z B ( p ) Z W x ∩{ s i + ≤ d C ≤ s i } f d µ ( x ) dy = Z B ( p ) Z { s i + ≤ d C ( y ) ≤ s i } χ i ( x , y ) f ( y ) d µ ( x ) dy (5.111) = Z { s i + ≤ d C ≤ s i } f ( y ) dy Z B ( p ) χ i ( x , y ) d µ ( x ) ≤ Z { s i + ≤ d C ≤ s i } f ( y ) µ ( B s i ( y )) dy ≤ C ( n , B ) s n − i Z { s i + ≤ d C ≤ s i } f ( y ) dy . Combining with the previous computation we obtain Z B ( p ) Z W x d − n C f d µ ≤ ∞ X i = C ( n , B ) Z { s i + ≤ d C ≤ s i } f ( y ) dy ≤ C Z f ≤ C Z N − f . (5.112) (cid:3) We are now in a position to prove Proposition 5.104:
Proof of Proposition 5.104.
Let us consider the two maximal functions m ( x ) ≡ sup r x ≤ r ≤ ? B rx ( x ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) , m ( x ) ≡ sup r x ≤ r ≤ r ? B rx ( x ) (cid:12)(cid:12)(cid:12) ∇ u (cid:12)(cid:12)(cid:12) . (5.113)We clearly have the estimate m ( x ) ≤ m ( x ) + m ( x ), so it is enough to estimate each of these individually.Let us begin by estimating m ( x ). By applying the scale invariant estimates of Theorem 5.36 with B − ǫ we have for every x ∈ C and r x ≤ r ≤ r ? B rx ( x ) (cid:12)(cid:12)(cid:12) ∇ u (cid:12)(cid:12)(cid:12) < B − ǫ , (5.114)and hence we clearly have the much weaker estimate Z B m ( x ) d µ = Z B sup r x ≤ r ≤ r ? B rx ( x ) (cid:12)(cid:12)(cid:12) ∇ u (cid:12)(cid:12)(cid:12) < ǫ . (5.115)The more challenging part of the Proposition is therefore the gradient estimate of m ( x ). To deal withthis we apply Theorem 5.89 with B − ǫ in order to see that for every x ∈ C and r x ≤ r ≤ ? B r ( x ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) < B − ǫ + C ( n , v) Z W x r − nh |∇ u | , (5.116)which of course gives the maximal function estimate m ( x ) < B − ǫ + C ( n , v) Z W x r − nh |∇ u | . (5.117)Now if we apply Lemma 5.107 we can then estimate Z B m ( x ) d µ < ǫ + C ( n , v) Z B (cid:16) Z W x r − nh |∇ u | (cid:17) d µ , ≤ ǫ + C ( n , v , B ) Z N − r − h |∇ u | . (5.118) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 55
To finish the proof let us now apply Theorem 5.55 with constant ǫ ′ = C ( n , v , B ) − ǫ in order to conclude Z B m ( x ) d µ ≤ ǫ + C ( n , v , B ) Z N − r − h |∇ u | , ≤ ǫ + ǫ = ǫ . (5.119)Combining this with our estimates on m ( x ) finishes the proof. (cid:3) Proof of Theorem 5.2 for Smooth Manifolds.
Now equipped with Proposition 5.104 we are now in aposition to finish the proof of Theorem 5.2 in the case when X = M is a smooth manifold. Indeed, let uspick δ, δ ′ < δ ( n , v , τ, B , ǫ ) such that Proposition 5.104 holds with ( ǫ ′ ) , where ǫ ′ = ǫ ′ ( n , B , ǫ ) will be chosenlater. That is, we have the estimate Z B m ( x ) d µ ( x ) < ( ǫ ′ ) , (5.120)where m ( x ) ≡ sup r x ≤ r ≤ ? B rx ( x ) (cid:12)(cid:12)(cid:12) h∇ u a , ∇ u b i − δ ab (cid:12)(cid:12)(cid:12) + r ? B rx ( x ) (cid:12)(cid:12)(cid:12) ∇ u (cid:12)(cid:12)(cid:12) ! . Let us now define C ǫ ⊆ C ∩ B to be the subset such that C ǫ ≡ { x ∈ C ∩ B : m ( x ) < ǫ ′ } . (5.121)Notice that because of our integral estimate on m ( x ) a weak L argument gives us for ǫ ′ < ǫ ′ ( n , B , ǫ ) that µ (cid:0)(cid:0) C ∩ B (cid:1) \ C ǫ (cid:1) < ǫ . (5.122)Let us now see that u : C ǫ → R n − is a 1 + ǫ -bilipschitz map. Indeed, let us take x , y ∈ C ǫ and let r = d ( x , y ).Now simply because we are in a ( δ, τ )-neck we already know that B r ( x ) is δ r -Gromov Hausdorff close to R n − × C ( S / Γ ). However, what we gain by our assumption that m ( x ) < ǫ ′ is that by theorem 2.21 we havethat we can take u to be the R n − part of our Gromov-Hausdorff map. More precisely, for ǫ ′ ≤ ǫ ′ ( n , ǫ ) wehave that ( u , π u ) : B r ( x ) → R n − × u − ( x ) , (5.123)is itself a 10 − ǫ r -GH map, where π u is the projection to the level set u − ( x ). Combining these points tells usthat u − ( x ), with the restricted metric, is ǫ r -GH close to C ( S / Γ ), and in particular we then have that u : B ǫ r (cid:0) L x , r (cid:1) → R n − , (5.124)is itself a ǫ r -GH map, where recall L x , r is the effective singular set as defined in Definition 3.1. But since x , y ∈ C ∩ B r ( x ) ⊆ B δ r (cid:0) L x , r (cid:1) this is precisely the statement that (cid:12)(cid:12)(cid:12) | u ( x ) − u ( y ) | − d ( x , y ) (cid:12)(cid:12)(cid:12) ≤ ǫ r . (5.125)But recall our choice of scale was that r = d ( x , y ), and therefore we have shown the desired bilipschitzestimate of u on C ǫ , which finishes the proof of Theorem 5.2. (cid:3) Proof of Theorem 5.2 for General Limit Spaces.
We wish to now use a relatively straight forwardlimiting argument based on the neck approximation of Theorem 3.19 in order to conclude the proof ofTheorem 5.2 for general limit spaces M nj → X .Thus let ǫ ′ << ǫ , which will later be fixed so that ǫ ′ = ǫ ′ ( n , τ, ǫ ), and let 2 δ, δ ′ be the correspondingconstants so that Theorem 5.2 holds on smooth manifolds with ǫ ′ . If N ⊆ B ( p ) is a ( δ, τ )-neck regionwith u : B ( p ) → R n − a δ ′ -harmonic splitting function, then by applying Theorem 3.19 let us consider asequence of neck regions N j → N which convergence in the sense of (1) → (4) of Theorem 3.19, and let u j : B ( p j ) → R n − a sequence of δ ′ -splitting functions with u j → u uniformly. Let C ǫ, j ⊆ C j be a sequencewhich satisfy the desired ǫ ′ -bilipschitz and measure estimates given by Theorem 5.2, and let us consider theHausdorff limit set C ǫ, j → C ǫ ⊆ C .Note that C ǫ satisfies the desired ǫ -bilipschitz control of Theorem 5.2 since u j → u uniformly. To finishthe proof we need to understand measure estimates on µ ( B \ C ǫ ). However recall condition (4) of our neckapproximation Theorem, which tells us that if µ j → µ ∞ then in particular µ ≤ C ( n , τ ) µ ∞ . Since B ( p ) \ C ǫ is an open set, we can conclude µ ( B \ C ǫ ) ≤ µ ∞ (cid:0) B \ C ǫ (cid:1) ≤ lim inf µ j (cid:0) B \ C ǫ, j (cid:1) < C ( n , τ ) ǫ ′ < ǫ , (5.126)where in the last line we have taken ǫ ′ < C ( n , τ ) − ǫ , which finishes the proof. (cid:3)
6. C
OMPLETING THE P ROOF OF T HEOREM
Proof of Lower Ahlfor’s Regularity bound of Theorem 3.10.1.
In this subsection we will prove thelower Ahlfor’s regularity bound of Theorem 3.10.1. Without any loss of generality we can focus on provingthe lower bound µ ( B ( p )) > A ( n ) − > , (6.1)as it will be clear the verbatim construction will tackle a general ball. Therefore with δ ′ << δ ′ -splitting function u : B ( p ) → R n − . The proof of (6.1) is based on a discrete degreetype argument. More precisely, let us begin by defining the discrete ǫ -Reifenberg map Φ : C → B , (6.2) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 57 given by Theorem 3.24. Note since both Φ and u are ǫ -GH maps, that after possibly composing u with anorthogonal transformation we can assume | Φ − u | < ǫ . Let us denote C ′ = C ′ ∪ C ′ + ≡ Φ ( C ) ⊆ B (0 n − ), andsince Φ is bih¨older by Theorem 3.24.1 we can consider the inverse mapping Φ − : C ′ → C ⊆ B ( p ). Let usnow compose this with the splitting function u in order to construct the map F ≡ u ◦ Φ − : C ′ ⊆ R n − → R n − . (6.3)Let us remark on some properties of the mapping F :(1) | F − Id | < ǫ ,(2) F (cid:0) B s x ′ ( x ′ ) (cid:1) ⊆ B r x ′ ( F ( x ′ )) for all x ′ ∈ C ′ + ,where s x ′ = s Φ ( x ) > Φ and u , while the second property follows from Theorem 3.24.2 together with thelipschitz bound |∇ u | ≤ R n − \ C ′ given by R n − \ C ′ ⊆ A , ∞ (0 n − ) ∪ [ x ∈ C + B s x (cid:0) Φ ( x ) (cid:1) = A , ∞ (0 n − ) ∪ [ x ′ ∈ C ′ + B s x ′ (cid:0) x ′ (cid:1) . (6.4)Let us consider a partition of unity φ ∪ { φ x ′ } with respect to this covering and then define the continuousmapping G : R n − → R n − given by G ( y ′ ) = φ ( y ′ ) · Id ( y ′ ) + P x ′ ∈ C ′ φ x ′ ( y ′ ) F ( x ′ ) if y ′ ∈ R n − \ C ′ , F ( y ′ ) if y ′ ∈ C ′ . Though we will not need it, it is not hard to check that G is a h¨older continuous map as well if the partition { φ x } is chosen with only a little care. Using the properties of F and Theorem 3.24 we can conclude thefollowing properties for G :(1) | G − F | < ǫ s x ′ on B s x ′ ( x ′ ) with x ′ ∈ C ′ ∩ B ,(2) | G − Id | < ǫ ,(3) G ( y ′ ) = Id ( y ′ ) = y ′ if y ′ < B (0 n − ),(4) G (cid:0) B s y ′ ( y ′ ) (cid:1) ⊆ B r y ′ ( G ( y ′ )) for all y ′ ∈ C ′ + ∩ B .The third condition above tells us that G is a degree one map from the ball B to itself, and in particularwe must have that G is onto. Combining with the second condition we in particular get that G (cid:0) B / (cid:1) ⊇ B / (0 n − ) . (6.5)Further combining this with the first and second conditions this tells us that B / (0 n − ) ⊆ G (cid:0) B / (cid:1) ⊆ G (cid:0) [ x ′ ∈ C ′ ∩ B / B s x ′ ( x ′ ) (cid:1) ⊆ [ x ′ ∈ C ′ + ∩ B / B r x ′ ( G ( x ′ )) ∪ (cid:0) G ( C ′ ) ∩ B / (cid:1) . (6.6) But this immediately leads to the estimate µ ( B ) = X x ∈ C + ∩ B r n − x + λ n − (cid:0) C ∩ B (cid:1) ≥ X x ′ ∈ C ′ ∩ B / r n − x ′ + λ n − (cid:16) G ( C ′ ) ∩ B / (cid:17) = C ( n ) X x ′ ∈ C ′ ∩ B / (30 r x ′ ) n − + λ n − (cid:16) G ( C ′ ) ∩ B / (cid:17) ≥ C ( n )Vol (cid:16) [ x ′ ∈ C ′ ∩ B / B r x ′ ( G ( x ′ )) ∪ ( G ( C ′ ) ∩ B / ) (cid:17) ≥ C ( n )Vol (cid:16) B / (0 n − ) (cid:17) ≡ A ( n ) − , (6.7)where we have used in the first line that |∇ u | ≤ λ n − (cid:0) C ∩ B (cid:1) ≥ λ n − (cid:0) u (cid:0) C ∩ B (cid:1)(cid:1) = λ n − (cid:0) G (cid:0) C ′ ∩ B (cid:1)(cid:1) ≥ λ n − (cid:16) G ( C ′ ) ∩ B / (cid:17) . This completes the proof of the lower volume bound. (cid:3) Induction Scheme and Proof of Theorem 3.10 on a Manifold.
Let us now focus our attention onfinishing the proof of Theorem 3.10 in the case when we are on a smooth manifold M . Let us begin bymaking several observations. First, if M is a smooth manifold then we have that inf r x >
0, see Remark 3.5.Therefore, as we have already shown the lower Ahlfor’s regularity bound we are left with just proving theupper Ahlfor’s regularity of Theorem 3.10.1 and the L curvature estimate of Theorem 3.10.3. Additionally,by Theorem 4.1 we have that the L estimate of Theorem 3.10.3 follows from the Ahlfor’s regularity of The-orem 3.10.1, and thus we can focus in this subsection on only proving the upper Ahlfor’s regularity estimate.Thus let us now outline our induction strategy. For α ∈ N let us consider the following statement:( α ) For x ∈ C with r x < r ≤ − α and B r ( x ) ⊆ B we have µ ( B r ( x )) < A r n − . Our strategy is to find A ≤ A ( n , τ ) such that we can prove statement ( α ) inductively. If we can prove theinductive statement for α = N = B \ B r x ( C + ) contains only balls ofpositive radius since we are on a manifold, and indeed since inf r x > r x > − α > , (6.8)for some α ∈ N sufficiently large. Therefore, we have that the inductive statement ( α ) trivially holds,since r x > − α for all x and thus there is no content to the statement.Now let us assume we have proved the inductive statement ( α + α ). Thus for the remainder of this subsection let x ∈ C with r x < r ≤ − α and B r ( x ) ⊆ B .Let us first deal with the easy case and observe that if r x > − r is relatively large compared to our ballsize, then we have both N ∩ B r ( x ) ⊆ B r ( x ) \ B − r ( C ) and that B r ( x ) \ B − r ( C ) is δ r -GH close to CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 59 B r \ B − r ( R n − × { } ) ⊆ R n − × C ( S / Γ ). Further, by the lipschitz condition ( n
4) we have for every ballcenter x i ∈ C ∩ B r ( x ) that r i > r x − δ r > − r . Thus using the definition of µ we see that ( α ) triviallyholds when A ( n , τ ) is taken to be the maximal number of disjoint balls of radius 10 − τ that one can fit in B (0 n − ).Therefore we may assume that r x ≤ − r without any loss, and our aim is to prove ( α ) for our ball B r ( x ).The beginning point is to first prove a weaker estimate. Thus let y ∈ C ∩ B r ( x ), and note by the lipschitzcondition ( n
4) of our neck region that r y < r x + δ r < − r . Now let us consider a radius r y < s ≤ r suchthat B s ( y ) ⊆ B r ( x ). If s ≤ r then B s ( y ) satisfies the hypothesis of ( α ) and therefore we have the upperbound µ ( B s ( y )) ≤ As n − . On the other hand, if s > r , then by a standard covering argument we may cover B s ( y ) ∩ C by at most C ( n ) balls { B r / ( x i ) } . Since the hypothesis holds for each of the balls B r / ( x i ) we canestimate to get the strictly weaker estimate( ∗ ) µ ( B s ( y )) ≤ X µ ( B r / ( x i )) ≤ C ( n ) A (cid:0) r (cid:1) n − ≤ C ( n ) As n − ≡ B ( n , τ ) s n − . (6.9)In particular, we see that for every y ∈ C ∩ B r ( x ) and r y < s such that B s ( y ) ⊆ B r ( x ) we have the weakerestimate µ ( B s ( y )) ≤ B s n − .In order to exploit this weaker estimate let us now observe that for δ ′ > δ ≤ δ ( n , τ, δ ′ )such that by theorem 2.21 there exists a ( n − , δ ′ )-splitting u : B r ( x ) → R n − . (6.10)If we consider Theorem 5.2 then we see if δ, δ ′ ≤ δ ( n , τ, v , B , ǫ ) = δ ( n , τ, v , ǫ ) then there exists a subset C ǫ ⊆ C ∩ B r ( x ) such that µ (cid:16)(cid:0) C ∩ B r ( x ) (cid:1) \ C ǫ (cid:17) < ǫ r n − , For all y , z ∈ C ǫ we have 1 − ǫ ≤ | u ( y ) − u ( z ) | d ( y , z ) ≤ + ǫ . (6.11)But let us now consider the consequences of this estimate. Restricting to the set C ǫ we have that { B τ r x ( x ) } x ∈ C ǫ are disjoint balls. However, using (6.11) this tells us that the image balls (cid:8) B − τ r x ( u ( x )) (cid:9) x ∈ C ǫ ⊆ B r (0 n − ) , (6.12)are also disjoint. But then we automatically get the packing estimate X x ∈ C ǫ r n − x = C ( n , τ ) X x ∈ C ǫ (cid:16) − τ r x (cid:17) n − ≤ C ( n , τ )Vol (cid:16) X x ∈ C ǫ B − τ r x ( u ( x )) (cid:17) ≤ C ( n , τ )Vol (cid:16) B r (0 n − ) (cid:17) ≤ C ( n , τ ) r n − ≡ A ( n , τ ) r n − . (6.13)But using the first estimate of (6.11) this then gives us µ (cid:0) B r ( x ) (cid:1) ≤ µ (cid:0) C ǫ (cid:1) + ǫ r n − = X x ∈ C ǫ r n − x + ǫ r n − ≤ A ( n , τ ) r n − . (6.14) This proves the estimate ( α ) for B r ( x ), and since B r ( x ) was arbitrary this completes the inductive step of theproof, and hence the proof of Theorem 3.10 itself. (cid:3) Proof of Theorem 3.10.
Let us now consider the case of a general limit space M nj → X . First observethat by the neck approximation of Theorem 3.19 we can consider a sequence of neck regions N j ⊆ B ( p j )such that N j → N in the sense of (1) → (4) of Theorem 3.19. In particular, if we consider the packingmeasures µ j of N j then we may limit µ j → µ ∞ so that µ ≤ C ( n , τ ) µ ∞ . (6.15)In particular, by applying the results of the previous two subsections we can immediately conclude that themeasure µ satisfies the required Ahlfor’s regularity of Theorem 3.10.1.Now if we apply Theorem 3.10.3 to each of the neck regions N j , then by observing that N j → N in C ,α ∩ W , p on all compact subsets, we immediately obtain the L estimate Z N ∩ B | Rm | < ǫ , (6.16)which finishes the proof of the L estimate of Theorem 3.10.3.To finish the proof we need to prove Theorem 3.10.2, which is to say we need to see that C is rectifiable.The main claim needed for this result is the following: Claim:
For each ǫ > δ < δ ( n , τ, v , ǫ ), then for x ∈ C with B r ( x ) ⊆ B there exists a closed subset R ǫ (cid:0) B r ( x ) (cid:1) ⊆ C ∩ B r ( x ) such that R ǫ is bilipschitz to an open subset of R n − and µ (cid:0) B r ( x ) ∩ ( C \ R ǫ ) (cid:1) < ǫ r n − .Indeed, for such a ball B r ( x ) ⊆ B let us choose a δ ′ -splitting function u : B r ( x ) → R n − , which for δ < δ ( n , δ ′ ) exists by theorem 2.21. Using Theorem 5.2 we see that if δ ′ , δ < δ ( n , τ, v , ǫ ) then there exists asubset C ǫ ⊂ C such that the restriction u : C ǫ → R n − is uniformly bilipschitz and µ (cid:0) B r ( x ) \ C ǫ (cid:1) < ǫ r n − . Wenow define R ǫ ≡ C ∩ C ǫ , and see this is our desired set. (cid:3) To finish the proof is now a measure theoretic argument. Indeed, let { y j } ⊆ C be a countable dense subsetand let us consider the set R ≡ [ B r ( y j ) ⊆ B : r ∈ Q R ǫ ( B r ( y j )) . (6.17)Clearly R ⊆ C is rectifiable, as it can be identified as a countable union of rectifiable sets. Notice thatbecause the sets R ǫ are closed we also have the identification R = [ B r ( y ) ⊆ B R ǫ ( B r ( y )) , (6.18) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 61 so that it does not matter if we union over all points or scales or just some countable dense collection.Now in order to conclude that C is rectifiable we need to see that µ ( C \ R ) = λ n − ( C \ R ) =
0. Thus letus assume that C \ R has positive n − n − λ n − ( C \ R ) > ǫ n > x ∈ C \ R there are radii r a → λ n − (cid:0) ( C \ R ) ∩ B r a ( x ) (cid:1) r n − a > ǫ n > . (6.19)But by taking ǫ < ǫ n and using the definition of R we immediately arrive at a contradiction, and therefore R ⊆ C is a full measure subset and C is rectifiable. This completes the proof of Theorem 3.10 (cid:3)
7. N
ECK D ECOMPOSITION T HEOREM
In this section we focus on decomposing our manifolds M n into two basic type of pieces, namely neckregions and ǫ -regularity regions with uniformly bounded curvature. The key aspect of this decompositionwill be our ability to control the number of pieces in this decomposition, a result which will depend heavilyon the results of Section 3 and is sharp. We begin with the decomposition in the case of smooth spaces: Theorem 7.1 (Neck Decomposition) . Let ( M ni , g i , p i ) → ( X , d , p ) satisfy Vol( B ( p i )) > v > and the Riccibound | Ric i | ≤ n − . Then for each δ > we can writeB ( p ) ∩ R ( X ) ⊆ [ a (cid:0) N a ∩ B r a (cid:1) ∪ [ b B r b ( x b ) , B ( p ) ∩ S ( X ) ⊆ [ a (cid:0) C , a ∩ B r a (cid:1) ∪ ˜ S ( X ) (7.2) such that(1) N a ⊆ B r a ( x a ) are δ -neck regions.(2) B r b ( x b ) satisfy r h ( x b ) > r b , where r h is the harmonic radius.(3) C , a ⊆ B r a ( x a ) \ N a is the singular set associated to N a .(4) ˜ S ( X ) is a singular set of n − measure zero.(5) P a r n − a + P b r n − b + H n − (cid:0) S ( X ) (cid:1) ≤ C ( n , v , δ ) . The proof will require a series of covering arguments, where we will first decompose our ball B ( p ) intofive types of pieces. Over the course of several subsections we will then break down these pieces until weare left with only the two which appear in the theorem itself. Notation and Ball Decomposition Types.
The proof of Theorem 7.1 will require multiple coveringsteps and lemmas, where we will first decompose B ( p ) into a much larger collection of balls, and thenproceed to recover those balls which are not either neck regions or smooth regions. In order to avoid asmuch confusion as is reasonable, we will introduce in this subsection a variety of notation which will holdthroughout this section. In particular, we will introduce and discuss the various ball types which will appear.Let us begin by introducing some notation which will play a role. Throughout the proof we have theunderlying constants δ and τ , however for rigor sake we will need several other constants floating aroundthat we can exploit. Throughout the proof these constants will eventually be fixed, however let us mentionthat they will roughly behave as0 < η ( n , δ ′ , τ ) << δ ′ ( n , δ, τ ) << δ << ǫ ( n , τ ) << τ << . (7.3)Let us now move to a distinct notational point. When studying a ball we will often be interested in how thevolume pinching behaves at any given point in comparison to a background reference value. This referencevalue will be denoted by V , and in practice will typically be given by V ≡ inf y ∈ B ( p ) V η − ( y ) . (7.4)Notice that it will be convenient to move up η − -scales in our volume measurements. In addition to this, wewill be interested in those points which remain volume pinched after we drop many scales. Thus, let us alsodefine the sets E η ( x , r ) ≡ { y ∈ B r ( x ) : V η r ( y ) < V + η } . (7.5)Therefore E η ( x , r ) represents those points of B r ( x ) whose value ratio has not increased to some definiteamount η more than our reference value. Using the cone splitting of theorem 2.11 this means that these arethe very 0-symmetric points of B r ( x ).Equipped with this terminology, which will be used consistently throughout this section, let us now con-sider the various ball types which will appear in the proof. The conventions below will allow us to keeporiented throughout the proof of the various conditions we will be forced to consider. After defining the setscarefully we will discuss in words their meaning:(a) A ball B r a ( x a ) with subscript a will be such that there exists a ( δ, τ )-neck region N a ⊆ B r a ( x a ).(b) A ball B r b ( x b ) with subscript b will be such that r b ( x b ) ≥ r b .(c) A ball B r c ( x c ) with subscript c will be such that Vol (cid:0) B δ r d E η ( x d , r d ) (cid:1) > ǫ δ r nd .(d) A ball B r d ( x d ) with subscript d will be such that Vol (cid:0) B δ r d E η ( x d , r d ) (cid:1) ≤ ǫ δ r nd .(e) A ball B r e ( x e ) with subscript e will be such that E η ( x e , r e ) = ∅ , that is we have the volume jumpinf y ∈ B re ( x e ) V η r e ( y ) ≥ V + η .(f) A ball B r f ( x f ) with subscript f will be a typical ball for which we have no apriori knowledge ofadditional structure. CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 63
Of the various ball types above, only the first two take part in the final result of Theorem 7.1. Noticealso that the ( c ), ( d ) and ( e ) balls require a background parameter V in their definition. Let us maybe take amoment to explain in words the import of these ball types which will appear in the proof. A c -ball is one forwhich there is a lot of points which are volume pinched on many scales, from a n − c -ball is δ ′ -GH close to R n − × C ( S / Γ ), and therefore is aball for which we expect a neck region to exist on, though maybe it has not yet been built. A d -ball is onefor which, from a n − d -balls, while on the other this makesthem easy to recover until we can get down to a scale we do understand. To explain the e -ball, let us nownote that the final proof of the decomposition theorem will be by an induction process, where we will inducton the lower volume of the ball. Therefore, an e -ball is a ball for which the volume has increased by somedefinite amount after dropping some scales, and thus we will be able to apply our inductive hypothesis todeal with it. Finally, an f -ball is one for which we may know nothing. Typically, the f -balls will not appearin the statements of results, but only in the proofs until we have categorized their behavior.The next several subsections will consider a sequence of covering lemmas which will allow us to buildthis final decomposition.7.2. Cone Splitting and Codimension Four.
In this subsection we prove a handful of preliminary results,which will be used to eventually help fix the constants δ ′ and η , in terms of τ and δ , throughout the proof.Our first result in this direction is essentially a new viewpoint on the notion of conesplitting in [ChNa13].Instead of trying to find independent points, we will instead measure the content of volume pinched points.In general these two notions need not coincide, however in the case of cone-splitting it is not hard to see thatthey do. The precise result is phrased in a manner most convenient for the applications of this section: Theorem 7.6 (Content Cone-Splitting) . Let ( M n , g , p ) satisfy Vol ( B ( p )) > v > with < δ, ζ ≤ δ ( n , v) and ǫ > . Then there exists η ( n , v , δ, ǫ, ζ ) > such that if(1) Ric ≥ − η ,(2) If V ≡ inf B ( p ) V η − ( y ) and E η ( p , ≡ { y ∈ B ( p ) : V η ( y ) < V + η } satisfies Vol (cid:0) B δ E η (cid:1) > ǫ δ k , (7.7) then there exists q ∈ B ( p ) such that B ζ − ( q ) is ( n − k , ζ ) -symmetric.Proof. We will prove the result inductively on k . If k = n , then this just follows from Cheeger-Colding’smetric cone theorem 2.10. Assume for k ≤ n that the theorem holds. We will prove the theorem for k − (cid:0) B δ E η (cid:1) > ǫ δ k − ≥ ǫ δ k . Let ζ ′ << ζ be fixed later. By induction, if η ≤ η ( n , v , ζ ′ , δ, ǫ ),there exists q ∈ B ( p ) such that B ζ ′− ( q ) is ( n − k , ζ ′ )-symmetric with respect to L k ζ ′ . By GH-approximationand a covering argument as in the proof of Proposition 4.3, it is easy to show Vol( B δ L k ζ ′ ∩ B ( p )) ≤ C ( n , v) δ k . Thus for δ ≤ δ ( n , v) small, we have Vol (cid:0) B δ E η \ B δ L k ζ ′ (cid:1) ≥ ǫ δ k − . In particular, there exists z ∈ E η \ B δ L k ζ ′ .For η small and metric cone theorem 2.10, we have B ζ ′− ( z ) is (0 , ζ ′ )-symmetric. By the cone-splitting oftheorem 2.11 and choosing ζ ′ = ζ ′ ( ζ, δ, n , v) small, we have that B ζ − ( q ) is ( n − k + , ζ )-symmetric. Thuswe finish the inductive step of the proof. (cid:3) The main application of the above in this paper will be to the k = ǫ -regularity oftheorem 2.18 in order to conclude that a ball which is sufficiently n − R n − × C ( S / Γ ), which is what will allow us to build our neck regions in subsequentsections. Our precise result is the following: Theorem 7.8.
Let ( M n , g , p ) satisfy Vol ( B ( p )) > v > with δ, ǫ > . Then for η ≤ η ( n , v , δ, ǫ ) withV ≡ inf B ( p ) V η − ( y ) and E η ( p , ≡ { y ∈ B ( p ) : V η ( y ) < V + η } , if we have | Rc | < η , Vol (cid:0) B δ E η (cid:1) > ǫ δ , (7.9) then either:(1) We have that r h ( p ) > δ − , or(2) There exists q ∈ B ( p ) such that B δ − ( q ) is pointed δ -GH close to B δ − (0 n − , y c ) ⊆ R n − × C ( S / Γ ) ,where Γ ≤ O (4) is nontrivial and acts freely away from the origin with y c the cone point.Proof. By Theorem 7.6 we have for η ≤ η ( n , v , δ, τ, ǫ ) that there exists q ∈ B ( p ) such that B δ − ( q ) is δ -Gromov Hausdorff close to R n − × C ( Z ), where C ( Z ) is a cone metric space. Let us begin with the following: Claim 1:
For η ≤ η ( n , v , δ ) sufficiently small we have that Z = S / Γ is a smooth three manifold.Indeed, assume this is not the case, then by applying Theorem 7.6 we have a sequence ( M nj , g j , q j )such that B δ − j ( q j ) is δ j -GH close to R n − × C ( Z j ), but we do not have that B δ − ( q j ) is δ -GH close to R n − × C ( S / Γ ). By passing to a limit we have that M j → X ≡ R n − × C ( Z ∞ ). Let us first see that Z ∞ isa smooth manifold. Indeed, if Z ∞ were not smooth, then we see that the singular set of X has codimensionat most three. However, by [ChNa15] the singular set of X is at most of codimension four, which tells usthat Z ∞ is indeed smooth. Additionally, since η j → X is Ricci flat on its smooth component.It is not difficult to check for a smooth manifold Z ∞ that C ( Z ∞ ) is Ricci flat iff Z ∞ has constant sectionalcurvature 1, which in particular implies that Z ∞ = S / Γ . Therefore for far enough in the sequence thismeans we have that B δ − ( q j ) is δ -GH close to R n − × C ( S / Γ ), which is a contradiction and so proves theclaim. (cid:3) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 65
Finally, to finish the proof we now distinguish between the cases when Γ is trivial or nontrivial. If Γ istrivial then B δ − ( q ) is δ -GH close to R n , which by theorem 2.18 implies r h ( q ) > δ − , and so r h ( p ) > δ − as in case (1). On the other hand if Γ is nontrivial then this is exactly case (2), which finishes the proof. (cid:3) Weak c -Ball Covering. Now recall that the essence of a ( δ, τ )-neck region N is that N is a smoothregion which on many scales looks like R n − × C ( S / Γ ). In order to prove the decomposition of Theorem7.1 with the sharp n − c -ball in the notation of Section 7.1.The resulting decomposition will produce one large ( δ, τ )-neck region, together with an n − δ, τ )-neck region will satisfy a form of maximal condition preventing it from being ex-tended further down, and the remaining balls will be such that for most points there is a large drop in thevolume ratio.Now this decomposition will take place over the next two subsections. The reason being is that in thissection we will first decompose a c -ball to produce one large weak neck region, as well as an n − weak neck region: Definition 7.10.
We call ˜ N ⊆ B ( p ) a weak ( δ, τ )-neck region if there exists a closed subset ˜ C = ˜ C ∪ ˜ C + = ˜ C ∪ { x i } and a radius function r : ˜ C → R + with r x > C + and r x = C such that ˜ N ≡ B \ B r x ( C )satisfies(˜ n { B τ r x ( x ) } ⊆ B ( p ) are pairwise disjoint.(˜ n
2) For each r x ≤ r ≤ δ r -GH map ι x , r : B δ − r (0 n − , y c ) ⊆ R n − × C ( S / Γ ) → B δ − r ( x ),where Γ ⊆ O (4) is nontrivial.(˜ n
3) For each r x ≤ r with B r ( x ) ⊆ B ( x ) we have that L x , r ≡ ι x , r (cid:0) B r (0 n − ) × { y c } (cid:1) ⊆ B τ ( r + r x ) ( C ). Remark . For precision sake, a weak neck region not only eliminates the lipschitz condition of ( n n
3) has changed slightly in order to accurately take this into account.
Remark . It is worth keeping an example in mind, as this clarifies the definition. Thus if one consideredExample 1.18 in the outline, then an example of a weak neck region is given by letting r x be any positivefunction which is not necessarily lipschitz. That is, a weak neck region is one for which all the usual conditions of a neck region hold, except thatpossibly the ball radii are not varying in a manner which has small lipschitz constant. The effect of thisis that it is possible there are nearby balls of uncontrollably different sizes. In practice, this will be quiteinconvenient for the analysis, and therefore in the next subsection we will see how to further refine the weak( δ, τ )-neck constructed in this subsection into an actual ( δ, τ )-neck. The main result of this subsection is thefollowing:
Proposition 7.13 (Weak c -Ball Covering) . Let ( M ni , g i , p i ) → ( X , d , p ) satisfy Vol( B ( p i )) > v > with δ > and < ǫ, τ < τ ( n ) , then for η ≤ η ( n , v , δ, ǫ, τ ) let us assume the following holds:(1) | Ric | ≤ η ,(2) If V ≡ inf B ( p ) V η − ( y ) and E η ≡ { x ∈ B ( p ) : V η ( x ) < V + η } then Vol (cid:0) B δ E η (cid:1) > ǫ · δ .(3) r h ( p ) < .then we can write B ⊆ C ∪ ˜ N ∪ S d B r d ( x d ) , where(a) ˜ N = B \ (cid:16) C ∪ S d B r d ( x d ) (cid:17) is a weak ( δ, τ ) -neck region,(d) For each d-ball B r d ( x d ) if E d ≡ { x ∈ B r d ( x d ) : V η r d ( x ) < V + η } , then we have Vol (cid:0) B δ r d E d (cid:1) ≤ ǫ · δ r nd .Remark . Condition (2) above says that B is a c -ball, while condition (3) above says that B is not a b -ball. Proof.
Let us consider δ ′ ≡ δ τ . To prove the result we will build on B ( p ) a sequence of weak ( δ, τ )-necks˜ N i , where each is a refinement of the last, with the eventual goal of arriving at a ( δ, τ )-neck which cannotbe extended any further. Before discussing the inductive procedure for this construction, we need to buildthe base neck region N . Precisely, by conditions (2) and (3) we can, for η ≤ η ( n , δ ′ , ǫ, τ ), apply Theorem7.8 to get that B δ ′− ( p ) is δ ′ -GH close to R n − × C ( S / Γ ). In particular, B ( p ) is ( n − , δ ′ )-symmetric withrespect to some L ≡ ι (cid:0) R n − × { y } (cid:1) ∩ B ( p ), where ι : B δ ′− (0 n − , y ) ⊆ R n − × C ( S / Γ ) → B δ ′− ( p ) is theGromov-Hausdorff map. Notice that L acts as an approximate singular set on the ball. Let us now considera covering L ⊆ [ B τ · δ ( x f ) ⊆ [ B δ ( x f ) , (7.15)where x f ∈ L and { B τ · δ ( x f ) } is a maximal disjoint collection. We then define˜ N ≡ B ( p ) \ [ B δ ( x f ) , (7.16)and it is easy to check that for η ≤ η ( n , ǫ, τ, δ ′ ) this is indeed a weak ( δ, τ )-neck. Before moving on to theinductive construction let us further separate the singular balls into a couple of better groups. To accomplishthis let us recall that for a ball B r ( x ) we can define the η -pinched subset E η by E η ( x , r ) ≡ { y ∈ B r ( x ) : V η r ( y ) < V + η } . (7.17)Thus for each ball B δ ( x f ) in our covering if we consider the pinched points E η ( x f , δ ) then we can considerone of two cases: CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 67 (c) Vol (cid:0) B δ · δ E η ( x f , δ ) (cid:1) > ǫ δ δ n ,(d) Vol (cid:0) B δ · δ E η ( x f , δ ) (cid:1) ≤ ǫ δ δ n .Of course, in case ( c ) above we have that B δ ( x f ) is a c -ball, while in case ( d )-above we have that B δ ( x f ) is a d -ball. Therefore, let us break out balls into these two subsets and write˜ N ≡ B ( p ) \ (cid:16) [ c B δ ( x c ) ∪ [ d B δ ( x d ) (cid:17) . (7.18)Now let us move on to discuss the inductive procedure for the construction of the weak ( δ, τ )-necks ˜ N i .Thus, let us assume we have constructed the weak ( δ, τ )-neck ˜ N i given by˜ N i ≡ B ( p ) \ (cid:16) [ c B δ i ( x ic ) ∪ [ j ≤ i [ d B δ j ( x jd ) (cid:17) , (7.19)such that the following additional conditions hold:i.1 Each c -ball B δ i ( x ic ) satisfies Vol (cid:0) B δ · δ i E η ( x ic , δ i ) (cid:1) > ǫ δ ( δ i ) n .i.2 Each d -ball B δ j ( x jd ) satisfies the volume condition Vol (cid:0) B δ · δ j E η ( x jd , δ j ) (cid:1) ≤ ǫ δ ( δ j ) n .i.3 For each d -ball B δ j ( x jd ) there exists a ball B δ j − ( y ) which satisfies condition ( c ) with x jd ∈ B δ j − ( y ).i.4 Every c -ball has radius δ i .We therefore wish to build a weak ( δ, τ )-neck ˜ N i + which satisfies ( i + . i + .
4. In order to build thisweak neck region we must therefore break apart the c -balls in our covering. More precisely, for each ball B δ i ( x ic ) let ι ic : B δ i (0 n − , y ) ⊆ R n − × C ( S / Γ ) → B δ i ( x ic ) be a pointed δ ′ δ i -GH map with L ic ≡ ι ic (cid:0) R n − ×{ y } (cid:1) the approximate singular set for which the ball is ( n − , δ ′ )-symmetric with respect to. Then we can considerthe total approximate singular set on scale δ i given by L i ≡ [ c L ic . (7.20)Let us now consider a collection of balls { B δ i + ( x i + f ) } such that L i \ [ j ≤ i B τ · δ j ( x jd ) ⊆ [ f B δ i + ( x i + f ) , x i + f ∈ L i \ [ j ≤ i B τ · δ j ( x jd ) , (cid:8) B τ · δ i + ( x i + f ) (cid:9) are disjoint . (7.21)Now each ball (cid:8) B δ i + ( x i + f ) (cid:9) is either a c ball which satisfies Vol (cid:0) B δ · δ i + E η ( x i + f , δ i + ) (cid:1) > ǫ δ ( δ i + ) n , or is a d -ball which satisfies the converse inequality Vol (cid:0) B δ · δ i + E η ( x i + f , δ i + ) (cid:1) ≤ ǫ δ ( δ i + ) n . Thus we can separatethe balls into these two types by (cid:8) B δ i + ( x i + f ) (cid:9) = (cid:8) B δ i + ( x i + c ) (cid:9) c ∪ (cid:8) B δ i + ( x i + d ) (cid:9) d . (7.22) This allows us to define the region ˜ N i + by˜ N i + ≡ B ( p ) \ (cid:16) [ c B δ i + ( x i + c ) ∪ [ j ≤ i + [ d B δ j ( x jd ) (cid:17) . (7.23)It is easy to check from the construction that for δ ′ ≤ δ τ this is indeed a weak ( δ, τ )-neck which satisfies( i + . i + .
4, as required.Now to finish the proof of the Proposition let us consider the closed discrete sets C ic ≡ S c { x ic } and noteby construction that C i + c ⊆ B δ i ( C ic ). Therefore we can define the Hausdorff limit C = lim i →∞ C i + c . (7.24)Taking the weak neck regions ˜ N i and letting i → ∞ we see we arrive at a weak neck region˜ N ≡ B ( p ) \ (cid:16) C ∪ [ j [ d B δ j ( x jd ) (cid:17) , (7.25)which finishes the construction of the weak neck region. (cid:3) Strong c -Ball Covering and Construction of Maximal Neck Regions. In this subsection we produceour more refined covering of a c -ball, which produces an actual ( δ, τ )-neck. Our precise theorem for thissubsection is the following: Proposition 7.26 ( c -Ball Covering) . Let ( M ni , g i , p i ) → ( X , d , p ) satisfy Vol( B ( p i )) > v > with δ > and < ǫ, τ < τ ( n ) , then for η ≤ η ( n , v , δ, ǫ, τ ) let us assume the following holds:(1) | Ric | ≤ η ,(2) If V ≡ inf B ( p ) V η − ( y ) and E η ≡ { x ∈ B ( p ) : V η ( x ) < V + η } then Vol (cid:0) B δ E η (cid:1) > ǫ · δ .(3) r h ( p ) < .then we can write B ⊆ C ∪ N ∪ S b B r b ( x b ) ∪ S c B r c ( x c ) ∪ S d B r d ( x d ) ∪ S e B r e ( x e ) , where(a) N = B \ (cid:16) C ∪ S b B r b ( x b ) ∪ S c B r c ( x c ) ∪ S d B r d ( x d ) ∪ S e B r e ( x e ) (cid:17) is a ( δ, τ ) -neck region,(b) For each b-ball B r b ( x b ) we have r h ( x b ) > r b ,(c) For each c-ball B r d ( x c ) , if E c ≡ { x ∈ B r c ( x c ) : V η r c ( x ) < V + η } , then we have Vol (cid:0) B δ r c E c (cid:1) > ǫ · δ r nc .(d) For each d-ball B r d ( x d ) , if E d ≡ { x ∈ B r d ( x d ) : V η r d ( x ) < V + η } , then we have Vol (cid:0) B δ r d E d (cid:1) ≤ ǫ · δ r nd .(e) For each e-ball B r e ( x e ) , if E e ≡ { x ∈ B r e ( x e ) : V η r e ( x ) < V + η } , then we have that E e = ∅ .Further, we have the content bounds X x b ∈ B / r n − b + X x d ∈ B / r n − d + X x e ∈ B / r n − e < C ( n , τ ) , X x c ∈ B / r n − c < C ( n , τ ) ǫ . (7.27) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 69
Proof.
Let us begin the proof by applying Proposition 7.13 with ( δ ′ , τ ) where δ ′ ≤ δ τ in order to get acovering B ⊆ ˜ C ∪ ˜ N ∪ [ d ∈ ˜ C + B ˜ r d ( ˜ x d ) , (7.28)where ˜ N ≡ B \ ˜ C ∪ S ˜ C + B ˜ r d ( ˜ x d ) is a weak ( δ ′ , τ )-neck region, and B ˜ r d ( ˜ x d ) are all d -balls with respect to η ′ as given from Proposition 7.13. We will eventually pick our η << η ′ .Now since ˜ N is a weak neck region, we have in particular that the collection of balls { B τ ˜ r d ( ˜ x d ) } are alldisjoint. Now let us define the following radius function: r y ≡ δ τ ˜ r d if y ∈ B τ ˜ r d ( ˜ x d ) ,δ d ( y , ˜ C ) if y < S B τ ˜ r d ( ˜ x d ) . Let us note that we have the gradient bound |∇ r y | ≤ δ , and if y ∈ B ˜ r d ( ˜ x d ) then r y ≤ δ ˜ r d .In order to build our new covering let us make several observations. To begin with for each y ∈ B let ˜ y ∈ ˜ C denote the center point which minimizes d ( y , ˜ C ). Recalling from Definition 3.1 that L x , r = ι x , r (cid:16) B r (0 n − ) × { y c } (cid:17) is the effective singular set on B r ( x ), we define the set˜ S ≡ (cid:8) y ∈ B : y ∈ B δ r y L ˜ y , τ − r y (cid:9) , (7.29)which is roughly to say that ˜ S are the set of points which belong to the effective singular set on their ownscale r y . Our first claim, which is immediate once you untangle the constants in the definition of ˜ S , is thatpoints of ˜ S look very close to R n − × C ( S / Γ ) at scales bigger than r y . Precisely, Claim:
Let y ∈ ˜ S with δ ′ ≤ δ ′ ( n , v , τ, δ ). Then for all r y < r ≤ B δ − r ( y ) is pointed δ r -GHclose to B δ − r (0 n − , y c ) ⊆ R n − × C ( S / Γ ). (cid:3) Let us now consider the covering of ˜ S given by˜ S ⊆ ˜ C ∪ [ y ∈ ˜ S B r y ( y ) , (7.30)and let { B r i ( y i ) } be a maximal Vitali subcovering such that { B τ r i ( y i ) } are disjoint. If we define N ≡ B \ ˜ C ∪ [ i B r i ( y i ) , (7.31)then it follows fairly easily from the previous remarks that N is a ( δ, τ )-neck region. By applying Theorem3.10 we even get the estimate H n − ( ˜ C ) + X x i ∈ B / r n − i ≤ C ( n , τ ) . (7.32)Now certainly each ball B r i ( x i ) is either a ( a ) → ( e ) ball, and therefore we can write n B r i ( x i ) o = n B r a ( x a ) o a ∪ n B r b ( x b ) o b ∪ n B r c ( x c ) o c ∪ n B r d ( x d ) o d ∪ n B r e ( x e ) o e . (7.33) The crucial aspect of the proof which is left is to show that we have the estimate X x c ∈ B / r n − c ≤ C ( n , τ ) ǫ , (7.34)which is to say that the c -balls have small content. In order to see this let us begin by observing that by thedefinition of r y and since ˜ N is a weak ( δ ′ , τ )-neck we have the covering˜ S ⊆ ˜ C [ d B τ ˜ r d ( ˜ x d ) . (7.35)Using Theorem 3.10 we have that µ ≡ P r n − i δ x i , which is the packing measure associated to the neck region N , is a doubling measure. In particular, since { B τ ˜ r d ( ˜ x d ) } are disjoint this gives us X ˜ x d ∈ B / ( p ) ˜ r n − d ≤ C ( n , τ ) X ˜ x d ∈ B / ( p ) µ ( B ˜ r d ( ˜ x d )) , ≤ C ( n , τ ) X ˜ x d ∈ B / ( p ) µ ( B τ ˜ r d ( ˜ x d )) , ≤ C ( n , τ ) µ ( B / ) ≤ C ( n , τ ) . (7.36)Now for each ball B ˜ r d ( ˜ x d ) let us consider the set C ˜ d ≡ { x i ∈ C ∩ B τ ˜ r d ( ˜ x d ) : s.t. B r i ( x i ) is a c -ball } . (7.37)The following is our main claim about this set: Claim:
We have the estimate µ ( C ˜ d ) ≤ C ( n , τ ) ǫ ˜ r n − d .In order to prove the claim let us first note that if x i ∈ C ˜ d then r i < δ ˜ r d and thus B r i ( x i ) ⊆ B ˜ r d ( ˜ x d ). Let usalso observe that if x i ∈ C ˜ d then there exists y i ∈ B r i ( x i ) such that V η r i ( y i ) < ¯ V + η . Then y i ∈ E ˜ d ,η ′ , hence B − δ ˜ r d ( x i ) ⊂ B − δ ˜ r d ( y i ), and therefore we have B − δ ˜ r d ( x i ) ⊆ B δ ˜ r d E ˜ d ,η ′ . Thus, let us choose a maximalcollection of balls { B δ ˜ r d ( x ′ i ) } N ′ with x ′ i ∈ C ˜ d such that { B − δ ˜ r d ( x ′ i ) } are disjoint. Note then that because B ˜ r d ( ˜ x d ) is a ˜ d -ball and since B − δ ˜ r d ( x ′ i ) ⊆ B δ ˜ r d E ˜ d ,η ′ we have the estimate N ′ δ n ˜ r nd ≤ X x ′ i Vol( B δ ˜ r d ( x ′ i )) ≤ C ( n ) X Vol( B − δ ˜ r d ( x ′ i )) ≤ C ( n )Vol (cid:0) B δ ˜ r d E ˜ d ,η ′ (cid:1) ≤ C ( n ) ǫ · δ ˜ r nd , (7.38)so that we get the estimate N ′ ≤ C ( n ) ǫδ − n as a bound for the number of balls in the covering { B δ ˜ r d ( x ′ i ) } .Using the Ahlfor’s regularity of Theorem 3.10 we can therefore estimate: µ ( C ˜ d ) ≤ N ′ X µ ( B δ ˜ r d ( x ′ i )) ≤ C ( n , τ ) δ n − ˜ r n − d N ′ ≤ C ( n , τ ) ǫ ˜ r n − d , (7.39)which finishes the proof of the Claim. (cid:3) Finally let us now consider the set C ≡ { x i ∈ B / ( p ) : x i is a c -ball } , (7.40) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 71 then we can combine (7.36) with the previous claim and the Ahlfor’s regularity of Theorem 3.10 in order toestimate X x c ∈ B / r n − c ≤ C ( n , τ ) µ ( C ) ≤ C ( n ) X ˜ x d ∈ B / µ ( C ˜ d ) ≤ C ( n , τ ) ǫ X ˜ x d ∈ B / ˜ r n − d ≤ C ( n , τ ) ǫ , (7.41)which finishes the proof of the Proposition. (cid:3) Refinement of Balls with Less than Maximal Symmetries.
In this subsection we will deal with ballsfor which the set of volume pinched points has small n − d -balls inthis subsection. In this case, we will simply recover our ball until we arrive at balls of other types. However,what we will gain is that for the ball types of the new covering which are not regular balls, the content willnot only be bounded, it will be small. This is crucial, as otherwise there could be a pile of errors if onewere forced to continually cover in this manner. As we will see in subsequent sections, the smallness of thecontent bound will allow us to produce a converging geometric series when considering these errors. Themain result of this subsection is the following: Proposition 7.42 ( d -ball Covering) . Let ( M ni , g i , p i ) → ( X , d , p ) satisfy Vol( B ( p i )) > v > with δ, τ > and < V ≤ inf y ∈ B ( p ) V η − ( y ) , then for ǫ ≤ ǫ ( n , v) , η ≤ η ( n , ¯ V , δ, ǫ, τ ) let us assume the following holds:(1) | Ric | ≤ η ,(2) If E η ≡ { y ∈ B ( p ) : V η ( y ) < ¯ V + η } , then Vol (cid:0) B δ E η (cid:1) < ǫ · δ .then we can write B = ˜ S ∪ S b B r b ( x b ) ∪ S c B r c ( x c ) ∪ S e B r e ( x e ) , where(s) ˜ S ⊆ S ( X ) and satisfies H n − ( ˜ S ) = .(b) For each b-ball B r b ( x b ) we have r h ( x b ) > r b ,(c) For each c-ball B r c ( x c ) , if E c ≡ { y ∈ B r c ( x c ) : V η r c ( y ) < ¯ V + η } then Vol (cid:0) B δ r c E c (cid:1) > ǫ · δ r nc .(e) For each e-ball B r e ( x e ) , E e ≡ { y ∈ B r e ( x e ) : V η r e ( y ) < ¯ V + η } then E e = ∅ .Further, we have the content estimates P b r n − b + P e r n − e < C ( n , δ ) and P c r n − c ≤ C ( n , v) ǫ .Proof. The procedure of the proof will be to iterate a certain covering construction and keep track of theestimates. To illustrate this let us begin by considering a Vitali covering of B ( p ) given by B ( p ) ⊆ [ f B δ ( x f ) , (7.43)with { B δ/ ( x f ) } disjoint. Recall that for any ball B r ( x ) we can define the η -pinched subset E η given by E η ( x , r ) ≡ { y ∈ B r ( x ) : V η r ( y ) < V + η } . (7.44) In this way we wish to separate the balls { B δ ( x f ) } into three types, depending on whether they are c -balls, d -balls or e -balls based on the conditions:(c) Vol (cid:0) B δ · δ E η ( x f , δ ) (cid:1) > ǫ δ δ n ,(d) Vol (cid:0) B δ · δ E η ( x f , δ ) (cid:1) ≤ ǫ δ δ n .(e) E η ( x f , δ ) = ∅ .Thus, by breaking up the balls { B δ ( x f ) } into these categories we can write our covering as B ( p ) ⊆ N c [ c = B δ ( x c ) ∪ N d [ d = B δ ( x d ) ∪ N e [ e = B δ ( x e ) . (7.45)Since B ( p ) is itself a d -ball, we can use the Vitali condition of the covering and assumption (2) toconclude by a standard covering argument that N e X e = δ n − ≤ C ( n , δ ) , N c X c = δ n − + N d X d = δ n − ≤ C ( n , v) ǫ . (7.46)This does not quite finish the Proposition because we still have a collection of d -balls in our covering,therefore we must recover them. Let us remark that the only aspect about B ( p ) used in the above coveringwas that B ( p ) was a d -ball which satisfied condition (2). Thus, for each d -ball B δ ( x d ) let us repeat thiscovering process just introduced for B ( p ). If we do this for every d -ball then we arrive at the covering N d [ d = B δ ( x d ) ⊆ N c [ c = B δ ( x c ) ∪ N d [ d = B δ ( x d ) ∪ N e [ e = B δ ( x e ) , (7.47)such that we have the estimates N d X d = ( δ ) n − ≤ C ( n , v) ǫ N d X d = δ n − ≤ (cid:0) C ( n , v) ǫ (cid:1) , N c X c = ( δ ) n − ≤ C ( n , v) ǫ N d X d = δ n − ≤ (cid:0) C ( n , v) ǫ (cid:1) , N e X e = ( δ ) n − ≤ C ( n , δ ) N d X d = δ n − ≤ C ( n , δ ) · C ( n , v) ǫ . (7.48) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 73
Combining this with the original covering we obtain a covering of B ( p ) given by B ( p ) ⊆ N c [ c = B δ ( x c ) ∪ N e [ e = B δ ( x e ) ∪ N c [ c = B δ ( x c ) ∪ N d [ d = B δ ( x d ) ∪ N e [ e = B δ ( x e ) , = N d [ d = B δ ( x d ) ∪ [ j ≤ N jc [ c = B δ j ( x jd ) ∪ [ j ≤ N je [ e = B δ j ( x je ) , (7.49)with N d X d = ( δ ) n − ≤ (cid:0) C ( n , v) ǫ (cid:1) , X j ≤ N jc X c = ( δ j ) n − ≤ C ( n , v) ǫ + ( C ( n , v) ǫ ) , X j ≤ N je X e = ( δ j ) n − ≤ C ( n , δ ) (cid:0) + C ( n , v) ǫ (cid:1) . (7.50)Of course, we may now proceed to cover the d -balls { B δ ( x d ) } in the same manner once again. In fact, afterrepeating this inductive covering i times we see we arrive at a covering B ( p ) ⊆ N id [ d = B δ i ( x id ) ∪ [ j ≤ i N jc [ c = B δ j ( x jc ) ∪ [ j ≤ i N je [ e = B δ j ( x je ) , (7.51)with the estimates N id X d = ( δ i ) n − ≤ (cid:0) C ( n , v) ǫ (cid:1) i , X j ≤ i N jc X c = ( δ j ) n − ≤ X ≤ j ≤ i (cid:0) C ( n , v) ǫ (cid:1) j , X j ≤ i N je X e = ( δ j ) n − ≤ C ( n , δ ) X ≤ j ≤ i (cid:0) C ( n , v) ǫ (cid:1) j . (7.52)To finish the proof consider the closed discrete sets ˜ S i = S d = { x id } . Note that by construction we have˜ S i + ⊆ B δ i (cid:0) ˜ S i (cid:1) and we can use (7.52) to conclude the estimateVol( B δ i (cid:0) ˜ S i (cid:1) ) ≤ (cid:0) C ( n , v) ǫ (cid:1) i ( δ i ) . (7.53)Taking the Hausdorff limit we can construct ˜ S = lim i →∞ ˜ S i + , (7.54) and we arrive at the covering B ( p ) ⊆ ˜ S ∪ [ j [ c B δ j ( x jd ) ∪ [ j [ e B δ j ( x je ) . (7.55)Using (7.52), (7.53) and the inclusion ˜ S ⊆ B δ i (cid:0) ˜ S i (cid:1) we see for ǫ ≤ ǫ ( n , v) and 0 < r ≤ r − Vol( B r (cid:0) ˜ S (cid:1) ) → r → , X j X c ( δ j ) n − ≤ C ( n , v) ǫ , X j ≤ i N je X e = ( δ j ) n − ≤ C ( n , δ ) , (7.56)which finishes the proof of Proposition 7.42. (cid:3) Inductive Covering.
In this subsection we combine the coverings of Proposition 7.13, and Proposition7.42 into a geometric series of coverings in order to take a generic ball and produce from it a covering byballs which are either necks, regularity regions, or for which the volume ratio increases by some definiteamount.Let us start with the following lemma, which in terms of our ball notation will take a generic ball andproduce from it a collection of a -balls, b -balls, and e -balls. Combined with a modest additional recoveringargument this will allow us to prove the inductive covering proposition of this subsection. This in turn willbe used in the next section to prove the neck decomposition theorem itself. Let us begin with the followinglemma: Lemma 7.57.
Let ( M ni , g i , p i ) → ( X , d , p ) satisfy Vol ( B ( p )) > v > and | Ric | ≤ n − with < δ and < τ ≤ τ ( n ) fixed constants and ¯ V ≡ inf y ∈ B V η − ( y ) . Then there exists η ( n , v , δ, τ ) > such that we canwrite B ⊆ ˜ S ∪ [ a (cid:0) C , a ∪ N a ∩ B r a (cid:1) ∪ [ b B r b ( x b ) ∪ [ e B r e ( x e ) , where(a) N a ⊆ B r a ( x a ) is a ( δ, τ ) -neck region with associated singular set C , a ,(b) For each b-ball B r b ( x b ) we have r h ( x b ) > r b ,(d) For each e-ball B r e ( x e ) , if E e ≡ { y ∈ B r e ( x e ) : V η r e < V + η } , then E e = ∅ .(s) ˜ S ⊆ S ( X ) with H n − ( ˜ S ) = .Further, we have the content bound P a r n − a + P b r n − b + P e r n − e < C ( n , τ, δ ) and radius bound r a , r b , r e ≤ η . CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 75
Proof.
Let us pick η as in Propositions 7.13 and 7.42. Note first that we can cover B ( p ) by the Vitalicovering B ( p ) ⊆ [ f B η ( x f ) , (7.58)where { B η / ( x f ) } are disjoint. Then by a standard covering argument there are at most C ( n , τ, δ ) balls inthis set. Therefore if we focus our covering on each of these balls individually, we may then take the unionand not effect the content bound by more than another factor.Thus, by rescaling one of these balls to scale one and focusing on it, we can assume without loss ofgenerality that | Ric | ≤ η on B ( p ). Now let us recall that for every ball B r ( x ) we can define the η -pinchedpoints E η ( x , r ) by E η ( x , r ) ≡ { y ∈ B r ( x ) : V η r ( y ) < V + η } . (7.59)In our decomposition we want to first distinguish between whether B ( p ) is a b -ball, c -ball, d -ball, or e -ball.In the cases where it is either a b -ball or e -ball we are of course done, therefore we can assume B ( p ) iseither a c -ball or d -ball. The handling of the two cases is almost verbatim, therefore we will assume B ( p )is a c -ball. In this case, we can apply Proposition 7.13 in order to build the covering B ⊆ C ∪ N ∪ [ b B r b ( x b ) ∪ [ c B r c ( x c ) ∪ [ d B r d ( x d ) ∪ [ e B r e ( x e ) , (7.60)where N ⊆ B is a ( δ, τ )-neck and we have the content estimates X b r n − b + X d r n − d + X e r n − e ≤ C ( n , τ ) , X c r n − c ≤ C ( n , τ ) ǫ . (7.61)What we have gained from the above is that the remaining c -balls have small n − c -balls and d -balls in this decomposition before the proof is complete. Let us first dealwith the r d -balls. Indeed, if we apply Proposition 7.42 to each d -ball B r d ( x d ) then we obtain the covering B ( p ) ⊆ ˜ S ∪ C ∪ N ∪ [ b B r b ( x b ) ∪ [ c B r c ( x c ) ∪ [ e B r e ( x e ) , (7.62)where ˜ S = S d ˜ S d is a countable union of n − n − X c r n − c ≤ C ( n , τ ) · ǫ , X b r n − b + X e r n − e ≤ C ( n , τ ) . (7.63)Let us remark that our only original assumption to construct this covering was that B ( p ) was a c -ball.Therefore, we can repeat this construction on each c -ball B r c ( x c ) in order to build a covering B ( p ) ⊆ ˜ S ∪ [ a (cid:0) C , a ∪ N a ∩ B r a (cid:1) ∪ [ b B r b ( x b ) ∪ [ c B r c ( x c ) ∪ [ e B r e ( x e ) , (7.64) with the estimates X a r n − a ≤ + C ( n , τ ) ǫ , X c r n − c ≤ (cid:16) C ( n , τ ) · ǫ (cid:17) , X b r n − b + X e r n − e ≤ C ( n , τ ) (cid:16) + C ( n , τ ) ǫ (cid:17) . (7.65)If we continue to recover the c -balls, then after i iterations we have the covering B ( p ) ⊆ ˜ S ∪ [ a (cid:0) C , a ∪ N a ∩ B r a (cid:1) ∪ [ b B r b ( x b ) ∪ [ c B r c ( x c ) ∪ [ e B r e ( x e ) , (7.66)with the estimates X a r n − a ≤ i X j = (cid:16) C ( n , τ ) ǫ (cid:17) j , X c r n − c ≤ (cid:16) C ( n , τ ) · ǫ (cid:17) i , X b r n − b + X e r n − e ≤ C ( n , τ ) i X j = (cid:16) C ( n , τ ) ǫ (cid:17) j . (7.67)Now we may consider the discrete sets ˜ S ic S c { x ic } and their Hausdorff limit ˜ S c = lim ˜ S ic . Arguing as inProposition 7.42 we see for ǫ ≤ ǫ ( n , τ ) that H n − ( ˜ S c ) =
0. Combining this with our previous ˜ S set we thenarrive at the covering B ( p ) ⊆ ˜ S ∪ [ a ( C , a ∪ N a ∩ B r a (cid:1) ∪ [ b B r b ( x b ) ∪ [ e B r e ( x e ) , (7.68)which for ǫ ≤ ǫ ( n , τ ) satisfies H n − ( ˜ S ) = , X a r n − a ≤ , X b r n − b + X e r n − e ≤ C ( n , τ ) , (7.69)which completes the proof of the Lemma. (cid:3) Let us now apply the above Lemma in order to prove the following corollary, which is the main result ofthis section:
Proposition 7.70 (Inductive Covering) . Let ( M ni , g i , p i ) → ( X , d , p ) satisfy | Ric | ≤ n − with V ≡ inf y ∈ B ( p ) V > v > . Let us fix < δ and < τ ≤ τ ( n ) , then there exists v ( n , v , δ, τ ) > such that we can writeB ⊆ ˜ S ∪ [ a (cid:0) C , a ∪ N a ∩ B r a (cid:1) ∪ [ b B r b ( x b ) ∪ [ v B r v ( x v ) , CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 77 where(a) N a ⊆ B r a ( x a ) is a ( δ, τ ) -neck region with associated singular set C , a ,(b) For each b-ball B r b ( x b ) we have r h ( x b ) > r b ,(d) For each v-ball B r v ( x v ) , if V v ≡ inf y ∈ B re V r e then V v ≥ V + v .(s) ˜ S ⊆ S ( X ) with H n − ( ˜ S ) = .Further, we have the content bound P a r n − a + P b r n − b + P v r n − v < C ( n , τ, δ ) .Proof. This Proposition is very similar to the previous Lemma, we essentially need only one additional cov-ering in order to deal with the fact that V here is coming from the volume pinching at scale one, instead ofscale η − .Now more precisely, let us begin by picking η as in Propositions 7.13 and 7.42 and let us cover B ( p ) bythe Vitali covering B ( p ) ⊆ [ f B η ( x f ) , (7.71)where { B η / ( x f ) } are disjoint. By a standard covering argument we have that there are at most C ( n , τ, δ )balls in this collection. For each ball B η ( x f ) let us apply Lemma 7.57 to find the covering given by B ( p ) ⊆ [ B η ( x f ) ⊆ ˜ S ∪ [ a (cid:0) C , a ∪ N a ∩ B r a (cid:1) ∪ [ b B r b ( x b ) ∪ [ e B r e ( x e ) , (7.72)such that we have the content bounds P a r n − a + P b r n − b + P e r n − e < C ( n , τ, δ ) and such that r e ≤ η . Wewill see that these e -balls will be the v -balls of the Proposition with v defined appropriately. To see this, letus be careful and note that the e -balls in this covering are with respect to one of the balls B η ( x f ). Thereforefor each e -ball we have the estimateinf B re ( x e ) V η ( y ) ≥ inf B η ( x f ) V η ( y ) + η ≥ inf B (1 + η ( p ) V ( y ) + η. (7.73)In order to finish the proof let us notice the simple estimateinf B (1 + η ( p ) V ( y ) ≥ Vol( B − )Vol( B − + η ) ) inf B ( p ) V ( y ) > (1 − c ( n ) η ) V > V − η , (7.74)where Vol( B − r ) is the volume of the ball of radius r in hyperbolic space, and the last inequality holds for η < η ( n ). Combining this with the previous inequality we arrive at the estimateinf B re ( x e ) V η ( y ) ≥ inf B (1 + η ( p ) V ( y ) + η ≥ V + η ≡ V + v , (7.75)which finishes the proof of the Proposition. (cid:3) Proof of Neck Decomposition of Theorem 7.1.
In this subsection we finish the proof of the NeckDecomposition Theorem. The proof will conclude by recursively applying the inductive covering of Propo-sition 7.70 a finite number of times until we have our desired covering.More precisely, let M satisfy the assumptions of the Theorem, and let V ≡ inf B ( p ) V ( y ) >
0. Let us thenfix v ( n , V , δ, τ ) as in Proposition 7.70. Then if we apply Proposition 7.70 we arrive at the covering B ⊆ ˜ S ∪ [ a (cid:0) ( C , a ∪ N a ) ∩ B r a (cid:1) ∪ [ b B r b ( x b ) ∪ [ v B r v ( x v ) , (7.76)such that we have the content estimates X a ( r a ) n − + X b ( r b ) n − + X v ( r v ) n − < C ( n , V , τ, δ ) . (7.77)and such that for each v -ball B r v ( x v ) we haveinf B r v ( x v ) V r v ( y ) > V + v . (7.78)Now let us observe that we may apply Proposition 7.70 to each v -ball B r v ( x v ) itself. In this case, weobtain a new covering of B ( p ) given by B ⊆ ˜ S ∪ [ a (cid:0) ( C , a ∪ N a ) ∩ B r a (cid:1) ∪ [ b B r b ( x b ) ∪ ˜ S ∪ [ a (cid:0) ( C , a ∪ N a ) ∩ B r a (cid:1) ∪ [ b B r b ( x b ) ∪ [ v B r v ( x v ) , = [ j ≤ ˜ S j ∪ [ j ≤ [ a (cid:0) ( C j , a ∪ N ja ) ∩ B r ja (cid:1) ∪ [ j ≤ [ b B r jb ( x jb ) ∪ [ v B r v ( x v ) , (7.79)such that we have the content estimates X a ( r a ) n − + X b ( r b ) n − + X v ( r v ) n − < C ( n , V , τ, δ ) X v ( r v ) n − ≤ C ( n , V , τ, δ ) , (7.80)and such that for each v -ball B r v ( x v ) we haveinf B r v ( x v ) V r v ( y ) > V + v . (7.81)By continuing this scheme and applying Proposition 7.70 to the v -balls i times we arrive at a covering B ⊆ [ j ≤ i ˜ S j ∪ [ j ≤ i [ a (cid:0) ( C j , a ∪ N ja ) ∩ B r ja (cid:1) [ j ≤ i [ a (cid:0) N ja ∩ B r ja (cid:1) ∪ [ j ≤ i [ b B r jb ( x jb ) ∪ [ v B r iv ( x iv ) , (7.82)such that we have the content estimates X j ≤ i X a ( r ja ) n − + X j ≤ i X b ( r jb ) n − + X v ( r iv ) n − ≤ C ( n , V , τ, δ ) i , (7.83)and such that for each v -ball B r iv ( x iv ) we haveinf B riv ( x iv ) V r iv ( y ) > V + iv . (7.84) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 79
In order to finish the proof let us observe that if i > v − then there must be no v -balls, as in this caseinf B riv ( x iv ) V r iv ( y ) > V + iv >
1, which is not possible. Thus for i > v − we have built our desired coveringand finished the proof of Theorem 7.1.8. P ROOF OF L -C URVATURE E STIMATE
In this short section we combine the neck region estimates of Theorem 3.10 with the neck decompositionof Theorem 7.1 in order to finish the proof of Theorem 1.6. Indeed, for δ > B ( p ) ⊆ [ a (cid:0) N a ∩ B r a (cid:1) ∪ [ b B r b ( x b ) , (8.1)where N a ⊆ B r a ( x a ) is a ( δ, τ )-neck, r h ( x b ) ≥ r b , and X a r n − a + X b r n − b < C ( n , v , δ ) . (8.2)Now since the Ricci curvature is uniformly bounded, as in the remark after Definition 2.12 we have scaleinvariant W , -estimates on the metric on B r b ( x b ). In particular, we have the estimate r − nb Z B rb ( x b ) | Rm | < C ( n ) . (8.3)On the other hand, if δ ≤ δ ( n , v), then by Theorem 3.10 we have the scale invariant L estimate on eachneck region N a given by: r − na Z N a ∩ B ra | Rm | < C ( n ) . (8.4)Combining these estimates with the content estimate of (8.2) we get the estimate Z B ( p ) | Rm | ≤ X a Z N a ∩ B ra | Rm | + X b Z B rb ( x b ) | Rm | ≤ C ( n ) (cid:16) X a r n − a + X b r n − b (cid:17) ≤ C ( n , v) , (8.5)which completes the proof of Theorem 1.6. (cid:3)
9. P
ROOF OF T HEOREM
Claim: there exists η ( n ) > C ( n ) > |∇| Rm || ≤ (cid:0) − η ( n ) (cid:1) |∇ Rm | + C ( n ) |∇ Ric | . (9.1)Let first us assume the claim and prove the theorem. By direct computation and the Kato inequality, we have ∆ p | Rm | + = ∆ | Rm | p | Rm | + − |∇| Rm | | | Rm | + / (9.2) ≥ η ( n ) |∇ Rm | p | Rm | + − C ( n ) |∇ Ric | p | Rm | + − C ( n ) | Rm | + ∇ Ric ∗ Rm p | Rm | + ≥ η ( n ) |∇ Rm | p | Rm | + − C ( n ) |∇ Ric | − C ( n ) | Rm | + ∇ Ric ∗ Rm p | Rm | + . (9.4)where one can find an explicit formula of the quadratic term ∇ Ric ∗ Rm in Proposition 2.4.1 of [To]. Choosea cutoff function φ as in [ChCo1] such that φ ≡ B ( p ) and φ ≡ B ( p ) with |∇ φ | + | ∆ φ | ≤ C ( n ).Multiplying φ to (9.2) and integrating by parts, we have Z B ( p ) |∇ Rm | p | Rm | + ≤ C ( n ) Z B ( p ) (cid:16) |∇ Ric | + p | Rm | + + | Rm | (cid:17) . (9.5)On the other hand, by Cauchy inequality, we have Z B ( p ) |∇ Rm | / ≤ Z B ( p ) |∇ Rm | p | Rm | + / Z B ( p ) ( | Rm | + ! / . (9.6)By the L curvature estimate of Theorem 1.6, we have proved Theorem 1.11 by assuming Claim.Now we wish to prove the claim, which is essentially just a repeated application of the second Bianchiidentity. For any fixed y ∈ M , choose a normal coordinate such that |∇| Rm || ( y ) = ∂ | Rm | ( y ). Then at y , wehave |∇| Rm | | = | ∂ | Rm | | = |h∇ Rm , Rm i| ≤ X i jkl R i jkl , / X i jkl R i jkl / . (9.7)To prove the claim, it suffices to show (1 + σ ( n )) P i jkl R i jkl , ≤ P i jklp R i jkl , p + C ( n ) P i jp Rc i j , p for some dimen-sional constant σ ( n ) and C ( n ). In fact, it suffices to show P i jkl R i jkl , ≤ C ( n ) (cid:16)P i jkl P p ≥ R i jkl , p + P i jp Rc i j , p (cid:17) : = C ( n ) Ξ . Hence we only need to prove R αβγδ, ≤ C ( n ) Ξ for any fixed α, β, γ, δ = , · · · , n .Case 1: For α, β, γ, δ ≥
2, then by second Bianchi identity and Cauchy inequality, we have R αβγδ, = (cid:16) R αβδ ,γ + R αβ γ,δ (cid:17) ≤ (cid:16) R αβδ ,γ + R αβ γ,δ (cid:17) ≤ Ξ . (9.8)Case 2: For α = β, γ, δ ≥
2, then by second Bianchi identity and Cauchy inequality, we have R βγδ, = (cid:16) ( R βδ ,γ + R β γ,δ (cid:17) ≤ (cid:16) R βδ ,γ + R β γ,δ (cid:17) ≤ Ξ . (9.9)Case 3: For α = δ = β, γ ≥
2, then by Cauchy inequality, we have R βγ , = Rc βγ, − X α ≥ R αβγα, ≤ n Rc βγ, + X α ≥ R αβγα, ≤ n Ξ + X α ≥ Ξ ≤ n (2 n − Ξ , (9.10) CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 81 where we have used the estimates in Case 1 to the second inequality.Thus, by the symmetric relations of curvature tensor, we have proved R αβγδ, ≤ C ( n ) Ξ . Hence, weprove P i jkl R i jkl , ≤ C ( n ) (cid:16)P i jkl P p ≥ R i jkl , p + P i jp Rc i j , p (cid:17) . This finishes the proof of the claim and then wecomplete the proof of Theorem 1.11. 10. P ROOF OF T HEOREM n − B ( p ) ∩ S ( X ) ⊆ [ a (cid:0) C , a ∩ B r a (cid:1) ∪ ˜ S δ ( X ) , (10.1)where C , a ⊆ B r a is the singular set of a ( δ, τ )-neck region N a ⊆ B r a and ˜ S δ has n − C , a is rectifiable, we therefore have that S ( X ) is rectifiable as well, asclaimed.In order to prove H n − ( S ( X )) < C ( n , v), let us observe by Theorem 3.10 that for τ < τ ( n ) and δ < δ ( n , v)we have H n − ( C , a ∩ B r a ) < C ( n , v) r n − a , and by Theorem 7.1 we have that P r n − a ≤ C ( n , v , δ, τ ) ≤ C ( n , v).Combining these we get the estimate H n − ( S ( X ) ∩ B ) ≤ X a H n − ( C , a ∩ B r a ) ≤ C ( n , v) X a r n − a ≤ C ( n , v) , (10.2)which proves the desired finiteness estimate.Finally, let us show that n − R n − × C ( S / Γ ). Indeed, by theneck decomposition we have for each δ < δ ( n , v) that x ∈ S ( X ) \ ˜ S δ lives in some C , a ∩ B r a , where C , a isthe singular set of a ( δ, τ )-neck region N a ⊆ B r a . From this we immediately get for any such x ∈ S ( X ) \ ˜ S δ that every tangent cone is δ -GH close to R n − × C ( S / Γ ). Now H n − ( ˜ S δ ) =
0, so let us define ˜ S ≡ S j ˜ S − j .Note that H n − ( ˜ S ) = x ∈ S ( X ) \ ˜ S we must therefore have that every tangent cone is actuallyisometric to R n − × C ( S / Γ ). This finishes the proof of Theorem 1.14.A CKNOWLEGMENTS
The first author would like to thank his advisor Gang Tian for constant encouragement and for usefulconversations during this work. Partial work was done while the first author was visiting the MathematicsDepartment at Northwestern University and he would like to thank the department for its hospitality and forproviding a good academic environment. The second author would like to thank NSF for its support undergrant DMS-1406259. R EFERENCES [A89] M.T. Anderson, Ricci curvature bounds and Einstein metrics on compact manifolds, Jour. Amer. Math. Soc, vol 2 (1989),455-490.[A90] M. T. Anderson, Convergence and rigidity of metrics under Ricci curvature bounds,
Invent. Math.
102 (1990), 429–445.[A93] M. T. Anderson, Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem,
Duke Math. J. , vol 68 (1993),67–82.[A94] M. T. Anderson, Einstein metrics and metrics with bounds on Ricci curvature,
Proceedings of the International Congressof Mathematicians , 1994, 443–452.[BKN89] S. Bando, A. Kasue and H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decayand maximal volume growth,
Invent. Math.
97, (1989), 313-349.[Ch01] J. Cheeger, Degeneration of Riemannian metrics under Ricci curvature bounds,
Lezioni Fermiane. [Fermi Lectures]Scuola Normale Superiore,
Pisa, 2001.[Ch2] J. Cheeger, Integral bounds on curvature, elliptic estimates, and rectifiability of singular set,
Geom. Funct. Anal. Ann. Math. (1), 189-237 (1996)[ChCo2] J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below. I.,
J. Differ. Geom. (3),406-480 (1997)[CCT02] J. Cheeger, T. H. Colding and G. Tian, On the singularities of spaces with bounded Ricci curvature Geom. Funct. Anal. ,12 No. 5 (2002) 873-914.[ChNa13] J. Cheeger and A. Naber, Lower bounds on Ricci curvature and Quantitative Behavior of Singular Sets,
Invent. Math.
191 (2013), 321–339.[ChNa15] J. Cheeger and A. Naber, Regularity of Einstein manifolds and the codimension 4 conjecture,
Ann. of Math. (2) 182(2015), no. 3, 1093–1165.[CNV15] J. Cheeger, A. Naber and D. Valtorta, Critical sets of elliptic equations,
Comm. Pure Appl. Math.
68 (2015), no. 2,173–209.[ChTi05] J. Cheeger and G. Tian, Anti-self-duality of curvature and degeneration of metrics with special holonomy,
Comm. Math.Phys.
255 (2005), no. 2, 391–417.[CD14] X.X. Chen and S.K. Donaldson, Integral bounds on curvature and Gromov-Hausdorff limits,
J. Topol.
Geom. and Functional Analysis
Vol 23, Issue 1 (2013), 134–148.[Ha93] R. Hamilton, A matrix Harnack estimate for the heat equation,
Comm. Anal. Geom.
Volume 1, (1993), 113-126.[Kot07] B. Kotschwar, Hamilton’s gradient estimate for the heat kernel on complete manifolds,
Proc. Amer. Math. Soc.
Acta Math.
156 (1986), no. 3-4, 153–201.[MN14] A. Mondino and A. Naber, Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds I, (preprint)
Proceedings of the International Congress of Mathematicians , Vol II 2014,911–938.[NV14] A. Naber and D. Valtorta, Volume estimates on the critical sets of solutions to elliptic PDEs, (preprint) (preprint)
Springer: Graduate Texts in Mathematics , 171.[Rei60] E. R. Reifenberg, Solution of the Plateau Problem for m-dimensional surfaces of varying topological type, Acta Math.104 (1960), 1–92.[SY] R. Schoen, S.T. Yau, Lectures on Differential Geometry,
International Press of Boston , 2010. CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE 83 [T90] G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class,
Invent. Math.
101 (1990), no. 1,101–172.[To] P. Topping, Lectures on the Ricci flow,
London Mathematical Society Lecture Note Series,
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