Existence of Typical Scales for Manifolds with Lower Ricci Curvature Bound
aa r X i v : . [ m a t h . DG ] M a r EXISTENCE OF TYPICAL SCALES FOR MANIFOLDSWITH LOWER RICCI CURVATURE BOUND
DOROTHEA JANSEN
Abstract.
For collapsing sequences of Riemannian manifolds whichsatisfy a uniform lower Ricci curvature bound it is shown that there is asequence of scales such that for a set of good base points of large measurethe pointed rescaled manifolds subconverge to a product of a Euclideanand a compact space. All Euclidean factors have the same dimension,all possible compact factors satisfy the same diameter bounds and theirdimension does not depend on the choice of the base point (along a fixedsubsequence).
Contents
1. Notation 52. Local construction 83. Global construction 36References 50If a sequence of Riemannian manifolds satisfies a uniform lower sectionalcurvature bound, this bound carries over to the possibly non-smooth Alexan-drov limit. If the limit is actually a smooth manifold, by Yamaguchi’s Fibra-tion Theorem, [Yam91], the manifolds fibre over the limit in the followingway: If M i is a sequence of n -dimensional manifolds with a uniform lowersectional curvature bound and a uniform upper diameter bound convergingto a compact manifold N of lower dimension, then for sufficiently large i ∈ N there are fibrations M i → N which are close to Riemannian submersions.Moreover, up to a finite covering, each fibre is the total space of a fibrationover a torus.For proving the latter, a crucial argument is the following: Consider thepre-image of some ball under the fibration M i → N . After rescaling themetric up, this pre-image converges to a product of a Euclidean and a com-pact space. In fact, it is possible to replace the rescaling factors by larger Date : March 29, 2017.2010
Mathematics Subject Classification.
Key words and phrases.
Lower Ricci curvature bound, Gromov-Hausdorff convergence.This work was supported by the Gottfried Wilhelm Leibniz-Preis of Prof. Dr. BurkhardWilking and the SFB 878: Groups, Geometry & Actions, at the University of Münster. ones such that the limit again has the form of such a product, but the Eu-clidean factor has higher dimension. This can be iterated until, finally, thecompact factor vanishes. Such scaling factors are called typical scales . Sim-ilar techniques have been used by e.g. Shioya and Yamaguchi in [SY00] andKapovitch, Petrunin and Tuschmann in [KPT10].If a sequence of manifolds only satisfies a uniform lower Ricci curvaturebound, Yamaguchi’s Fibration Theorem might fail. This was proven by An-derson in [And92]. However, in recent years Cheeger and Colding obtaineddeep structure results for limits of such sequences, [CC97, CC00a, CC00b],by using measured Gromov-Hausdorff convergence : After renormalizing themeasure of the manifolds and passing to a subsequence, the manifolds con-verge to a metric measure space such that the (renormalised) measures con-verge to the limit measure. The uniform lower Ricci curvature bound carriesover to the limit in the sense that the limit measure still satisfies the Bishop-Gromov Theorem.Another difficulty occurring with only a lower Ricci curvature bound isthe following: Unlike for lower section curvature bounds, tangent cones ofthe limit space need not be metric cones. In fact, in the case of a collapsingsequence, the tangent cones at some point may depend on the choice of therescaling sequence, cf. [CC97]. However, Colding and Naber [CN12] provedthat any limit of a sequence of manifolds with uniform lower Ricci curvaturebound contains a connected subset of full measure such that for each pointin this subset the tangent cone is unique and a Euclidean space of a fixeddimension k ∈ N . This k is called the dimension of the limit space. Noticethat this dimension is at most the Hausdorff-dimension of the space. Inparticular, k < n .If a collapsing sequence of manifolds M i satisfies the lower Ricci curvaturebound − ε i , where ε i → , and if this sequence converges to a Euclideanspace, then the Rescaling Theorem of Kapovitch and Wilking in [KW11]already provides the existence of one sequence of typical scales. For thissequence, the blow-ups of the manifolds split into products of this Euclideanspace and a compact factor. The main theorem of this paper generalisesthis statement by allowing arbitrary limit spaces and a uniform lower Riccicurvature bound. Main Theorem.
Let ( M i , p i ) i ∈ N be a collapsing sequence of pointed com-plete connected n -dimensional Riemannian manifolds which satisfy the uni-form lower Ricci curvature bound Ric M i ≥ − ( n − and converge to a limit ( X, p ) of dimension k < n in the measured Gromov-Hausdorff sense. Thenfor every ε ∈ (0 , there exist a subset of good points G ( p i ) ⊆ B ( p i ) satis-fying vol( G ( p i )) ≥ (1 − ε ) · vol( B ( p i )) , a sequence λ i → ∞ and a constant D > such that the following holds:For any choice of base points q i ∈ G ( p i ) and every sublimit ( Y, q ) of ( λ i M i , q i ) i ∈ N there is a compact metric space K of dimension l ≤ n − k and XISTENCE OF TYPICAL SCALES 3 diameter D ≤ diam( K ) ≤ D such that Y splits isometrically as a product Y ∼ = R k × K. Moreover, for any base points q i , q ′ i ∈ G ( p i ) such that, after passing to asubsequence, both ( λ i M i , q i ) → ( R k × K, · ) and ( λ i M i , q ′ i ) → ( R k × K ′ , · ) asbefore, dim( K ) = dim( K ′ ) . Observe that dim( K ) < n − k might occur in the situation of the theorem:Consider the sequence of flat tori M i := S × S ( i ) × S ( i ) where S ( r ) denotes a circle of radius r > . In this example, it is easy to imagine the lasttwo factors ‘collapsing to a point’ in the limit, although the very last factorcollapses faster than central one. Hence, M i converges to S . For λ i = i , therescaled manifolds λ i M i converge to R × S . Using the notation of the maintheorem, one has n = 3 , k = 1 and l = 1 < n − k .Furthermore, note that the theorem does not prove that all possible com-pact spaces need to have the same dimension, but that the dimension onlydepends on the regarded subsequence and not on the choice of the basepoints: Let M i = S ( i ) for i ∈ N , M i = S ( i ) for i ∈ N +1 and λ i = i . Then M i collapses to a point, but ( λ i M i ) i ∈ N → R ×{ pt } and ( λ i M i ) i ∈ N +1 → R × S .Now turn to the proof of the main theorem. Let ( M i , p i ) i ∈ N be a collapsingsequence as in the main theorem and let ( X, p ) denote its limit. First willbe proven that for points q i ∈ M i and a small radius r > the conclusionof the main theorem holds correspondingly on B r ( q i ) for a subset of goodpoints G r ( q i ) and a sequence of scales λ i → ∞ , i.e. a ‘local’ version of themain theorem will be established.Recall that the set of good points G r ( q i ) ∈ B r ( q i ) and the scales λ i → ∞ have to satisfy the following: Any point x i ∈ G r ( q i ) needs to have the prop-erty that each sublimit of ( λ i M i , x i ) is the product of R k with a compactmetric space where the compact spaces (essentially) all have the same di-mension. This is achieved in two steps.First, let G r ( q i ) ⊆ B r ( q i ) denote the set of all points x i such that allsublimits of ( µ i M i , x i ) split off an R k -factor where µ i → ∞ is an arbitrarysequence. In particular (once λ i → ∞ is constructed), for any x i ∈ G r ( q i ) ,any limit of the sequence ( λ i M i , x i ) splits off an R k -factor. That these sets G r ( q i ) have large volume inside B r ( q i ) is obtained by generalising results ofCheeger, Colding [CC00a] and Kapovitch, Wilking [KW11] involving mod-ified distance functions coming from splitting theorems. For more details,see subsection 2.1.Next, define scales λ i → ∞ and another subset G r ( q i ) ⊆ B r ( q i ) of largevolume as the set of points x i such that ( λ i M i , x i ) has small distance toa product of R k and a compact metric space of diameter . The exis-tence of such λ i → ∞ and G r ( q i ) is obtained by using the Rescaling The-orem of Kapovitch and Wilking in [KW11]. For further explanations, seesubsection 2.2. D. JANSEN
These scales λ i → ∞ and the intersection G r ( q i ) = G r ( q i ) ∩ G r ( q i ) givethe splitting result in the local version of the main theorem.In order to finish the local version, prove that two such limits have thesame dimension. First, the following special case is investigated: Supposethat two points x i , y i ∈ G r ( q i ) are connected by an integral curve of a vectorfield whose flow is measure preserving and bi-Lipschitz (on a set of largeenough volume). In this situation, via Gromov-Hausdorff convergence oneobtains a bi-Lipschitz map between subsets of positive volume inside of thelimit spaces of ( λ i M i , x i ) and ( λ i M i , y i ) , respectively. In particular, theselimits need to have the same dimension.For the general situation, recall that the flow of any divergence-free vectorfield is measure preserving. Moreover, using (slight generalisations of) resultsin [KW11], it is bi-Lipschitz if its derivative is small—in some L α -norm, α > , and taking some average value (in volume sense). In fact every twopoints of the set G r ( q i ) are connected by the curve of such a vector field (orare at least sufficiently close to its start and end point). For more details,see subsection 2.3.For verifying the main theorem, i.e. defining G ( p i ) and λ i → ∞ , fix r > and finitely many sequences ( q i ) i ∈ N such that the union of the subsets of goodpoints G r ( q i ) ⊆ B r ( q i ) has sufficiently large volume in B ( p i ) . Define G ( p i ) as the union of these G r ( q i ) . Now difficulties arise since each sequence ( q i ) i ∈ N provides its own sequence of scales λ i → ∞ and these sequences might bepairwise different, but the main theorem requires only a single sequence ofscales. This problem can be solved if, given two such sequences ( q i ) i ∈ N and ( q i ) i ∈ N and their corresponding scales λ i → ∞ and λ i → ∞ , the localversion of the main theorem still holds for ( q i ) i ∈ N and λ i → ∞ (instead of λ i → ∞ ). Indeed, this is achieved by a clever choice of the finitely many ( q i ) i ∈ N utilizing the Hölder continuity result of Colding and Naber [CN12].For more intuitive details on this approach, see the introduction to section 3.This paper is structured as follows: First, notation is fixed in section 1.The subsequent sections deal with the proof of the main theorem: First,section 2 proves the above mentioned local version of the main theorem: Ifpoints q i and some small r > satisfy that the rescaled manifolds ( r M i , q i ) are sufficiently close to R k , then the statement of the main theorem holds onthe ball B r ( q i ) analogously. As explained before, using finitely many of suchsequences q i and taking the union of the obtained subsets G r ( q i ) ⊆ B r ( q i ) will be used in section 3 to prove the main theorem. Acknowledgements.
The results of this paper are part of the author’s PhDthesis . The author would like to thank her advisor Burkhard Wilking forhis guidance and expertise. Available at https://miami.uni-muenster.de.
XISTENCE OF TYPICAL SCALES 5 Notation
For the sake of clarifying notation, recall the following theorem.
Theorem 1.1 (Bishop-Gromov Theorem, [Pet06, Chapter 9, Lemma 1.6]) . Let M be a complete n -dimensional Riemannian manifold with lower Riccicurvature bound Ric M ≥ ( n − · κ for some κ ∈ R and let p ∈ M . Thenthe map r vol M ( B r ( p )) V nκ ( r ) is monotonically decreasing with limit as r → , where V nκ ( r ) is the volumeof an r -ball in the n -dimensional space form of sectional curvature κ .In particular, for R ≥ r > , vol M ( B R ( p )) ≤ V nκ ( R ) V nκ ( r ) · vol M ( B r ( p )) . This factor is independent of M and denoted byC BG ( n, κ, r, R ) := V nκ ( R ) V nκ ( r ) . Throughout this paper, the Bishop-Gromov Theorem will always be ap-plied using the notion C BG ( n, κ, r, R ) . Regarding C BG as a function of radiior curvature bound, it has the following properties: Fix any n ∈ N , c ≥ , y > , − ≤ κ ≤ and R ≥ r > . Then C BG ( n, − , r, cr ) ≤ C BG ( n, − , R, cR ) , C BG ( n, κ, r, R ) ≤ C BG ( n, , r, R ) and lim x →∞ C BG ( n, − , x, x + y ) = e ( n − y .In the following, all metric spaces are assumed to be proper, i.e. closedballs are compact. Recall that a sequence ( X i , p i ) of pointed metric spacesconverges to ( X, p ) (in the pointed Gromov-Hausdorff sense) if d GH ( B Xr ( p ) , B Yr ( q )) → for all r > where d GH ( B Xr ( p ) , B Yr ( q )) denotes the pointed Gromov-Haus-dorff distance of the (compact) balls ( ¯ B r ( p ) , p ) and ( ¯ B r ( q ) , q ) . In this case, ( X, p ) is called (pointed Gromov-Hausdorff) limit of ( X i , p i ) . If ( X, p ) occursas limit of a convergent subsequence, it is called sublimit , and the sequence ( X i , p i ) is said to subconverge . Note that limits are unique up to isometry.If d GH ( B X / ε ( p ) , B Y / ε ( q )) ≤ ε for some ε > , this is denoted by d GH (( X, p ) , ( Y, q )) ≤ ε . Recall the following correspondence of Gromov-Hausdorff convergence andGromov-Hausdorff approximations where maps f : X → Y and g : Y → X between compact metric spaces are called ( ε -)Gromov-Hausdorff approxima-tions or ε -approximations between the spaces ( X, p ) and ( Y, q ) if f ( p ) = q , g ( q ) = p and the following holds for all x, x , x ∈ X and y, y , y ∈ Y : | d X ( x , x ) − d Y ( f ( x ) , f ( x )) | < ε, d X ( g ◦ f ( x ) , x ) < ε, | d Y ( y , y ) − d X ( g ( y ) , g ( y )) | < ε, d Y ( f ◦ g ( y ) , y ) < ε . D. JANSEN
Proposition 1.2.
Let ( X, d X , p ) and ( X i , d X i , p i ) , i ∈ N , be pointed metricspaces. If the X i and X are length spaces, the following are equivalent: a) ( X i , p i ) → ( X, p ) . b) There is a sequence ε i → and ε i -approximations ( f i , g i ) between theballs ( ¯ B X i / ε i ( p i ) , p i ) and ( ¯ B X / ε i ( p ) , p ) . c) For any function g : R > → R > with lim x → g ( x ) = 0 there exists ε i → with d GH ( B X i / ε i ( p i ) , B X / ε i ( p )) ≤ g ( ε i ) . For convergent spaces ( X i , p i ) → ( X, p ) , maps as in b) are always implic-itly fixed. In this situation, recall that a sequence of points q i ∈ ¯ B X i / ε i ( p i ) issaid to converge to a point q ∈ X , denoted by q i → q , if f i ( q i ) converges to q (in X ). In particular, p xi := g i ( x ) → x . In this situation, ( X i , p xi ) → ( X, x ) as well.Further, note the following fact: If ( X, p ) and ( Y, q ) are pointed lengthspaces and R ≥ r > , then d GH ( B Xr ( p ) , B Yr ( q )) ≤ · d GH ( B XR ( p ) , B YR ( q )) .In particular, convergence d GH ( B X i R ( p i ) , B XR ( p )) → for some R > impliesconvergence d GH ( B X i r ( p i ) , B Xr ( p )) → for all r ≤ R .For the sake of simpler notation, instead of passing to a subsequenceoccasionally ultralimits will be used. Recall that a non-principal ultrafilteron N is a finitely additive probability measure ω on N such that all subsets S ⊆ N are ω -measurable with value ω ( S ) ∈ { , } and ω ( S ) = 0 if S is finite,and that for any bounded sequence ( a i ) i ∈ N of real numbers there exists aunique real number lim ω a i such that ω ( { i ∈ N | | a i − lim ω a i | < ε } ) = 1 for every ε > .Given a non-principal ultrafilter ω and pointed metric spaces ( X i , d X i , p i ) ,the metric space lim ω ( X i , p i ) := ( X ω , d ω ) is called ultralimit of ( X i , p i ) where X ω := { [( x i ) i ∈ N ] | x i ∈ X i and sup i ∈ N d X i ( x i , p i ) < ∞} for ( x i ) i ∈ N ∼ ( y i ) i ∈ N if and only if lim ω d X i ( x i , y i ) = 0 and d ω ([( x i ) i ∈ N ] , [( y i ) i ∈ N ]) := lim ω d X i ( x i , y i ) . Lemma 1.3.
Let ( X i , d X i , p i ) and ( Y i , d Y i , q i ) , i ∈ N , be pointed lengthspaces. a) Let ω be a non-principal ultrafilter on N . Then lim ω ( X i , p i ) is a sub-limit in the pointed Gromov-Hausdorff sense. Concretely, there existsa subsequence ( i j ) j ∈ N such that both ( X i j , p i j ) → lim ω ( X i , p i ) and ( Y i j , q i j ) → lim ω ( Y i , q i ) as j → ∞ in the pointed Gromov-Hausdorff sense. XISTENCE OF TYPICAL SCALES 7 b) The sublimit of a sequence of pointed length spaces in the pointed Gro-mov-Hausdorff sense is the ultralimit with respect to a non-principalultrafilter. To be more precise: If ( X, d X , p ) and ( Y, d Y , q ) are pointedlength spaces and ( i j ) j ∈ N is a subsequence such that both ( X i j , p i j ) → ( X, p ) and ( Y i j , q i j ) → ( Y, q ) as j → ∞ in the pointed Gromov-Hausdorff sense, then there exists anon-principal ultrafilter ω on N such that there are isometries lim ω ( X i , p i ) ∼ = ( X, p ) and lim ω ( Y i , q i ) ∼ = ( Y, q ) . Let ( M i , p i ) i ∈ N be a sequence of pointed complete connected n -dimension-al Riemannian manifolds with lower Ricci curvature bound Ric M i ≥ − ( n − .If vol M i ( B ( p i )) → as i → ∞ , this sequence is said to be collapsing . In thissituation, renormalised limit measures are used, cf. [CC97, section 1]: Let ( M i , p i ) i ∈ N be a collapsing sequence as above. Then ( M i , p i ) subconvergesto a metric space ( X, p ) such that a ‘renormalisation’ of the measures vol M i converges to a limit measure vol X . In fact, the following is true. Theorem 1.4 ([CC97, Theorem 1.6, Theorem 1.10]) . Let ( M i , p i ) i ∈ N be asequence of pointed complete connected n -dimensional Riemannian manifoldssatisfying the uniform lower Ricci curvature bound Ric M i ≥ − ( n − . Then ( M i , p i ) subconverges to a metric space ( X, p ) in the pointed Gromov-Haus-dorff sense and there exists a Radon measure vol X on X such that for all x ∈ X , x i → x and r > , vol M i ( B M i r ( x i ))vol M i ( B M i ( p i )) → vol X ( B Xr ( x )) as i → ∞ . Moreover, for any R ≥ r > and x ∈ X , vol X ( B XR ( x ))vol X ( B Xr ( x )) ≤ C BG ( n, − , r, R ) . This vol X is called (renormalised) limit measure . Observe that the limit measure of a sequence ( M i , p i ) depends on thechoice of the base points and the considered subsequence, cf. again [CC97,section 1]. Moreover, observe the following: Gromov’s Pre-compactness The-orem ensures subconvergence (for pointed Riemannian manifolds of the samedimension with a lower Ricci curvature bound), but the above theorem guar-antees more, namely subconvergence (for the same class) including conver-gence of the (renormalised) measures. Throughout this paper, only mea-sured Gromov-Hausdorff convergence will be used, i.e. whenever a sequence ( M i , p i ) i ∈ N converges to a limit space ( X, p ) , this limit is equipped with ameasure vol X as in the above theorem.Let the complete pointed metric space ( X, p ) be the pointed Gromov-Hausdorff limit of a sequence of pointed connected n -dimensional Riemann-ian manifolds ( M i , p i ) satisfying the uniform lower Ricci curvature bound D. JANSEN
Ric M i ≥ − ( n − . As introduced in [CC97, p. 408], a tangent cone at x ∈ X is a Gromov-Hausdorff limit of ( λ i X, x ) where λ i → ∞ as i → ∞ . In gen-eral, this limit depends on the choice of x ∈ X and the sequence λ i → ∞ . Ifthe limit is independent of the choice of λ i → ∞ , it is denoted by C x X . If C x X = R k , this point x is called k -regular and the set of all k -regular pointsis denoted by R k . Furthermore, R = S k R k denotes the set of all regularpoints.Moreover, Cheeger and Colding proved that there are points such thatnon-unique tangent cones of different dimensions occur, cf. [CC97, Example8.80]. In particular, there are points that are not regular. However, theyproved that for any renormalised limit measure the complement X \ R hasmeasure , i.e. almost all points are regular, cf. [CC97, Theorem 2.1]. Evenmore, it was conjectured that there is some k such that R \ R k has measure as well, i.e. almost all points are k -regular. This conjecture was proven byColding and Naber in [CN12]. Theorem 1.5 ([CN12, Theorem 1.18 and p. 1185]) . Let ( M i , p i ) i ∈ N be asequence of pointed complete connected n -dimensional Riemannian manifoldswhich satisfy the uniform lower Ricci curvature bound Ric M i ≥ − ( n − andconverge to a limit ( X, p ) . Then there is k = k ( X ) ∈ N such that R k hasfull measure and is connected. This k is called the dimension of X , a k -regular point is called generic and X gen := R k denotes the set of all generic points. Note that k < n if thesequence is collapsing. 2. Local construction
For a collapsing sequence of pointed complete connected n -dimensionalRiemannian manifolds ( M i , p i ) satisfying the uniform lower Ricci curvaturebound Ric M i ≥ − ( n − , the main proposition of this section provides acondition on points q i ∈ M i such that on balls around these points withsufficiently small radius a ‘local version’ of the main theorem holds: In fact,the statement of the main theorem holds on B r ( q i ) if the rescaled manifolds ( r M i , q i ) are sufficiently close to the Euclidean space. Applying this resultto finitely many sequences of such points q i and radii r will prove the maintheorem in section 3.This local result follows from generalising several theorems of Cheeger andColding in [CC96, CC00a] and Kapovitch and Wilking in [KW11]. Thoseresults make statements assuming a sequence of manifolds to converge to aEuclidean space R k . The generalisations do not assume such a convergencebut that the manifolds are sufficiently close to R k , and then make similarstatements as the mentioned theorems.In the situation of a sequence ( M i , p i ) converging to a limit ( X, p ) as inthe main proposition, there is no reason why the manifolds should alreadybe sufficiently close to R k . On the other hand, there is hope that this istrue after rescaling all manifolds with (the same) factor: For a generic point XISTENCE OF TYPICAL SCALES 9 x ∈ X , the rescaled limit space ( λX, x ) converges to R k as λ → ∞ . Inparticular, ( λX, x ) is close to R k for sufficiently large λ > . Moreover,given any sequence of points x i ∈ M i converging to x ∈ X , the equallyrescaled manifolds ( λM i , x i ) converge to ( λX, x ) . Hence, the ( λM i , x i ) areclose to R k for sufficiently large λ and i .So, one can expect to be able to use the above explained generalisationsfor the rescaled manifolds. In fact, those (generalised) theorems make state-ments about balls of radius . Applied to the rescaled manifolds λM i , thiscorresponds to balls of radius λ in the unscaled manifolds M i . Thus, in thefollowing, instead of λ the notation r will be used, where r > is sufficientlysmall, and statements about balls of radius r will be obtained. This leads tothe following local version of the main theorem, where the choice of notation ˆ ε and ˆ δ —while seemingly artificial—will turn out to be helpful when provingthe main theorem by applying the ‘local version’. Proposition 2.1.
Let ( M i ) i ∈ N be a sequence of complete connected n -di-mensional Riemannian manifolds with uniform lower Ricci curvature bound Ric M i ≥ − ( n − , and let k < n . Given ˆ ε ∈ (0 , , there is ˆ δ = ˆ δ (ˆ ε ; n, k ) > such that for any < r ≤ ˆ δ and q i ∈ M i with d GH (( r − M i , q i ) , ( R k , ≤ ˆ δ there are a family of subsets of good points G r ( q i ) ⊆ B r ( q i ) with vol( G r ( q i )) ≥ (1 − ˆ ε ) · vol( B r ( q i )) and a sequence λ i → ∞ such that the following holds: a) For every choice of base points x i ∈ G r ( q i ) and every sublimit ( Y, · ) of ( λ i M i , x i ) there exists a compact metric space K of dimension l ≤ n − k satisfying ≤ diam( K ) ≤ such that Y splits isometrically as aproduct Y ∼ = R k × K. b) If x i , x i ∈ G r ( q i ) are base points such that, after passing to a subse-quence, ( λ i M i , x ji ) → ( R k × K j , · ) for ≤ j ≤ as before, then dim( K ) = dim( K ) . The idea of the proof is to construct two families of sets G r ( q i ) and G r ( q i ) ,where r is sufficiently small, with the following properties: For any choice ofpoints x i ∈ G r ( q i ) and for any rescaling sequence λ i → ∞ , every (sub)limitof the sequence ( λ i M i , x i ) splits off an R k -factor. The second family of sets G r ( q i ) is constructed together with a rescaling sequence λ i → ∞ such that forall large enough i and any point x i ∈ G r ( q i ) each single rescaled manifold ( λ i M i , x i ) is close to the product of R k and a compact space, where thiscompact space depends on the choice of the regarded i and the base point x i . After fixing this sequence λ i → ∞ , the intersection of those two setsgives the result. This section is structured as follows: First, G r ( q i ) and G r ( q i ) are con-structed in subsection 2.1 and subsection 2.2, respectively. Next, subsection 2.3establishes a basis for proving that blow-ups have the same dimension. Fi-nally, Proposition 2.1 is proven in subsection 2.4.2.1. Construction of G r ( q i ) . This subsection deals with finding familiesof subsets G r ( q i ) ⊆ B r ( q i ) such that all blow-ups of M i with base pointsfrom G r ( q i ) split off an R k -factor. Recall that a blow-up is the limit ofthe sequence of rescaled manifolds µ i M i for a sequence of scales µ i → ∞ .Thus, the natural question is under which condition such a splitting can beguaranteed.By modifying certain distance functions, Cheeger and Colding obtainedharmonic functions which were used to prove the following splitting theorem. Theorem 2.2 ([CC96, Theorem 6.64]) . Let ( M i , p i ) i ∈ N be a sequence ofpointed n -dimensional Riemannian manifolds and let R i → ∞ and ε i → besequences of positive real numbers such that B M i R i ( p i ) has Ricci curvature atleast − ( n − · ε i . Assume ( B M i R i ( p i ) , p i ) to converge to some pointed metricspace ( Y, y ) in the pointed Gromov-Hausdorff sense.If Y contains a line, then Y splits isometrically as Y ∼ = R × Y ′ . Assuming that the limit space already is the Euclidean space R n (of thesame dimension as the manifolds of the convergent sequence), Colding provedconvergence of the volume of balls of radius in the manifolds to the volumeof the -ball in R n by using n of such function (for every manifold), cf. [Col97,Lemma 2.1]. Using both the observations there and the proof of the abovesplitting theorem, Cheeger and Colding obtained the following statementwhich is stated here as noted in [KW11, Theorem 1.3]. Theorem 2.3 ([CC00a, section 1]) . Let ( M i , p i ) → ( R k , be a sequenceof pointed n -dimensional Riemannian manifolds which satisfy the uniformlower Ricci curvature bound Ric M i ≥ − i . Then there are harmonic functions b i , . . . , b ik : B ( p i ) → R and a constant C ( n ) ≥ such that a) |∇ b ij | ≤ C ( n ) for all i and j and b) − R B ( p i ) P kj,l =1 |h∇ b ij , ∇ b il i− δ jl | + P kj =1 k Hess b ij k dV M i → as i → ∞ .Moreover, the maps Φ i = ( b i , . . . , b ik ) : B ( p i ) → R k provide ε i -Gromov-Hausdorff approximations between B ( p i ) and B (0) with ε i → . Conversely, Kapovitch and Wilking proved the following in [KW11]: Ifthere exist k functions with analogous properties on balls with radii r i → ∞ ,then the sublimit splits off an R k -factor. Theorem 2.4 (Product Lemma, [KW11, Lemma 2.1]) . Let ( M i , p i ) i ∈ N bea pointed sequence of n -dimensional manifolds with Ric M i > − ε i for a se-quence ε i → and let r i → ∞ such that ¯ B r i ( p i ) is compact for all i ∈ N .Assume for every i ∈ N and ≤ j ≤ k there are harmonic functions XISTENCE OF TYPICAL SCALES 11 b ij : B r i ( p i ) → R which are L -Lipschitz and fulfil − Z B R ( p i ) k X j,l =1 |h∇ b ij , ∇ b il i − δ jl | + k X j =1 k Hess b ij k dV M i → for all R > . Then ( B r i ( p i ) , p i ) subconverges in the pointed Gromov-Hausdorff sense to ametric product ( R k × X, p ∞ ) for some metric space X . Moreover, ( b i , . . . , b ik ) converges to the projection onto the Euclidean factor. The above theorems will be generalised to the following statements: Ifall manifolds are sufficiently close to R k , then there exist harmonic func-tions similar to those of Theorem 2.3 such that the average integral doesnot converge to zero but only is bounded, cf. Lemma 2.5. Consequently, anadaptation of the Product Lemma will be established: Under the followingweaker hypothesis, the same conclusion holds, cf. Lemma 2.7: Only the av-erage integral about the norms of the Hessian vanishes when passing to thelimit whereas the average integrals about the scalar products of the gradientsare bounded by a small constant.First, maps similar to those in Theorem 2.3 will be constructed. A crucialstep of the proof will be the rescaling of maps. Lemma 2.5.
Given n ∈ N , there exists L = L ( n ) ≥ such that the fol-lowing holds: For arbitrary ˆ ε > , R > , k ≤ n and g : R + → R + with lim x → g ( x ) = 0 there exists ˆ δ = ˆ δ (ˆ ε, g, R ; n, k ) ∈ (0 , such that thefollowing is true for every δ ≤ ˆ δ : For every pointed complete connected n -di-mensional Riemannian manifold ( M, p ) with Ric M ≥ − ( n − · δ and d GH (( M, p ) , ( R k , ≤ g ( δ ) there exist harmonic functions f , . . . , f k : B MR ( p ) → R such that |∇ f j | ≤ L and − Z B MR ( p ) k X j,l =1 |h∇ f j , ∇ f l i − δ jl | + k X j =1 k Hess( f j ) k dV M < ˆ ε. Proof.
Let L := C ( n ) be the constant of Theorem 2.3. The proof is doneby contradiction: Assume the statement is false and let ˆ ε , R , k and g becontradicting. For every i ∈ N , let ˆ δ i := ( i · ( n − − / ∈ (0 , and choosethe contradicting δ i ≤ ˆ δ i and ( M i , p i ) with Ric M i ≥ − ( n − · δ i ≥ − i and d GH (( M i , p i ) , ( R k , ≤ g ( δ i ) such that for all harmonic Lipschitz maps f i , . . . , f ik : B M i ( p i ) → R with |∇ f ij | ≤ L , − Z B Mi ( p i ) k X j,l =1 |h∇ f ij , ∇ f il i − δ jl | + k X j =1 k Hess( f ij ) k dV M i ≥ ˆ ε. Since g ( δ i ) → as i → ∞ , ( M i , p i ) → ( R k , and so ( R M i , p i ) → ( R k , as well. By Theorem 2.3, there are harmonic functions ˜ f ij : B R − M i ( p i ) → R , ≤ j ≤ k , with |∇ ˜ f ij | ≤ L and − Z B R − Mi ( p i ) k X j,l =1 |h∇ ˜ f ij , ∇ ˜ f il i − δ jl | + k X j =1 k Hess( ˜ f ij ) k dV R − M i → as i → ∞ . In particular, the rescaled functions f ij := R · ˜ f ij : B M i R → R satisfy |∇ f ij | ≤ L and − Z B MiR ( p i ) k X j,l =1 |h∇ f ij , ∇ f il i − δ jl | + k X j =1 k Hess( f ij ) k dV M i = − Z B R − Mi ( p i ) k X j,l =1 |h∇ ˜ f ij , ∇ ˜ f il i − δ jl | + 1 R · k X j =1 k Hess( ˜ f ij ) k dV M i ≤ (cid:16) R (cid:17) · (cid:16) − Z B R − Mi ( p i ) k X j,l =1 |h∇ ˜ f ij , ∇ ˜ f il i − δ jl | + k X j =1 k Hess( ˜ f ij ) k dV M i (cid:17) → as i → ∞ . This is a contradiction. (cid:3)
In order to generalise the Product Lemma, the following result of Cheegerand Colding is used. Again, the theorem is stated using the notation of[KW11, Theorem 1.5].
Theorem 2.6 (Segment Inequality, [CC96, Theorem 2.11]) . Given any di-mension n ∈ N and radius r > , there exists τ = τ ( n, r ) such that thefollowing holds: Let M be an n -dimensional Riemannian manifold which sat-isfies the lower Ricci curvature bound Ric M ≥ − ( n − and g : M → R + bea non-negative function. Then for r ≤ r , − Z B r ( p ) × B r ( p ) Z d ( z ,z )0 g ( γ z ,z ( t )) dt dV ( z , z ) ≤ τ · r · − Z B r ( p ) g ( q ) dV ( q ) , where γ z ,z denotes a minimising geodesic from z to z . The following lemma is a generalisation of the Product Lemma where theaverage integral of scalar products of the gradients does not have to vanish,but only needs to be bounded.
Lemma 2.7.
Let ( M i ) i ∈ N be a sequence of connected n -dimensional Rie-mannian manifolds with Ric M i ≥ − ( n − · ε i where ε i → . Let r i → ∞ and q i ∈ M i be points such that the balls ¯ B r i ( q i ) are compact. Furthermore,let k ≤ n and assume that for every ≤ j ≤ k there is a harmonic L -Lipschitz map b ij : B r i ( q i ) → R satisfying − R B r ( q i ) P kj =1 k Hess( b ij ) k dV → and − R B r ( q i ) P kj,l =1 |h∇ b ij , ∇ b il i − δ jl | dV ≤ − n for all r ≤ r i . XISTENCE OF TYPICAL SCALES 13
Then every sublimit of ( B r i ( q i ) , q i ) is isometric to a product ( R k × X, q ∞ ) for some metric space X and some point q ∞ ∈ R k × X .Proof. Let ( Y, y ) be an arbitrary sublimit of ( B r i ( q i ) , q i ) . Without loss ofgenerality, assume convergence ( B r i ( q i ) , q i ) → ( Y, y ) . The concept of theproof is the following: For well chosen ˆ q i ∈ B / ( q i ) , c ijl := h∇ b ij , ∇ b il i (ˆ q i ) and after passing to a subsequence, − R B ( q i ) |h∇ b ij , ∇ b il i − c ijl | dV → . Ina second step, the corresponding statement for balls of arbitrary radiuswill be shown. Finally, after passing to a subsequence such that every ( c ijl ) i ∈ N converges to some limit c jl and defining h jl via the matrix identity (cid:0) ( h jl ) ≤ j,l ≤ k (cid:1) = (cid:0) ( c jl ) ≤ j,l ≤ k (cid:1) − , the linear combinations d ij := P kl =1 h jl b il satisfy the hypothesis of the Product Lemma, and thus, prove the claim. First step:
Fix ≤ j, l ≤ k and suppose there exists ε > such that forevery N ∈ N there is i ≥ N with − Z B ( q i ) Z ( k Hess( b ij ) k + k Hess( b il ) k )( γ ˆ q i x i ( t )) dt dV ( x i ) ≥ ε for all ˆ q i ∈ B / ( q i ) where γ ˆ q i x i denotes a minimising geodesic from ˆ q i to x i .For such an i , − Z B ( q i ) × B ( q i ) Z ( k Hess( b ij ) k + k Hess( b il ) k )( γ ˆ q i x i ( t )) dt dV (ˆ q i , x i ) ≥ − R B / ( q i ) × B ( q i ) R ( k Hess( b ij ) k + k Hess( b il ) k )( γ ˆ q i x i ( t )) dt dV (ˆ q i , x i ) C BG ( n, − , , ≥ ε C BG ( n, − , , > . On the other hand, the Segment Inequality provides τ = τ ( n, such that − Z B ( q i ) × B ( q i ) Z ( k Hess( b ij ) k + k Hess( b il ) k )( γ ˆ q i x i ( t )) dt dV (ˆ q i , x i ) ≤ τ · − Z B ( q i ) ( k Hess( b ij ) k + k Hess( b il ) k ) dV ≤ τ · (cid:16) − Z B ( q i ) k X j =1 k Hess( b ij ) k dV (cid:17) / → as i → ∞ . This is a contradiction. Therefore, after passing to a subsequence, there exist ˆ q i ∈ B / ( q i ) with − R B ( q i ) R ( k Hess( b ij ) k + k Hess( b il ) k )( γ ˆ q i q ( t )) dt dV ( q ) → . Define c ijl := h∇ b ij , ∇ b il i (ˆ q i ) . Then − Z B ( q i ) |h∇ b ij , ∇ b il i ( x i ) − c ijl | dV ( x i ) ≤ − Z B ( q i ) Z (cid:12)(cid:12)(cid:12) ddt | t = τ h∇ b ij ( γ ˆ q i q ( t )) , ∇ b il ( γ ˆ q i q ( t )) i (cid:12)(cid:12)(cid:12) d τ dV ( q )= − Z B ( q i ) Z (cid:12)(cid:12)(cid:12) h Hess( b ij ) · ˙ γ ˆ q i x i ( τ ) , ∇ b il i ( γ ˆ q i x i ( τ ))+ h∇ b ij , Hess( b il ) · ( ˙ γ ˆ q i x i ( τ )) i ( γ ˆ q i x i ( τ )) (cid:12)(cid:12)(cid:12) d τ dV ( x i ) ≤ − Z B ( q i ) L · Z ( k Hess( b ij ) k + k Hess( b il ) k ) ◦ γ ˆ q i x i ( τ ) d τ dV ( x i ) → as i → ∞ . Second step:
Fix an arbitrary
R > . Analogously to the first step, one canprove the existence of ¯ q i ∈ B ( q i ) such that |h∇ b ij , ∇ b il i (¯ q i ) − c ijl | → and − Z B R ( q i ) Z ( k Hess( b ij ) k + k Hess( b il ) k )( γ ¯ q i x i ( t )) dt dV ( x i ) → as i → ∞ . As in the first step, − Z B R ( q i ) |h∇ b ij , ∇ b il i ( q ) − c ijl | dV ( q ) ≤ − Z B R ( q i ) Z (cid:12)(cid:12)(cid:12) ddt | t = τ h∇ b ij , ∇ b il i ◦ γ ¯ q i q ( t ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) h∇ b ij , ∇ b il i ( ¯ q i ) − c ijl (cid:12)(cid:12)(cid:12) d τ dV ( q ) → as i → ∞ . Third step:
As the b ij are L -Lipschitz, c ijl := h∇ b ij , ∇ b il i (ˆ q i ) ∈ [ − L , L ] isa bounded sequence, and thus, has a convergent subsequence. Pass to asubsequence such that all sequences ( c ijl ) i ∈ N converge and denote the limitsby c jl := lim i →∞ c ijl ∈ [ − L , L ] . Then | c jl − δ jl | ≤ lim i →∞ − Z B R ( q i ) | c ijl − h∇ b ij , ∇ b il i| dV + − Z B R ( q i ) |h∇ b ij , ∇ b il i − δ jl | dV ≤ lim i →∞ − Z B R ( q i ) k X j,l =1 |h∇ b ij , ∇ b il i − δ jl | dV ≤ − n . Hence, the matrix C := ( c jl ) ≤ j,l ≤ k is invertible, symmetric and positivedefinite. In particular, its inverse C − is diagonalisable with positive eigen-values. Let C − D denote the diagonal matrix and S the invertible matrixwith C − = S · C − D · S − and define C − / D as the diagonal matrix whose XISTENCE OF TYPICAL SCALES 15 entries are the square roots of the diagonal entries of C − D . Then the ma-trix H := ( h jl ) jl := S · C − / D · S − satisfies H = C − . Now define d ij := P kl =1 h jl b il .Obviously, these are Lipschitz and harmonic. Furthermore, it is straight-forward to see − Z B R ( q i ) k X j ,j =1 (cid:12)(cid:12) h∇ d ij , ∇ d ij i − δ j j (cid:12)(cid:12) + k X j =1 k Hess( d ij ) k dV → as i → ∞ . By the Product Lemma, after passing to a subsequence, ( B r i ( q i ) , q i ) con-verges to ( R k × X, (0 , q ∞ )) for some metric space X and q ∞ ∈ X . Since ( B r i ( q i ) , q i ) converges to ( Y, y ) , this proves that Y is isometric to R k × X . (cid:3) Applying the previous two lemmata proves that for sufficiently small ballsthere is a subset of good base points of arbitrary good volume such that allsublimits of sequences with respect to those base points split off an R k -factor.In order to verify this, the following statement, which in its first form wasproven by Stein in [Ste93, p. 13], is needed for estimating the volume of aset where the so called ρ -maximum function is bounded from above. Again,the notation of [KW11, Lemma 1.4b)] is used. Theorem 2.8 (Weak type 1-1 inequality) . Let M be an n -dimensional Rie-mannian manifold with lower Ricci curvature bound Ric M ≥ − ( n − . For anon-negative function f : M → R and ρ > , define the ρ -maximum functionof f as Mx ρ f ( p ) := sup r ≤ ρ − Z B r ( p ) f. Especially, put Mx f ( p ) = Mx f ( p ) . Then there is C ( n ) > such that forany non-negative function f ∈ L ( M ) and c > , vol( { x ∈ M | Mx ρ f ( x ) > c } ) ≤ C ( n ) c Z M f dV M . In the following, C ( n ) will always denote the constant of the weak type1-1 inequality.Now the first set needed for Proposition 2.1 can be constructed. Lemma 2.9.
Let ( M i ) i ∈ N be a sequence of complete connected n -dimension-al Riemannian manifolds which satisfies the uniform lower Ricci curvaturebound Ric M i ≥ − ( n − , and let k < n . For every ˆ ε ∈ (0 , there exists ˆ δ = ˆ δ (ˆ ε ; n, k ) > such that for all < r ≤ ˆ δ and q i ∈ M i with d GH (( r − M i , q i ) , ( R k , ≤ ˆ δ there is a family of subsets of good points G r ( q i ) ⊆ B r ( q i ) satisfying vol M i ( G r ( q i )) ≥ (1 − ˆ ε ) · vol M i ( B r ( q i )) such that for every choice of x i ∈ G r ( q i ) , every λ i → ∞ and every sublimit ( Y, · ) of ( λ i M i , q i ) there exists Y ′ such that Y ∼ = R k × Y ′ isometrically. Proof.
Define C ( n ) := C ( n ) · n · C BG ( n, − , , , and let ˆ δ = ˆ δ (ˆ ε ; n, k ) be the constant ˆ δ ( ˆ εC ( n ) , id , n, k ) and L ( n ) be as in Lemma 2.5. Further,let < r ≤ ˆ δ and q i ∈ M i satisfy d GH (( r − M i , q i ) , ( R k , ≤ ˆ δ . Then there are harmonic and L -Lipschitz functions f ij : B r − M i ( q i ) → R , ≤ j ≤ k , satisfying − Z B r − Mi ( q i ) ψ ∇ ( f i ) + ψ H ( f i ) dV r − M i ≤ ˆ εC ( n ) where ψ ∇ ( f i ) := P kj,l =1 |h∇ f ij , ∇ f il i − δ jl | and ψ H ( f i ) := P kj =1 k Hess( f ij ) k .Define G i := { x i ∈ B r − M i ( q i ) | Mx r − M i ( ψ ∇ ( f i ) + ψ H ( f i ))( x i ) < − n } where the -maximum function is taken with respect to d r − M i = r d M i .Using Theorem 2.8, the volume of this set can be estimated by vol r − M i ( B r − M i ( q i ) \ G i ) ≤ C ( n ) C BG ( n, − , , · ˆ εC ( n ) · vol r − M i ( B r − M i ( q i )) ≤ ˆ ε · vol r − M i ( B r − M i ( q i )) . Hence, regarding G r ( q i ) := G i as a subset of M i , vol M i ( G r ( q i ))vol M i ( B M i r ( q i )) = vol r − M i ( G i )vol r − M i ( B r − M i ( q i )) ≥ − ˆ ε. Now let x i ∈ G r ( q i ) and λ i → ∞ be arbitrary. Define r i := λ i · r → ∞ andlet < ρ ≤ r i . Since B λ i M i r i ( x i ) = B r − M i ( x i ) ⊆ B r − M i ( q i ) , the rescaledmaps ˜ f ij := r i · f ij : B λ i M i r i ( x i ) → R are well defined, harmonic and L -Lipschitz. It is straightforward to see − Z B λiMiρ ( x i ) ψ ∇ ( ˜ f i ) dV λ i M i ≤ − n and − Z B λiMiρ ( x i ) ψ H ( ˜ f i ) dV λ i M i ≤ − n r i → as i → ∞ . By Lemma 2.7, any sublimit of ( λ i M i , x i ) has the form ( R k × Y ′ , · ) for somemetric space Y ′ . (cid:3) XISTENCE OF TYPICAL SCALES 17
Construction of G r ( q i ) and λ i . The aim of this subsection is to find arescaling sequence λ i → ∞ and a family of subsets G r ( q i ) ⊆ B r ( q i ) with thefollowing two properties: On the one hand, every single rescaled manifold λ i M i (with a base point from G r ( q i ) ) is close to a product of R k and a com-pact metric space. On the other hand, the sublimits of sequences ( λ i M i , x i ) with base points x i ∈ G r ( q i ) have the same dimension (depending not onthe base points but only on the choice of the subsequence).Before motivating the procedure, recall the definition of time-dependentvector fields, cf. [Lee03]: A time-dependent vector field on a Riemannianmanifold M is a continuous map X : I × M → T M , where I ⊆ R isan interval, such that X t is a vector field for all t ∈ I , i.e. X t satisfies X tp := X t ( p ) := X ( t, p ) ∈ T p M for all ( t, p ) ∈ I × M . Such a time-dependentvector field X : I × M → T M is called piecewise constant in time if thereexist disjoint sub-intervals I = I ∐ . . . ∐ I n such that X s = X t for all ≤ i ≤ n and s, t ∈ I i .For arbitrary s ∈ I and I − s := { τ − s | τ ∈ I } , a curve c : I − s → M is called s -integral curve of X if c ′ ( t ) = X s + tc ( t ) for all t ∈ I − s . A -integralcurve is also called integral curve of X .Furthermore, there exist an open set Ω ⊆ S s ∈ I { s } × ( I − s ) × M anda map Φ : Ω → M satisfying the following: for any ( s, p ) ∈ I × M , theset Ω ( s,p ) := { t ∈ I − s | ( s, t, p ) ∈ Ω } is an open interval which contains , and for any fixed ( s, p ) ∈ I × M the map c : Ω ( s,p ) → M defined by c ( t ) := Φ( s, t, p ) is the unique maximal s -integral curve of X with startingpoint p . Using the notation ϕ st := Φ( s, t, · ) , this is equivalent to ϕ beinga maximal solution of ddt | t = t ϕ st ( p ) = X s + t ϕ st ( p ) and ϕ s = id . Such a Φ iscalled flow of X . Moreover, the following is true: If p ∈ M and s, t, u aretimes with ( s, t, p ) ∈ Ω and ( s + t, u, ϕ st ( p )) ∈ Ω , then ( s, t + u, p ) ∈ Ω and ϕ s + tu ◦ ϕ st ( p ) = ϕ st + u ( p ) . In particular, if defined, ϕ s + t − t is the inverse of ϕ st .A time-dependent vector field X has compact support if there exists acompact set K ⊆ M such that for all t ∈ I the vector fields X t have support K . In this case, the flow Φ exists for all times.In order to prove that two blow-ups have the same dimension, the followingwill be established and used: Let X i : [0 , × M i → T M i be time-dependentvector fields with R (Mx r ( k∇ .X ti k / )( c i ( t ))) / dt < ˆ ε for all i ∈ N where ˆ ε > and the c i are integral curves. Moreover, let the X i be divergencefree, i.e. the flows are measure preserving. Then any blow-ups coming fromthe sequences with base points c i (0) and c i (1) , respectively, have the samedimension, i.e. if λ i → ∞ with ( λ i M i , c i (0)) → ( Y , y ) and ( λ i M i , c i (1)) → ( Y , y ) , then dim( Y ) = dim( Y ) . This will be proven in subsection 2.3.Since Gromov-Hausdorff convergence is preserved by shifting base points alittle bit, the same statement is true if the base points c i (0) and c i (1) , respec-tively, are replaced by points x i and y i , respectively, where λ i · d ( c i (0) , x i ) < C and λ i · d ( c i (1) , y i ) < C for some C > (independent of i ). This motivatesthe following definition. Definition 2.10.
Let M be a complete connected n -dimensional Riemann-ian manifold and r, C, ˆ ε > . A subset B r ( q ) ′ ⊆ B r ( q ) has the C ( M, r, C, ˆ ε ) -property if the following holds:For all pairs of points x, y ∈ B r ( q ) ′ there exists a time-dependent vectorfield X : [0 , × M → T M which is piecewise constant in time and hascompact support and an integral curve c : [0 , → M satisfying the followingconditions:a) The vector field X t is divergence free on B r ( c ( t )) for all ≤ t ≤ ,b) d ( x, c (0)) < C , d ( y, c (1)) < C andc) R (Mx r ( k∇ .X t k / )( c ( t ))) / dt < ˆ ε .Note the following: If a subset B r ( q ) ′ ⊆ B r ( q ) has the C ( M, r, C, ˆ ε ) -property and is regarded as a subset B r ( q ) ′ ⊆ B λMλr ( q ) , where λ > , thenit has the C ( λM, λr, λC, ˆ ε ) -property. For this reason, the notation of the C -property contains the manifold M .In order to construct the subset G r ( q i ) , the following statement is used:There is a rescaling factor such that, if a manifold is sufficiently close to R k ,the rescaled manifold is close to a product. This statement will be proven bycontradiction using the following theorem of Kapovitch and Wilking wherethe first part is the first part of the original theorem and the second is takenfrom its proof. Theorem 2.11 (Rescaling Theorem [KW11, Theorem 5.1]) . Let ( M i , p i ) i ∈ N be a sequence of pointed n -dimensional Riemannian manifolds and r i → ∞ and µ i → be sequences of positive real numbers such that B M i r i ( p i ) has Riccicurvature larger than − µ i and ¯ B M i r i ( p i ) is compact. Suppose that ( M i , p i ) converges to ( R k , for some k < n . After passing to a subsequence, thereis a compact metric space K with diam( K ) = 10 − n , a family of subsets G ( p i ) ⊆ B ( p i ) with vol( G ( p i ))vol( B ( p i )) → , a sequence λ i → ∞ and a sequence ˆ ε i → such that the following holds: a) For all q i ∈ G ( p i ) , the isometry type of the limit of any convergentsubsequence of ( λ i M i , q i ) is given by the metric product R k × K . b) The set G ( p i ) has the C ( M i , , n λ i , ˆ ε i ) -property. Lemma 2.12.
For every ˆ ε ∈ (0 , , R > , η > and k < n there existsa bound ˆ δ = ˆ δ (ˆ ε, R, η ; n, k ) > such that for all pointed n -dimensional Rie-mannian manifolds ( M, p ) with Ric M ≥ − ( n − · ˆ δ satisfying that ¯ B / ˆ δ ( p ) is compact and d GH (( M, p ) , ( R k , ≤ ˆ δ there is a factor λ > such that the following holds: XISTENCE OF TYPICAL SCALES 19 a) There are a subset of good points G ( p ) ⊆ B ( p ) satisfying vol M ( G ( p )) ≥ (1 − ˆ ε ) · vol M ( B ( p )) and a compact metric space K of diameter such that for all q ∈ G ( p ) there is a point ˜ q ∈ R k × K with d GH ( B λMR ( q ) , B R k × KR (˜ q )) ≤ η. b) The set G ( p ) has the C ( M, , n · n λ , ˆ ε ) -property.Proof. This is a straightforward contradiction argument rescaling both thesequence λ i → ∞ and the compact space K occurring in the Rescaling Theorem 2.11by a factor n . (cid:3) Now rescaling the sequence M i such that each element is close enough to R k and applying the previous result, one obtains factors λ i which basicallyare the sought-after rescaling sequence. However, the lemma does provide λ i for every i , but does not give any information about whether or not λ i → ∞ as i → ∞ . In order to prove λ i → ∞ , the fact is needed thatspaces of different dimensions are not close. This in turn follows from thefact that sequences of limit spaces do not increase dimension. For this, thefollowing lemma is needed which states that, given a converging sequence ofproper length spaces, there exists a rescaling sequence such that the rescaledsequence converges to a tangent cone. Lemma 2.13.
Let ( X i , p i ) → ( X, p ) be a converging sequence of properlength spaces. Then there exists µ i → ∞ such that for all λ i → ∞ with λ i ≤ µ i , ( λ i X i , p i ) subconverges to a tangent cone of ( X, p ) .Proof. For ε i → such that d GH (( X i , p i ) , ( X, p )) ≤ ε i , let µ i := ε − / i . Forfixed r > , let i be large enough such that r ≤ ε − / i . Then rµ i ≤ ε i and d GH ( B X i r/µ i ( p i ) , B Xr/µ i ( p )) ≤ · ε i → as i → ∞ . After passing to a subsequence, ( µ i X, p ) converges to a tangent cone ( Y, q ) .Then d GH ( B µ i X i r ( p i ) , B Yr ( q )) ≤ µ i · d GH ( B X i r/µ i ( p i ) , B Xr/µ i ( p )) + d GH ( B µ i Xr ( p ) , B Yr ( q )) ≤ · ε i · µ i + d GH ( B µ i Xr ( p ) , B Yr ( q )) → , and this proves that ( µ i X i , p i ) subconverges to ( Y, q ) .Now let λ i → ∞ with λ i ≤ µ i . After passing to a further subsequence, λ i µ i → α for some α ≤ , ( λ i X, p ) → ( αY, q ) and ( λ i M i , p i ) → ( αY, q ) . Inparticular, ( λ i X i , p i ) subconverges to a tangent cone of ( X, p ) . (cid:3) Let X n denote the class of all pointed metric spaces that can occur asGromov-Hausdorff limit of a sequence of pointed complete connected n -di-mensional Riemannian manifolds M i with Ric M i ≥ − ( n − . Lemma 2.14.
Let ( X i , p i ) → ( X, p ) be converging spaces in X n such thatall X i have the same dimension dim( X i ) = k . Then dim( X ) ≤ k .Proof. In order to estimate l := dim( X ) , take a generic point x ∈ X gen andconstruct a tangent cone R l . The idea of this construction is to considersequences of manifolds M ij converging to X i . For large i , these are suffi-ciently close to X , and applying Lemma 2.5 and Lemma 2.7 will give theclaim. So, let ( M ij , p ij ) → ( X i , p i ) as j → ∞ . Without loss of generality,assume ( X, p ) = ( R l , and Ric M ij ≥ − ( n − · δ i for some monotonicallydecreasing sequence δ i → .Choose ε i → such that d GH ( B X i / ε i ( p i ) , B R l / ε i (0)) ≤ ε i . Without loss ofgenerality, d GH ( B M ij / ε i ( p ij ) , B X i / ε i ( p i )) ≤ ε i for all j ∈ N . Hence, d GH ( B M ij / ε i ( p ij ) , B R l / ε i (0)) ≤ ε i . Define g : R + → R + by g ( x ) := ( ε i if δ i ≤ x < δ i − , if x ≥ δ ,c = c ( n ) := 2 · C ( n ) · C BG ( n, − , , and choose ˆ δ = ˆ δ ( − n c , g, n, l ) as in Lemma 2.5. Let i ∈ N be sufficiently large such that δ i ≤ ˆ δ . Then Ric M ij ≥ − ( n − · δ i and, since g ( δ i ) = ε i , d GH ( B M ij /g ( δ i ) ( p ij ) , B R l /g ( δ i ) (0)) ≤ g ( δ i ) . So, there are L = L ( n ) and harmonic L -Lipschitz maps f ijh : B M ij ( p ij ) → R , ≤ h ≤ l , such that − Z B Mij ( p ij ) l X h ,h =1 |h∇ f ijh , ∇ f ijh i − δ h h | + l X h =1 k Hess( f ijh ) k dV M ij < − n c . In order to simplify notation, let F ij := l X h ,h =1 |h∇ f ijh , ∇ f ijh i − δ h h | + l X h =1 k Hess( f ijh ) k . With the usual argumentation, the set G ij := { p ∈ ¯ B M ij / ( p ij ) | Mx / ( F ij )( p ) ≤ − n } is compact with vol( G ij ) ≥ · vol( B M ij / ( p ij )) and the sequence ( G ij ) j ∈ N subconverges to a set G i ⊆ ¯ B X i / ( p i ) with positive volume. In particular, theintersection with ( X i ) gen is nonempty. Without loss of generality, assume XISTENCE OF TYPICAL SCALES 21 that ( G ij ) j ∈ N itself already converges to G i , and choose q ij ∈ G ij convergingto a q i ∈ ( X i ) gen .Since ( M ij , q ij ) → ( X i , q i ) , there exists µ ij → ∞ as in Lemma 2.13 suchthat ( µ ij M ij , q ij ) → ( R k , as j → ∞ . On the other hand, the rescaled maps ˜ f ij := µ ij f ij : B µ ij M ij µ ij ( q ij ) → R are harmonic and L -Lipschitz. Furthermore, for arbitrary r > and j largeenough such that r < µ ij , − Z B µij Mijr ( q ij ) l X h ,h =1 |h∇ ˜ f ijh , ∇ ˜ f ijh i − δ h h | dV µ ij M ij ≤ − n and − Z B µij Mijr ( q ij ) l X h =1 k Hess( ˜ f ijh ) k dV µ ij M ij → as j → ∞ . By Lemma 2.7, there exists a metric space Z and a point z ∈ R l × Z suchthat ( µ ij M ij , q ij ) → ( R l × Z, z ) as j → ∞ . In particular, R k ∼ = R l × Z , and thus, k ≥ l . (cid:3) Lemma 2.15.
For all k < n there is ε = ε ( n, k ) ∈ (0 , ) such that thefollowing is true: If ( X, p ) , ( R k × K, q ) ∈ X n for a compact metric space K with diam( K ) = 1 and dim( X ) = k , then d GH ( B X / ε ( p ) , B R k × K / ε ( q )) > ε . Proof.
Assume the statement is false and let k < n be contradicting. Forevery i ∈ N with i > choose sequences ( M ij , p ij ) → ( X i , p i ) ∈ X n and ( N ij , q ij ) → ( R k × K i , q i ) ∈ X n as j → ∞ with diam( K i ) = 1 , dim( X i ) = k and d GH (( X i , p i ) , ( R k × K i , q i )) ≤ i .For every i ∈ N there is J ( i ) ∈ N such that d GH (( M ij , p ij ) , ( X i , p i )) ≤ i forall j ≥ J ( i ) . Define inductively j := J (1) and j i +1 := max { J ( i + 1) , j i + 1 } .In particular, j i → ∞ as i → ∞ and d GH (( M ij i , p ij i ) , ( X i , p i )) ≤ i .After passing to a subsequence, ( M ij i , p ij i ) converges to some ( X, p ) ∈ X n as i → ∞ . For arbitrary r > , d GH ( B X i r ( p i ) , B Xr ( p )) ≤ d GH ( B X i r ( p i ) , B M iji r ( p ij i )) + d GH ( B M iji r ( p ij i ) , B Xr ( p )) → as i → ∞ . Hence, ( X i , p i ) → ( X, p ) . With analogous argumentation, after passing toa further subsequence, ( R k × K i , q i ) → ( R k × K, q ) ∈ X n for some compactmetric space K with diam( K ) = 1 . On the other hand, for r > and i ≥ r , d GH ( B R k × K i r ( q i ) , B Xr ( p )) ≤ d GH ( B R k × K i r ( q i ) , B X i r ( p i )) + d GH ( B X i r ( p i ) , B Xr ( p )) → as i → ∞ . Hence, ( R k × K i , q i ) → ( X, p ) . In particular, X ∼ = R k × K and dim( X ) > k .This is a contradiction to dim( X ) ≤ k by Lemma 2.14. (cid:3) Using this lemma, the sought-after rescaling sequence and family of setscan finally be constructed.
Lemma 2.16.
Let ( M i , p i ) i ∈ N be a sequence of complete connected n -dimen-sional Riemannian manifolds which satisfy the uniform lower Ricci curvaturebound Ric M i ≥ − ( n − , and let k < n . For every ˆ ε ∈ (0 , there exists ˆ δ = ˆ δ (ˆ ε ; n, k ) > such that for all < r ≤ ˆ δ and q i ∈ M i with d GH (( r − M i , q i ) , ( R k , ≤ ˆ δ there are a family of subsets of good points G r ( q i ) ⊆ B r ( q i ) with vol( G r ( q i )) ≥ (1 − ˆ ε ) · vol( B r ( q i )) and a sequence λ i → ∞ which satisfy: a) For each x i ∈ G r ( q i ) there is a compact metric space K i with diameter and a point ˜ x i ∈ { } × K i such that d GH ( B λ i M i / ε ( x i ) , B R k × K i / ε (˜ x i )) ≤ ε for ε = ε ( n, k ) as in Lemma 2.15. b) The sets G r ( q i ) have the C ( M i , r, n · n λ i , ˆ ε ) -property.Proof. For ˆ ε ∈ (0 , , let ˆ δ = ˆ δ (ˆ ε ; n, k ) denote the ˆ δ (ˆ ε, ε , ε ; n, k ) ofLemma 2.12. Fix an arbitrary < r ≤ ˆ δ and points q i ∈ M i satisfying d GH (( r − M i , q i ) , ( R k , ≤ ˆ δ .For these ( r − M i , q i ) , let ˜ λ i > , ˜ G ( q i ) ⊆ B r − M i ( q i ) and K i be as inLemma 2.12, define λ i := ˜ λ i r and regard G r ( q i ) := ˜ G ( q i ) as a subset of M i .Then vol M i ( G r ( q i ))vol M i ( B M i r ( q i )) = vol r − M i ( ˜ G ( q i ))vol r − M i ( B r − M i ( q i )) ≥ − ˆ ε, for every x i ∈ G r ( q i ) there is ˜ x i ∈ { } × K i with d GH ( B λ i M i / ε ( x i ) , B R k × K i / ε (˜ x i )) ≤ ε and G r ( q i ) has the C ( M i , r, n · n λ i , ˆ ε ) -property.Assume that the sequence ( λ i ) i ∈ N is bounded. After passing to a subse-quence, λ i → α and ( λ i M i , x i ) → ( αX, q ) for some q ∈ X . Then d GH ( B αX / ε ( q ) , B R k × K i / ε (˜ x i )) ≤ ε for all i large enough in contradiction to Lemma 2.15. Hence, λ i → ∞ . (cid:3) This concludes the construction of G r ( q i ) , G r ( q i ) and λ i . XISTENCE OF TYPICAL SCALES 23
The C -property and the dimension of blow-ups. In order to provethat the blow-ups with base points x i and y i , respectively, have the samedimension, a crucial argument is that the flow of a time-dependent vectorfield as in the definition of the C -property is bi-Lipschitz on some smallset. This result and the implication about dimensions are proven in thissubsection.For the proof it is important to know under which conditions large subsetsof two balls intersect. The following lemmata deal with this question. Lemma 2.17.
Let ( X, d, vol) be a metric measure space and let A ′ ⊆ A ⊆ X and B ′ ⊆ B ⊆ X be measurable subsets with vol( A ′ ) ≥ (1 − ˆ ε ) · vol( A ) , vol( B ′ ) ≥ (1 − ˆ ε ) · vol( B ) , vol( A ∩ B ) > ε · max { vol( A ) , vol( B ) } for some ˆ ε > . Then vol( A ′ ∩ B ′ ) > , in particular, A ′ ∩ B ′ = ∅ .Proof. Observe vol(( A ∩ B ) \ A ′ ) ≤ vol( A \ A ′ ) ≤ ˆ ε · vol( A ) < · vol( A ∩ B ) by hypothesis. Analogously, vol(( A ∩ B ) \ B ′ ) < · vol( A ∩ B ) . Thus, vol( A ′ ∩ B ′ ) = vol( A ∩ B ) − vol(( A ∩ B ) \ ( A ′ ∩ B ′ )) ≥ vol( A ∩ B ) − (vol(( A ∩ B ) \ A ′ ) + vol(( A ∩ B ) \ B ′ )) > . (cid:3) Lemma 2.18.
Let ( M, g ) be a complete connected n -dimensional Riemann-ian manifold with lower Ricci curvature bound Ric M ≥ − ( n − . Given < ˆ ε < and s > , there exist d ( n, ˆ ε, s ) > , δ ( n, ˆ ε, s ) > such that ( δ ( n ) , sd − ) is non-empty and for δ ∈ ( δ ( n ) , sd − ) , points p, q ∈ M with distance d := d ( p, q ) < d and R := d + δd < s , vol( B R ( p ) ∩ B R ( q )) > ε · max { vol( B R ( p )) , vol( B R ( q )) } . Moreover, this δ can be chosen to be monotonically increasing in d .Proof. Let p, q ∈ M be arbitrary, γ : [0 , d ] → M be a shortest geodesicconnecting p and q and define m := γ ( d ) as the midpoint of this geodesic,i.e. d ( p, m ) = d ( q, m ) = d .First, let r > be arbitrary. Since B r ( m ) ⊆ B d/ r ( p ) ∩ B d/ r ( q ) and B d/ r ( p ) ⊆ B d + r ( m ) , vol( B d/ r ( p ) ∩ B d/ r ( q ))vol( B d/ r ( p )) ≥ vol( B r ( m ))vol( B d + r ( m )) ≥ C BG ( n, − , r, d + r ) . Now let C := ε > and ˆ r := ˆ r ( n, ˆ ε, s ) := min (cid:8) ln( C )2( n − , s (cid:9) . For any fixed < d < r , define ˜ δ ( n, ˆ ε, d ) := inf { δ ′ > | ∀ δ > δ ′ : f n,δ ( δd ) < C } ∈ [0 , ∞ ] where for δ > and r > , f n,δ ( r ) := C BG (cid:0) n, − , r, (cid:16) δ (cid:17) · r (cid:1) . In fact, this ˜ δ ( n, ˆ ε, d ) is finite and monotonically increasing in d as will beproven next: Assume ˜ δ ( n, ˆ ε, d ) = ∞ , i.e. there exists δ m → ∞ such that f n,δ m ( δ m d ) ≥ C . Then f n,δ m ( δ m d ) = C BG ( n, − , δ m d, δ m d + d ) → e ( n − d as m → ∞ , and this implies e ( n − d ≥ C . On the other hand, e ( n − d < e n − r ≤ C .This is a contradiction. Thus, ˜ δ ( n, ˆ ε, d ) < ∞ .Now let d < d and δ > ˜ δ ( n, ˆ ε, d ) . Since f n,δ is monotonically increasingin r , C > f n,δ ( δd ) ≥ f n,δ ( δd ) , i.e. δ ≥ ˜ δ ( n, ˆ ε, d ) , and this proves the monotonicity of ˜ δ ( n, ˆ ε, · ) .Hence, ˜ δ ( n, ˆ ε, d ) decreases for decreasing d whereas sd − increases.Therefore, there exists d = d ( n, ˆ ε, s ) ≤ r such that ˜ δ ( n, ˆ ε, d ) ≤ sd − for d ≤ d . Let δ = δ ( n, ˆ ε, s ) := ˜ δ ( n, ˆ ε, d ( n, ˆ ε, s )) = max { ˜ δ ( n, ˆ ε, d ) | < d ≤ d } where the monotonicity of ˜ δ is used in the last equality. Finally, for d ≤ d and δ < δ < sd − , let R := d + δd = (cid:16) + δ (cid:17) · d < (cid:16) + sd − (cid:17) · d = s .Then vol( B R ( p ) ∩ B R ( q ))vol( B R ( p )) ≥ C BG ( n, − , δd, d + δd ) = 1 f n,δ ( δd ) > C = 2ˆ ε. (cid:3) The next lemma will only be needed in section 3 but is already given heresince its statement and the proof are similar to the previous one.
Lemma 2.19.
Let ( M, g ) be a complete connected n -dimensional Riemann-ian manifold with lower Ricci curvature bound Ric M ≥ − ( n − . For all < ˆ ε < and R > there is d = d ( n, ˆ ε, R ) > such that for all p, q ∈ M with d ( p, q ) < d , vol( B R ( p ) ∩ B R ( q )) > ε · max { vol( B R ( p )) , vol( B R ( q )) } . Proof.
Similarly to the proof of Lemma 2.18, for arbitrary points p, q ∈ M with distance d := d ( p, q ) < R , observe vol( B R ( p ) ∩ B R ( q ))vol( B R ( p )) ≥ C BG ( n, − , R − d , R + d ) . As C BG ( n, − , R − d , R + d ) → as d → , there is d = d ( n, ˆ ε, R ) ∈ (0 , R ) such that C BG ( n, − , R − d , R + d ) < ε for all d ≤ d . In particular, forpoints p, q ∈ M with d ( p, q ) < d , vol( B R ( p ) ∩ B R ( q )) > ε · max { vol( B R ( p )) , vol( B R ( q )) } . (cid:3) An important notion for investigating the C -property is the distortion ofa function which describes how much a function changes the distance of twopoints. In particular, it will be important to know how much the flow of a XISTENCE OF TYPICAL SCALES 25 vector field changes the distance of two points up to some fixed time. Recallthat the distortion of a map f : M → N between Riemannian manifolds isthe function dt f : M × M → [0 , ∞ ) defined by dt f ( p, q ) := | d N ( f ( p ) , f ( q )) − d M ( p, q ) | . If Φ is the flow of a time-dependent vector field on M , t ∈ [0 , , p, q ∈ M and r ≥ , denote ϕ t := Φ(0 , t, · ) and define dt ( t )( p, q ) := max { dt ϕ τ ( p, q ) | ≤ τ ≤ t } and dt r ( t )( p, q ) := min { r, dt ( t )( p, q ) } . The subsequent lemma generalises [KW11, Lemma 3.7] and can be provencompletely analogously to it.
Lemma 2.20.
For ˜ α ∈ (1 , there exist C = C ( n, ˜ α ) and ˆ C = ˆ C ( n, ˜ α ) suchthat the following holds for any < R ≤ : Let M be an n -dimensionalRiemannian manifold with Ric M ≥ − ( n − and X : [0 , × M → T M be atime dependent, piecewise constant in time vector field with compact supportand flow Φ , define ϕ st := Φ( s, t, · ) and let c : [0 , → M be an integral curveof X such that X t is divergence free on B R ( c ( t )) for all t ∈ [0 , .Let ˜ ε := R (Mx R ( k∇ .X t k )) ◦ c ( t ) dt. Then for any r ≤ R , − Z B r ( c ( s )) × B r ( c (0)) dt r (1)( p, q ) dV ( p, q ) ≤ Cr · ˜ ε. Furthermore, there is a subset B r ( c (0)) ′ ⊆ B r ( c (0)) with c (0) ∈ B r ( c (0)) ′ such that vol( B r ( c (0)) ′ ) ≥ (1 − C ˜ ε ) · vol( B r ( c (0))) . Finally, for any t ∈ [0 , , ϕ t ( B r ( c (0)) ′ ) ⊆ B ˜ αr ( c ( t )) and vol( B r ( c ( t ))) ≤ ˆ C · vol( B r ( c (0))) . Proof.
The proof can be done completely analogously to the one of [KW11,Lemma 3.7] by replacing r in the induction by rm where m = 2 · ˜ α +1˜ α − > .Again, the constants C and ˆ C can be made explicit in terms of the constantappearing in the Bishop-Gromov Theorem. (cid:3) The following lemma states that the flow of a time dependent vector fieldas in the definition of the C -property is Lipschitz on certain small sets. Lemma 2.21.
Given α ∈ (1 , , there exist constants C = C ( n, α ) and C ′ = C ′ ( n, α ) such that for < ˆ ε < C and < R ≤ there is a number ˆ r = ˆ r ( n, ˆ ε, α, R ) < R α satisfying the following:Let M be an n -dimensional Riemannian manifold with Ric M ≥ − ( n − , X : [0 , × M → T M a time dependent, piecewise constant in time vector fieldwith compact support and flow Φ , ϕ st := Φ( s, t, · ) , ϕ t := ϕ t , c : [0 , → M an integral curve of X such that X t is divergence free on B R ( c ( t )) for all t ∈ [0 , and R (Mx R ( k∇ .X t k / )( c ( t ))) / dt < ˆ ε . Let p := c (0) and < r < ˆ r . Then there exists a subset B r ( p ) ′′ ⊆ B r ( p ) containing p with vol( B r ( p ) ′′ ) > (1 − C ′ √ ˆ ε ) · vol( B r ( p )) such that ϕ t is α -bi-Lipschitz on B r ( p ) ′′ for any t ∈ [0 , .Proof. Define ˜ α := α +12 ∈ (1 , ) ⊆ (1 , and fix the following constants: • Let C = C ( n, α ) be the C ( n, ˜ α ) and ˆ C = ˆ C ( n, α ) be the ˆ C ( n, ˜ α ) appearing in Lemma 2.20. • Let ˜ C = ˜ C ( n ) > be the constant of [KW11, formula (6)] satisfying Mx ρ [Mx ρ ( f )]( x ) ≤ ˜ C ( n ) · (cid:0) Mx ρ ( f / )( x ) (cid:1) / for any f ∈ L / ( M ) and < ρ ≤ . • Let C = C ( n, α ) := ˜ C · C . • Let ˆ C = ˆ C ( n, α ) := 2 ˆ C · C BG ( n, − , , α ) . • Let C ′ = C ′ ( n, α ) := ˆ C + q C .Fix < ˆ ε < min (cid:8) C , (cid:9) and < R ≤ . First, observe ˜ ε := Z (Mx R ( k∇ .X t k ))( c ( t )) dt ≤ Z Mx R (Mx R ( k∇ .X t k ))( c ( t )) dt ≤ ˜ C · Z Mx R ( k∇ .X t k / ) / ( c ( t )) dt < ˜ C ˆ ε. In particular, C ˜ ε < C ˆ ε < . By Lemma 2.20, for all r ≤ R , − Z B r ( p ) × B r ( p ) dt r (1) dV ≤ Cr · ˜ ε < C ˆ ε · r and there is a subset B r ( p ) ′ ⊆ B r ( p ) containing p with vol( B r ( p ) ′ ) ≥ (1 − C ˆ ε ) · vol( B r ( p )) > · vol( B r ( p )) and ϕ t ( B r ( p ) ′ ) ⊆ B αr ( c ( t )) for all t ∈ [0 , . Furthermore, for r ≤ R , vol( B αr ( c ( t )))vol( B r ( p ) ′ ) ≤ C BG ( n, − , r, αr ) · vol( B r ( c ( t )))vol( B r ( p ) ′ ) ≤ C BG (cid:16) n, − , , α (cid:17) · ˆ C · vol( B r ( p ))vol( B r ( p ) ′ ) ≤ ˆ C . Moreover, − Z B r ( p ) ′ Z Mx R ( k∇ .X t k ) ◦ ϕ t ( x ) dt dV ( x )= Z B r ( p ) ′ ) · Z ϕ t ( B r ( p ) ′ ) Mx R ( k∇ .X t k )( x ) dV ( x ) dt XISTENCE OF TYPICAL SCALES 27 ≤ Z vol( B αr ( c ( t )))vol( B r ( p ) ′ ) · − Z B αr ( c ( t )) Mx R ( k∇ .X t k )( x ) dV ( x ) dt ≤ ˆ C · Z max ≤ ρ ≤ R − Z B ρ ( c ( t )) Mx R ( k∇ .X t k )( x ) dV ( x ) dt ≤ ˆ C · ˜ C Z (Mx R ( k∇ .X t k / )) / ( c ( t )) dt < ˆ C · ˜ C ˆ ε. Define B r ( p ) ′′ := { x ∈ B r ( p ) ′ | Z Mx R ( k∇ .X t k ) ◦ ϕ t ( x ) dt < ˜ C · √ ˆ ε } . Note p ∈ B r ( p ) ′′ due to R Mx R ( k∇ .X t k ) ◦ c ( t ) dt < ˜ C · ˆ ε ≤ ˜ C · √ ˆ ε and ϕ t ( p ) = c ( t ) . Furthermore, vol( B r ( p ) ′′ ) > (1 − ˆ C √ ˆ ε ) · (1 − C ˆ ε ) · vol( B r ( p )) > (1 − C ′ √ ˆ ε ) · vol( B r ( p )) using C ′ = ˆ C + q C = ˆ C + C · q C > ˆ C + C √ ˆ ε .Moreover, points in B r ( p ) ′′ satisfy the following: Fix t ∈ [0 , , a ∈ B r ( p ) ′′ and let ˜ a := ϕ t ( a ) . In particular, R Mx R ( k∇ .X t k ) ◦ ϕ t ( a ) dt < ˜ C · √ ˆ ε and, by Lemma 2.20, for any ρ ≤ R there are subsets B ρ ( a ) ′ ⊆ B ρ ( a ) and B ρ (˜ a ) ′ ⊆ B ρ (˜ a ) such that vol( B ρ ( a ) ′ ) ≥ (1 − C √ ˆ ε ) · vol( B ρ ( a )) and ϕ t ( B ρ ( a ) ′ ) ⊆ B ˜ αρ (˜ a ) , vol( B ρ (˜ a ) ′ ) ≥ (1 − C √ ˆ ε ) · vol( B ρ (˜ a )) and ϕ t − t ( B ρ (˜ a ) ′ ) ⊆ B ˜ αρ ( a ) where C √ ˆ ε < C · ˆ ε < .Let d = d ( n, ˆ ε, α, R ) and δ = δ ( n, ˆ ε, α, R ) , respectively, denote theconstants d ( n, C √ ˆ ε, R ) and δ ( n, C √ ˆ ε, R ) , respectively, of Lemma 2.18.This d < R can be chosen so small that δ ≤ − α +1 = − α < . Define ˆ r = ˆ r ( n, ˆ ε, α, R ) := d α < R α . From now on, assume r < ˆ r and let b ∈ B r ( p ) ′′ be another point. Inparticular, d := d ( a, b ) < r < d α < d . For arbitrary δ < δ < , let ρ := ( + δ ) d < R . By Lemma 2.17 and Lemma 2.18, there exists a point z ∈ B ρ ( a ) ′ ∩ B ρ ( b ) ′ . Thus, d ( ϕ t ( a ) , ϕ t ( b )) ≤ d ( ϕ t ( a ) , ϕ t ( z )) + d ( ϕ t ( z ) , ϕ t ( b )) < · ˜ αρ = ˜ α · (2 δ + 1) d. Since δ > δ was arbitrary and (2 δ + 1) · ˜ α ≤ α , this proves d ( ϕ t ( a ) , ϕ t ( b )) ≤ α · d ( a, b ) . As before, let ˜ a = ϕ t ( a ) and ˜ b = ϕ t ( b ) . These points have distance d (˜ a, ˜ b ) = d ( ϕ t ( a ) , ϕ t ( b )) ≤ α · r < d , and an analogous argumentationgives d ( a, b ) = d ( ϕ t − t (˜ a ) , ϕ t − t (˜ b )) ≤ α · d (˜ a, ˜ b ) = α · d ( ϕ t ( a ) , ϕ t ( b )) . Thus, ϕ t is α -bi-Lipschitz on B r ( p ) ′′ for r < ˆ r . (cid:3) If a sequence of manifolds satisfies the previous lemma and the rescaledmanifolds endowed with the end points of the integral curve as base pointsconverge, the limits have the same dimension.
Proposition 2.22.
Let ( M i ) i ∈ N be a sequence of n -dimensional Riemannianmanifolds with Ric M i ≥ − ( n − . For every i ∈ N , let X i : [0 , × M → T M be a time dependent, piecewise constant in time vector field with compactsupport and flow Φ i , ϕ ti := Φ i (0 , t, · ) , c i : [0 , → M i be an integral curveof X i such that X ti is divergence free on B r ( c i ( t )) for all t ∈ [0 , and R (Mx r ( k∇ .X ti k / )( c i ( t ))) / dt < ˆ ε for some < r ≤ and ˆ ε > .Assume that x ′ i := c i (0) and y ′ i := c i (1) satisfy d ( x ′ i , y ′ i ) ≤ and let λ i → ∞ be scales such that ( λ i M i , x ′ i ) → ( X, x ∞ ) and ( λ i M i , y ′ i ) → ( Y, y ∞ ) as i → ∞ . Then dim( X ) = dim( Y ) .Proof. The proof consists of three steps: First, for any radius r > , constructa bi-Lipschitz map between subsets of ¯ B Xr ( x ∞ ) and ¯ B Yαr ( y ∞ ) , cf. Figure 1.Next, observe that these subsets have positive volume. In particular, theyintersect the set of generic points. Finally, repeating the argument for thelimit spaces proves the claim.Choose any α ∈ (1 , . Without loss of generality, let i ∈ N be large enoughsuch that r < λ i · ˆ r where ˆ r = ˆ r ( α ) is the constant from Lemma 2.21.Furthermore, let B M i r/λ i ( x ′ i ) ′′ ⊆ B M i r/λ i ( x ′ i ) and ϕ i : B M i r/λ i ( x ′ i ) ′′ → B M i αr/λ i ( y ′ i ) be as in Lemma 2.21. Since ϕ i is α -bi-Lipschitz, it can be extended to an α -bi-Lipschitz map ϕ i : B M i r/λ i ( x ′ i ) ′′ → ¯ B M i αr/λ i ( y ′ i ) .In order to regard ϕ i as a map λ i M i → λ i M i instead of M i → M i , let G i denote this closure B M i r/λ i ( x ′ i ) ′′ regarded as a subset of ¯ B λ i M i r ( x ′ i ) ⊆ λ i M i .Correspondingly, define f i : G i → B λ i M i αr ( y ′ i ) by f i ( q ) := ϕ i ( q ) , cf. Figure 1.By definition, this map is α -bi-Lipschitz and measure preserving.After passing to a subsequence, the G i converge to a compact subset S r ⊆ ¯ B Xr ( x ∞ ) and an α -bi-Lipschitz homeomorphism f r : S r → f r ( S r ) suchthat f r ( S r ) is the limit of the f i ( G i ) , cf. again Figure 1.Now find a point x ∈ S r such that both x and f r ( x ) are generic: Thereexists a constant C > such that vol Y ( f r ( S r ∩ X gen ) ∩ Y gen ) = vol Y ( f r ( S r ∩ X gen ))= C · vol X ( S r ∩ X gen )= C · vol X ( S r ) > . XISTENCE OF TYPICAL SCALES 29 ¯ B M i r/λ i ( x ′ i ) ∼ = ¯ B λ i M i r ( x ′ i ) ¯ B Xr ( x ∞ ) B M i r/λ i ( x ′ i ) ′′ ∼ = G i S r ¯ B M i αr/λ i ( y ′ i ) ∼ = ¯ B λ i M i αr ( y ′ i ) ¯ B Yαr ( y ∞ ) ⊆ ⊆ ⊆ i → ∞ i → ∞ i → ∞ ϕ i f i f r Figure 1.
Sets and maps used to construct f r : S r → ¯ B αr ( y ∞ ) .Hence, there exists x ∈ S r ∩ X gen with image f r ( x ) ∈ Y gen . By sim-ilar arguments as before, for λ → ∞ , the sets ( λS r , x ) ⊆ ( λX, x ) and ( λf r ( S r ) , f r ( x )) ⊆ ( λY, f r ( x )) , respectively, (sub)converge to limits S ∞ and S ′∞ , respectively. Moreover, for λf r ( x ) := f r ( x ) , the α -bi-Lipschitzmaps λf r : ( λS r , x ) → ( λf r ( S r ) , f r ( x )) (sub)converge to an α -bi-Lipschitzmap f : S ∞ → S ′∞ as λ → ∞ . Since x and f r ( x ) are generic, one has S ∞ ⊆ R dim( X ) and S ′∞ ⊆ R dim( Y ) . Furthermore, vol( S ∞ ) > . This implies dim( X ) = dim( Y ) . (cid:3) Proof of Proposition 2.1.
The idea is to intersect the sets constructedin Lemma 2.9 and Lemma 2.16. For fixed base points x i in the intersec-tion and the λ i of Lemma 2.16, the ( λ i M i , x i ) are both close to products ( R k × K i , · ) and converging to a product ( R k × Y, · ) where the K i are com-pact with diameter and Y is some metric space. The following (technical)lemmata show that this space Y in fact is compact. Subsequently, the mainproposition can be proven.A map f : ( X, d X ) → ( Y, d Y ) between two metric spaces is called ε -isometry , where ε > , if | d Y ( f ( p ) , f ( q )) − d X ( p, q ) | < ε for all p, q ∈ X . Lemma 2.23.
Fix
R > r ≥ , k ∈ N , S R := S R k R (0) , ε > , and let f : S R → ¯ B R k R (0) \ B R k R − r (0) be a continuous ε -isometry with ε < · ( R − r ) .Define pr : ¯ B R k R (0) \ B R k R − r (0) → S R by pr( p ) := R k p k · p . Then pr ◦ f : S R → S R is surjective.Proof. Denote the distance function on R k by d and distinguish the two casesof r = 0 and r > : First, let r = 0 , i.e. f ( S R ) ⊆ S R and pr = id . Assumethat there exists a point p ∈ S R \ f ( S R ) and define j : S R \ { p } → R k − asthe stereographic projection. Then the composition j ◦ f : S R → R k − iscontinuous and, by the theorem of Borsuk-Ulam, there exist ± q ∈ S R suchthat j ◦ f ( q ) = j ◦ f ( − q ) . Since j is a homeomorphism, f ( q ) = f ( − q ) . Hence, ε > | d ( f ( q ) , f ( − q )) − d ( q, − q ) | = 2 R . This is a contradiction. Therefore, f is surjective. Now let r > be arbitrary. For any p, q ∈ ¯ B R k R (0) \ B R k R − r (0) , | d (pr ◦ f ( p ) , pr ◦ f ( q )) − d ( p, q ) |≤ | d (pr ◦ f ( p ) , pr ◦ f ( q )) − d ( f ( p ) , f ( q )) | + | d ( f ( p ) , f ( q )) − d ( p, q ) |≤ d (pr ◦ f ( p ) , f ( p )) + d (pr ◦ f ( q ) , f ( q )) + ε ≤ r + ε . Therefore, pr ◦ f is a continuous r + ε -isometry and, by the first part, sur-jective. (cid:3) The following lemma states that, if two products R k × X and R k × Y aresufficiently close and X is compact, then Y is compact as well with similardiameter as X . Lemma 2.24.
Let ( X, d X ) and ( Y, d Y ) be complete length spaces, X becompact, x ∈ X , y ∈ Y and define rad Y ( y ) := sup { d Y ( y, y ) | y ∈ Y } . Let k ∈ N , R > diam( X ) and ε > . Then the following is true: a) If diam( X )+4 ε ≤ R and d GH ( B R k × XR ((0 , x )) , B R k × YR ((0 , y ))) < ε ,then min { R, rad Y ( y ) } ≤ diam( X ) + 2 ε · diam( X ) + 4 ε ( R + ε ) . b) If diam( X ) = 0 and d GH ( B R k × XR ((0 , x )) , B R k × YR ((0 , y ))) < R , then Y is compact with diam( Y ) < R . c) If diam( X ) = 1 and d GH ( B R k × XR ((0 , x )) , B R k × YR ((0 , y ))) < R forsome R ≥ , then Y is compact with c ≤ diam( Y ) ≤ c for a smallconstant c > . d) If · diam( X ) ≤ R < · diam( Y ) , then d GH ( B R k × XR ((0 , x )) , B R k × YR ((0 , y ))) ≥ · diam( X ) . Proof. a) The idea is to approximate the diameter of Y by taking points y ′ as far away from y as possible, to map both the set S R × { y } and thepoints (0 , y ′ ) to R k × X via ε -approximations, to take the projection onto theEuclidean factor and to find an upper and lower estimate for the distanceof the obtained set and point. Finally, comparison of this upper and lowerbound gives the result.Let ( f, g ) be ε -approximations between the balls ¯ B R k × XR ((0 , x )) and ¯ B R k × YR ((0 , y )) with f ((0 , x )) = (0 , y ) and g ((0 , y )) = (0 , x ) . Let d := diam( X ) and δ := min { R, rad Y ( y ) } . For each n ∈ N , n ≥ , choose y n ∈ Y such that δ − n ≤ δ n := d Y ( y n , y ) ≤ δ .(If rad Y ( y ) > R , choose y n ∈ ∂ ¯ B R (0) which is nonempty since Y is alength space; otherwise, by definition, there exists a sequence ¯ y n satisfying d Y (¯ y n , y ) > δ − n .) In particular, δ n is convergent with limit δ . XISTENCE OF TYPICAL SCALES 31 ∂ ¯ B R k R (0) × { y } = S R × { y } ⊆ B R k × YR ((0 , y )) g ( S R × { y } ) = S ⊆ B R k × XR ((0 , x )) gg Figure 2.
Definition of S .Let S R := ∂ ¯ B R k R (0) ⊆ R k and S := g ( S R × { y } ) ⊆ R k × X , cf. Figure 2.Since d R k × X ( g (0 , y n ) , p )= q d R k (pr R k ( g (0 , y n )) , pr R k ( p )) + d X (pr X ( g (0 , y n )) , pr X ( p )) ≤ d R k (pr R k ( g (0 , y n )) , pr R k ( S )) + d for every p ∈ S , p R + δ n − ε = d R k × Y ((0 , y n ) , S R × { y } ) − ε ≤ d R k × X ( g (0 , y n ) , S ) ≤ q d R k (pr R k ( g (0 , y n )) , pr R k ( S )) + d and this proves the lower bound d R k (pr R k ( g (0 , y n )) , pr R k ( S )) ≥ q ( p R + δ n − ε ) − d . In order to find the upper bound, choose a natural number m ≥ ε andlet ∆ be a spherical triangulation of S R such that the set of vertices ˜Γ of ∆ is a finite m -net in S R and each two vertices of a simplex have (spherical)distance at most m . (Notice that their Euclidean distance is at most m aswell.) Define Γ := ˜Γ × { y } and h := pr R k ◦ g : Γ → R k and extend h to acontinuous map H : S R × { y } → ¯ B R k R (0) \ B R k R − ( d +3 ε ) (0) by mapping each(spherical) simplex of ∆ with vertices γ i continuously to the corresponding(Euclidean) simplex in R k with vertices h ( γ i ) , cf. Figure 3. Since Γ is finite, H is continuous.Then h (Γ) defines an ( m + ε ) -net in H ( S R × { y } ) : Since each two verticesof a simplex in ∆ have (Euclidean) distance at most m , their images havedistance at most m + ε . Hence, each point x ∈ H ( S R × { y } ) is containedin a Euclidean simplex whose vertices have pairwise distance at most m + ε .Recall that, since the simplex is Euclidean, x has distance at most m + ε toeach of these vertices. Let h ( γ ) denote one of those vertices. In particular, x ∈ ¯ B /m + ε ( h ( γ )) , hence, H ( S R × { y } ) ⊆ S { ¯ B /m + ε ( γ ′ ) | γ ′ ∈ H (Γ) } . S R × { y } ⊇ Γ = ˜Γ × { y } ⊆ ∆( γ , . . . , γ m ) H ( S R × { y } ) ⊆ R k ⊇ h (Γ) ⊆ ∆( h ( γ ) , . . . , h ( γ m )) h = pr R k ◦ gH Figure 3.
Definition of H .Furthermore, H is a (5 ε + d ) -isometry: Let p, q ∈ S R be arbitrary. Choosepoints γ p , γ q ∈ ˜Γ such that d R k ( p, γ p ) ≤ m and d R k ( q, γ q ) ≤ m . By construc-tion, d R k ( H ( p, y ) , h ( γ p , y )) ≤ m + ε, and thus, | d R k ( H ( p, y ) , H ( q, y )) − d R k × Y (( p, y ) , ( q, y )) |≤ | d R k ( H ( p, y ) , H ( q, y )) − d R k ( h ( γ p , y ) , h ( γ q , y )) | + | d R k ( h ( γ p , y ) , h ( γ q , y )) − d R k × X ( g ( γ p , y ) , g ( γ q , y )) | + | d R k × X ( g ( γ p , y ) , g ( γ q , y )) − d R k × Y (( γ p , y ) , ( γ q , y )) | + | d R k ( γ p , γ q ) − d R k ( p, q ) |≤ d R k ( H ( p, y ) , h ( γ p , y )) + d R k ( H ( q, y ) , h ( γ q , y ))+ (cid:0) d R k (pr R k ◦ g ( γ p , y ) , pr R k ◦ g ( γ q , y )) + d X (pr X ◦ g ( γ p , y ) , pr X ◦ g ( γ q , y )) (cid:1) / − d R k (pr R k ◦ g ( γ p , y ) , pr R k ◦ g ( γ q , y ))+ ε + d R k ( p, γ p ) + d R k ( q, γ q ) ≤ · (cid:16) m + ε (cid:17) + d X (pr X ◦ g ( γ p , y ) , pr X ◦ g ( γ q , y )) + ε +2 · m ≤ ε + d. Finally, verify H ( S R × { y } ) ⊆ ¯ B R (0) \ B R − ( d +3 ε ) (0) : Let p ∈ S R bearbitrary and choose γ p ∈ Γ such that d ( p, γ p ) ≤ m . Then d R k ( h ( γ p , y ) , pr R k (0 , x ))= q d R k × X ( g ( γ p , y ) , g (0 , y )) − d X (pr X ◦ g ( γ p , y ) , pr X ◦ g (0 , y )) ≥ q ( d R k × Y (( γ p , y ) , (0 , y )) − ε ) − d = p ( R − ε ) − d , XISTENCE OF TYPICAL SCALES 33 R k ( g (0 , y n )) pq ∂ ¯ B R (0) H ( S R × { y } ) σ Figure 4.
Points used to estimate d R k (pr R k ( g (0 , y n )) , pr R k ( S )) .and hence, d R k ( H ( p, y ) , ≥ d R k ( h ( γ p , y ) , pr R k (0 , x )) − d R k ( H ( p, y ) , h ( γ p , y )) ≥ p ( R − ε ) − d − (cid:16) m + ε (cid:17) ≥ R − ( d + 3 ε ) . Then the image of H intersects each segment σ from the origin to a pointin ∂ ¯ B R (0) : By Lemma 2.23, the segment σ intersects ∂ ¯ B R (0) in a pointcontained in pr ◦ H ( S R × { y } ) where pr is the radial projection to the spheredefined as in Lemma 2.23. Since the projection is radial, the segment σ intersects H ( S R × { y } ) as well.Let p be this intersecting point for the segment through pr R k ( g (0 , y n )) ,cf. Figure 4. Since h (Γ × { y } ) is a ( m + ε ) -net in H ( S R ) , there exists apoint q ∈ h (Γ) such that d R k ( p, q ) ≤ m + ε . Thus, using that the segmentfrom pr R k ( g (0 , y n )) to p is part of a segment connecting the origin and theboundary of the R -ball and pr R k ( S ) = pr R k ◦ g ( S R × { y } ) ⊇ h (Γ) ∋ q , d R k (pr R k ( g (0 , y n )) , pr R k ( S )) ≤ d R k (pr R k ( g (0 , y n )) , q ) ≤ d R k (pr R k ( g (0 , y n )) , p ) + d R k ( p, q ) ≤ R + (cid:16) m + ε (cid:17) . Now m → ∞ proves q ( p R + δ n − ε ) − d ≤ d R k (pr R k ( g (0 , y n )) , pr R k ( S )) ≤ R + ε, and thus, δ n ≤ q ( p ( R + ε ) + d + ε ) − R = q R ε + d + 2 ε +2 p ( R ε + ε ) + ( ε d ) ≤ p d + 2 ε d + 4 ε ( R + ε ) . Since this is true for all n and δ n → δ as n → ∞ , this proves the claim. b) Let ε := R . Then diam( X ) + 4 ε = R , and by a), min { R, rad Y ( y ) } ≤ ε ( R + ε ) = 2425 · R < R . Thus, rad Y ( y ) < R , and this implies diam( Y ) ≤ · rad Y ( y ) < R .c) Let ε := R . Then diam( X ) + 4 ε ≤ < R · π and a) can be applied.Since R ε = = 2 · − and ε ≤ · − , min { R, rad Y ( y ) } ≤ diam( X ) + 2 ε diam( X ) + 4 ε ( R + ε ) ≤ diam( X ) + 4 · − · diam( X ) + 8 · − + 4 · − . Using diam( X ) = 1 , min { R, rad Y ( y ) } ≤ · − + 10 − + 10 − < . < R . In particular, diam( Y ) ≤ · rad Y ( y ) < · .
05 = .On the other hand, by permuting X and Y ,
14 = (cid:16) diam( X )2 (cid:17) ≤ rad X ( x ) = min { R, rad X ( x ) } ≤ diam( Y ) + 4 · − · diam( Y ) + 8 · − + 16 · − , and this implies diam( Y ) ≥ =: c .d) Assume d GH ( B R k × XR ((0 , x )) , B R k × YR ((0 , y ))) < d for d := diam( X ) .Let ε := 40 d . By choice of R , diam( X ) + 4 ε = 161 · d ≤ · R < R .Furthermore, Y ( y ) ≥ diam( Y ) > R . By a), R = min { R, rad Y ( y ) } ≤ d + 2 ε d + 4 ε ( R + ε ) ≤ R + 2 R · + 4 R · R
25 = 16648110 · R < R . This is a contradiction. (cid:3)
Using these results, the main proposition of this section finally can beproven.
Proposition 2.1.
Let ( M i ) i ∈ N be a sequence of complete connected n -di-mensional Riemannian manifolds with uniform lower Ricci curvature bound Ric M i ≥ − ( n − , and let k < n . Given ˆ ε ∈ (0 , , there is ˆ δ = ˆ δ (ˆ ε ; n, k ) > such that for any < r ≤ ˆ δ and q i ∈ M i with d GH (( r − M i , q i ) , ( R k , ≤ ˆ δ there are a family of subsets of good points G r ( q i ) ⊆ B r ( q i ) with vol( G r ( q i )) ≥ (1 − ˆ ε ) · vol( B r ( q i )) and a sequence λ i → ∞ such that the following holds: XISTENCE OF TYPICAL SCALES 35 a) For every choice of base points x i ∈ G r ( q i ) and every sublimit ( Y, · ) of ( λ i M i , x i ) there exists a compact metric space K of dimension l ≤ n − k satisfying ≤ diam( K ) ≤ such that Y splits isometrically as aproduct Y ∼ = R k × K. b) If x i , x i ∈ G r ( q i ) are base points such that, after passing to a subse-quence, ( λ i M i , x ji ) → ( R k × K j , · ) for ≤ j ≤ as before, then dim( K ) = dim( K ) .Proof. Given ˆ ε ∈ (0 , , let ˆ δ = ˆ δ (ˆ ε ; n, k ) > be the ˆ δ (cid:16) ˆ ε n, k (cid:17) of Lemma 2.9, ˆ δ = ˆ δ (ˆ ε ; n, k ) > be the ˆ δ (cid:16) ˆ ε n, k (cid:17) of Lemma 2.16,and define ˆ δ = ˆ δ (ˆ ε ; n, k ) := 116 · min { ˆ δ , ˆ δ } > . Furthermore, let ε = ε ( n, k ) ∈ (0 , ) be as in Lemma 2.15. Let < r ≤ ˆ δ and q i ∈ M i with d GH (( r − M i , q i ) , ( R k , ≤ ˆ δ. Let G r ( q i ) ⊆ B r ( q i ) be as in Lemma 2.9 and G r ( q i ) ⊆ B r ( q i ) and λ i → ∞ as in Lemma 2.16. Define G r ( q i ) := G r ( q i ) ∩ G r ( q i ) ⊆ B r ( q i ) . Clearly, vol( G r ( q i )) ≥ (1 − ˆ ε ) · vol( B r ( q i )) .Let x i ∈ G r ( q i ) and ( Y, y ) be a sublimit of ( λ i M i , x i ) . Using x i ∈ G r ( q i ) ,there are a metric space Y ′ and y ′ ∈ { }× Y ′ such that ( Y, y ) ∼ = ( R k × Y ′ , y ′ ) .On the other hand, since x i ∈ G r ( q i ) , d GH ( B λ i M i / ε ( x i ) , B R k × K i / ε (˜ x i )) ≤ ε for some compact metric space K i with diameter and ˜ x i ∈ { }× K i . Hence,by the triangle inequality and for i large enough, d GH ( B R k × Y ′ / ε ( y ′ ) , B R k × K i / ε (˜ x i )) < ε . By Lemma 2.24 c), there exists a constant c > such that Y ′ is compactas well with c ≤ diam( Y ′ ) ≤ c , and after rescaling with c this finishes thefirst part of the claim.So let x i , y i ∈ G r ( q i ) and K and K be compact metric spaces such that,after passing to a subsequence, ( λ i M i , x i ) → ( R k × K , x ∞ ) and ( λ i M i , y i ) → ( R k × K , y ∞ ) . Because of x i , y i ∈ G r ( q i ) , there is a time-dependent, piecewise constantin time vector field X i with compact support and an integral curve c i suchthat the vector field X ti is divergence free on B r ( c i ( t )) for all ≤ t ≤ , d ( x i , c i (0)) < c ( n ) λ i and d ( y i , c i (1)) < c ( n ) λ i for c ( n ) = 9 n · n and Z (Mx r ( k∇ .X ti k / )( c i ( t ))) / dt < ˆ ε . Let x ′ i := c i (0) and y ′ i := c i (1) . Because of d λ i M i ( x i , x ′ i ) ≤ c ( n ) and d λ i M i ( y i , y ′ i ) ≤ c ( n ) , there exists x ′∞ ∈ R k × K such that, after passingto a subsequence, ( λ i M i , x ′ i ) → ( R k × K , x ′∞ ) . After passing to a further subsequence, ( λ i M i , y ′ i ) → ( R k × K , y ′∞ ) for some y ′∞ ∈ R k × K . Then Proposition 2.22 implies dim( K ) = dim( R k × K ) − k = dim( R k × K ) − k = dim( K ) . (cid:3) Global construction
Based on the ‘local’ version (Proposition 2.1) established in the last sec-tion, the proof of the following main result can now be given.
Theorem 3.1.
Let ( M i , p i ) i ∈ N be a collapsing sequence of pointed completeconnected n -dimensional Riemannian manifolds which satisfy the uniformlower Ricci curvature bound Ric M i ≥ − ( n − and converge to a limit ( X, p ) of dimension k < n in the measured Gromov-Hausdorff sense. Then forevery ε ∈ (0 , there exist a subset of good points G ( p i ) ⊆ B ( p i ) satisfying vol( G ( p i )) ≥ (1 − ε ) · vol( B ( p i )) , a sequence λ i → ∞ and a constant D > such that the following holds:For any choice of base points q i ∈ G ( p i ) and every sublimit ( Y, q ) of ( λ i M i , q i ) i ∈ N there is a compact metric space K of dimension l ≤ n − k anddiameter D ≤ diam( K ) ≤ D such that Y splits isometrically as a product Y ∼ = R k × K. Moreover, for any base points q i , q ′ i ∈ G ( p i ) such that, after passing to asubsequence, both ( λ i M i , q i ) → ( R k × K, · ) and ( λ i M i , q ′ i ) → ( R k × K ′ , · ) asbefore, dim( K ) = dim( K ′ ) . The idea of the proof is to take (finitely many) sequences ( q i ) i ∈ N satisfyingthe hypothesis of Proposition 2.1 for some r > and to define G ( p i ) as theunion of the G r ( q i ) obtained from Proposition 2.1. Instantly, the followingquestion occurs: XISTENCE OF TYPICAL SCALES 37 (1) Why do sequences ( q i ) i ∈ N satisfying the hypothesis of Proposition 2.1exist?It will turn out that sequences p xi → x , where x ∈ X is a generic point, arecandidates for these ( q i ) i ∈ N : If x ∈ X is generic, then ( r X, x ) is close to ( R k , for sufficiently small r > and so is ( r M i , p xi ) for sufficiently large i ∈ N . In fact, decreasing r only improves the situation.Now assume that x, x ′ are two such generic points, let r > be smallenough and λ i → ∞ and λ ′ i → ∞ , respectively, be the sequences given byProposition 2.1. These sequences might be different, but Theorem 3.1 callsfor one single rescaling sequence. This gives rise to the following question:(2) Does Proposition 2.1 still hold for ( p x ′ i ) i ∈ N if λ ′ i → ∞ is replaced by λ i → ∞ ?In order to answer this question, first consider the special case λ i = 2 λ ′ i :Obviously, if q i ∈ G r ( p x ′ i ) and ( R k × K, · ) is a sublimit of ( λ ′ i M i , q i ) , then ( R k × K, · ) is a sublimit of ( λ i M i , q i ) = (2 λ ′ i M i , q i ) . Conversely, everysublimit of ( λ i M i , q i ) has the form ( R k × K, · ) for a sublimit ( R k × K, · ) of ( λ ′ i M i , q i ) . It turns out that such a correspondence holds whenever thesequence (cid:0) λ ′ i λ i (cid:1) i ∈ N is bounded. In this way, λ ′ i indeed can be replaced by λ i ifone allows weaker diameter bounds for the compact factors of the sublimits.Therefore, the question (2) can be reformulated in the following way:(2’) Under which condition is the quotient (cid:0) λ ′ i λ i (cid:1) i ∈ N of two such rescalingsequences bounded?In fact, one can prove the following: If the subsets G r ( p xi ) and G r ( p x ′ i ) havenon-empty intersection, then (cid:0) λ ′ i λ i (cid:1) i ∈ N is bounded. An obvious approach forcomparing points where these subsets do not intersect is to connect thepoints by a curve consisting of generic points only and to cover this curveby balls B r j ( y j ) such that for every two subsequent points y j and y j +1 (andsufficiently large i ∈ N ) the subsets intersect. If this can be done using onlyfinitely many y j , an inductive argument proves the boundedness of (cid:0) λ ′ i λ i (cid:1) i ∈ N .Usually, such covers are constructed by using a compactness argument: Let r y denote the minimal radius such that all r ≤ r y and p yi satisfy the hypothe-sis of Proposition 2.1. If this r y is continuous in y , there exists r > that canbe used at each point of the (compact image of the) curve. Unfortunately,there is no reason for this r y to depend continuously on y . It will turn outthat a similar approach to compare λ i and λ ′ i can be performed if x and x ′ lie in the interior of a minimising geodesic such that all the points of thisgeodesic lying between x and x ′ are generic. The Hölder continuity result ofColding and Naber [CN12, Theorem 1.2] then allows a cover similar to theone described above. The subsequent question(3) Does there exists a minimising geodesic such that x, x ′ lie in its inte-rior? can be answered affirmatively for a set of full measure (in X × X ) by applyingfurther results of [CN12].This section is subdivided into several subsections answering the abovequestions: First, subsection 3.1 investigates generic points x ∈ X and an-swers question (1) by applying Proposition 2.1 to the sequence p xi → x . Bothquestions (2) and (2’) are dealt with in subsection 3.2, which discusses thecomparison of the different λ i . Afterwards, subsection 3.3 treats question (3)by proving that the necessary conditions for performing the comparison aregiven on a set of full measure. Finally, subsection 3.4 deals with the proofof Theorem 3.1.3.1. Application to generic points.
A very important property of genericpoints is that, after rescaling, the manifolds with base points converging toa generic point are in some sense close to the Euclidean space.
Lemma 3.2.
Let ( X, p ) be the limit of a collapsing sequence of pointed com-plete connected n -dimensional Riemannian manifolds which satisfy the uni-form lower Ricci curvature bound − ( n − , k = dim( X ) < n , x ∈ X gen and p xi → x . For fixed R > and ε > there exists λ = λ ( x, R, ε ) such thatfor all λ ≥ λ , d GH ( B λXR ( x ) , B R k R (0)) ≤ ε . Proof.
Since x is a generic point, all tangent cones at x equal R k , i.e. for everychoice of λ → ∞ , the sequence ( λX, x ) converges to ( R k , . In particular,the R -balls converge and for sufficiently large λ the distance of these balls isbounded from above by ε . This proves that there exists λ ( x ) := min { λ ≥ | ∀ µ ≥ λ : d GH ( B µXR ( x ) , B R k R (0)) ≤ ε } < ∞ . (cid:3) Notation.
From now on, for given k < n and ˆ ε ∈ (0 , , let ˆ δ = ˆ δ (ˆ ε ; n, k ) be as in Proposition 2.1. For r > , define X r (ˆ ε ; n, k ) := n x ∈ X gen | d GH ( B r − X / ˆ δ ( x ) , B R k / ˆ δ (0)) ≤ ˆ δ o . Lemma 3.3.
Let ( M i , p i ) i ∈ N be a collapsing sequence of pointed completeconnected n -dimensional Riemannian manifolds which satisfy the uniformlower Ricci curvature bound Ric M i ≥ − ( n − and converge to a limit ( X, p ) of dimension k < n and let ˆ ε ∈ (0 , . a) For every x ∈ X gen there is < r x = r (ˆ ε, x ; n, k ) ≤ ˆ δ such that x ∈ X r (ˆ ε ; n, k ) for any < r ≤ r x . b) For < r ≤ ˆ δ , x ∈ X r (ˆ ε ; n, k ) and p xi → x there is i ∈ N such thatfor i ≥ i there are a subset of good points G r ( p xi ) ⊆ B r ( p xi ) with vol( G r ( p xi )) ≥ (1 − ˆ ε ) · vol( B r ( p xi )) and a sequence λ i → ∞ satisfying the following: XISTENCE OF TYPICAL SCALES 39 (i)
For any choice of base points x i ∈ G r ( p xi ) and all sublimits ( Y, · ) of ( λ i M i , x i ) there exists a compact metric space K of dimension l ≤ n − k and diameter ≤ diam( K ) ≤ such that Y splitsisometrically as a product Y ∼ = R k × K. (ii) If x i , x i ∈ G r ( p xi ) are base points such that, after passing to asubsequence, ( λ i M i , x ji ) → ( R k × K j , · ) for ≤ j ≤ as before, then dim( K ) = dim( K ) .Moreover, if ω is a fixed ultrafilter on N , there exists l ∈ N such thatthe following holds: Given q i ∈ G r ( p xi ) , the ultralimit of ( λ i M i , q i ) isa product R k × K such that K is compact with ≤ diam( K ) ≤ and dim( K ) = l. Proof. a) Let λ = λ (ˆ ε ; n, k ) be the λ ( x, δ , ˆ δ ) appearing in Lemma 3.2.Then r x := r (ˆ ε, x ; n, k ) := min (cid:8) ˆ δ, λ (cid:9) > proves the claim.b) Let x ∈ X r (ˆ ε ; n, k ) be arbitrary, i.e. d GH ( B r − X / ˆ δ ( x ) , B R k / ˆ δ (0)) ≤ ˆ δ . Since ( r M i , p xi ) → ( r X, x ) , there is i ∈ N with d GH ( B r − M i / ˆ δ ( p xi ) , B r − X / ˆ δ ( x )) ≤ ˆ δ for all i ≥ i . In particular, d GH ( B r − M i / ˆ δ ( p xi ) , B R k / ˆ δ (0)) ≤ ˆ δ by triangle in-equality and Proposition 2.1 implies the claim. (cid:3) Notation.
For < r ≤ ˆ δ and x ∈ X r (ˆ ε ; n, k ) , let λ ˆ ε,xi ( r ) and G ˆ εr ( p xi ) beas in Lemma 3.3, i.e. for q i ∈ G ˆ εr ( p xi ) the sublimits of ( λ ˆ ε,xi ( r ) M i , q i ) areisometric to products ( R k × K, · ) where the K are compact metric spaceswith diam( K ) ∈ (cid:2) , (cid:3) . Moreover, for x ∈ X gen , let r x (ˆ ε ; n, k ) be as inLemma 3.3, i.e. x ∈ X r (ˆ ε ; n, k ) for all < r ≤ r x (ˆ ε ; n, k ) .Furthermore, for a non-principal ultrafilter ω on N , let l ˆ ε,xω ( r ) be as inLemma 3.3, i.e. for q i ∈ G ˆ εr ( p xi ) , lim ω ( λ ˆ ε,xi ( r ) M i , q i ) = ( R k × K, · ) and dim( K ) = l ˆ ε,xω ( r ) for some K as above.All notations will be used throughout the remaining section without refer-ring to Lemma 3.3 explicitly. Occasionally, if they are fixed, the dependenceson n , k and ˆ ε will be suppressed.3.2. Comparison of rescaling sequences.
Throughout this subsection,let k < n and ˆ ε ∈ (0 , ) be fixed and use the notation introduced insubsection 3.1.The following lemma states that for each two rescaling sequences corre-sponding to limit spaces which are products of the same Euclidean and acompact set, the quotient of these rescaling sequences is bounded. Espe-cially, this holds in the situation of Lemma 3.3. Lemma 3.4.
Let ( M i , p i ) i ∈ N be a collapsing sequence of pointed completeconnected n -dimensional Riemannian manifolds which satisfy the uniformlower Ricci curvature bound Ric M i ≥ − ( n − and converge to a limit ( X, p ) of dimension k < n . a) Let q i ∈ M i satisfy that for any λ i → ∞ and every sublimit ( Y, · ) of ( λ i M i , q i ) there exists a metric space Y ′ such that Y ∼ = R k × Y ′ isometrically. For ≤ j ≤ , let λ ji → ∞ and K j be compact with ( λ ji M i , q i ) → ( R k × K j , · ) as i → ∞ . Then the sequence (cid:0) λ i λ i (cid:1) i ∈ N isbounded. b) For ≤ j ≤ , let r j > , x j ∈ X r j , p ji := p x j i and λ ji := λ ˆ ε,x j i ( r j ) .Moreover, assume G r ( p i ) ∩ G r ( p i ) = ∅ . Then the sequence (cid:0) λ i λ i (cid:1) i ∈ N is bounded.Proof. a) The proof is done by contradiction: Without loss of generality,assume λ ji > for ≤ j ≤ and all i ∈ N . Obviously, the sequence (cid:0) λ i λ i (cid:1) i ∈ N is bounded from below by . Assume the sequence is not bounded fromabove, i.e. λ i λ i → ∞ and, without loss of generality, λ i < λ i for all i ∈ N .There exists λ i → ∞ satisfying λ i < λ i < λ i such that ( λ i M i , q i ) → ( R k × N, y ) for some unbounded metric space N : Let µ i → ∞ be as in Lemma 2.13 suchthat ( µ ′ i λ i M i , q i ) subconverges to a tangent cone of ( R k × K , x ) for any µ ′ i → ∞ with µ ′ i ≤ µ i . Define µ ′ i := min (cid:26) µ i , s λ i λ i (cid:27) → ∞ and λ i := µ ′ i · λ i → ∞ . Without loss of generality, assume µ i > . Thus, λ i < λ i < λ i . Furthermore,there exist a metric space N and q ∈ R k × N with ( λ i M i , q i ) → ( R k × N, q ) and this is a tangent cone of ( R k × K , x ) . Hence, for a sequence α i → ∞ , ( R k × α i K , x ) → ( R k × N, q ) . Assume this N is compact and let N ′ := N ) · N and β i := N ) · α i .Then ( R k × β i K , x ) → ( R k × N ′ , q ) , and, for sufficiently large i , d GH ( B R k × β i K ( x ) , B R k × N ′ ( q )) ≤ − . By Lemma 2.24 c), the sequence (diam( β i K )) i ∈ N is bounded. This is acontradiction to β i → ∞ . Hence, N is unbounded.Now let D := diam( K ) and fix R > D . As N is unbounded, R < diam( N ) and d GH ( B R k × NR ( q ) , B R k × K R ( x )) ≥ D by Lemma 2.24 XISTENCE OF TYPICAL SCALES 41 d). Thus, for i large enough, d GH ( B λ i M i R ( q i ) , B λ i M i R ( q i )) ≥ d GH ( B R k × NR ( q ) , B R k × K R ( x )) − d GH ( B λ i M i R ( q i ) , B R k × NR ( q )) − d GH ( B λ i M i R ( q i ) , B R k × K R ( x )) ≥ D. Since the maps h i : (0 , ∞ ) → (0 , ∞ ) defined by h i ( µ i ) := d GH ( B µ i M i R ( q i ) , B λ i M i R ( q i )) are continuous with h i ( λ i ) = 0 and h i ( λ i ) ≥ D , by the intermediate valuetheorem, there is a maximal λ i < λ ′ i ≤ λ i < λ i such that h i ( λ ′ i ) = 5 D . Afterpassing to a subsequence, ( λ ′ i M i , q i ) → ( R k × Y, y ) for some metric space Y . In particular, d GH ( B λ ′ i M i R ( q i ) , B λ i M i R ( q i )) → d GH ( B R k × YR ( y ) , B R k × K R ( x )) as i → ∞ . Hence, d GH ( B R k × YR ( y ) , B R k × K R ( x )) = 5 D . Furthermore, d GH ( B R k × K R ( x ) , B R k R (0)) ≤ diam( K ) = D and the triangle inequality implies D ≤ d GH ( B R k × YR ( y ) , B R k R (0)) ≤ D < R . By Lemma 2.24 b), Y is compact. Moreover, diam( Y ) ≥ d GH ( B R k × YR ( y ) , B R k R (0)) ≥ D > D, in particular, Y is not a point.Next, prove λ i λ ′ i → ∞ : Assume the quotient is bounded. Hence, afterpassing to a subsequence, λ i λ ′ i → α and α ≥ due to λ ′ i ≤ λ i . Then ( λ i M i , q i ) = (cid:16) λ i λ ′ i · λ ′ i M i , q i (cid:17) → ( R k × αY, y ) ∼ = ( R k × K , x ) . In particular, diam( Y ) ≤ α · diam( Y ) = diam( K ) = D , and this is acontradiction.Thus, λ i λ ′ i → ∞ . Analogously to the previous argumentation, there existssome maximal λ ′ i < ˜ λ i < λ i such that h i (˜ λ i ) = 5 D . This is a contradictionto the maximal choice of λ ′ i .b) Let q i ∈ G r ( p i ) ∩ G r ( p i ) and α i := λ i λ i . Assume α i → ∞ and choose asubsequence ( i j ) j ∈ N such that α i j > j for all j ∈ N . After passing to a furthersubsequence, there exist compact metric spaces K m , where ≤ m ≤ , such that ( λ mi j M i jl , q i j ) → ( R k × K m , · ) . By a), the sequence ( α i j ) j ∈ N is bounded.This is a contradiction. (cid:3) The following lemma gives a statement about the limit of such a boundedsequence of quotients.
Lemma 3.5.
Let ( M i , p i ) i ∈ N be a collapsing sequence of pointed completeconnected n -dimensional Riemannian manifolds satisfying the uniform lowerRicci curvature bound Ric M i ≥ − ( n − and let λ i , µ i > such that (cid:0) λ i µ i (cid:1) i ∈ N is bounded. If ( λ i M i , p i ) → ( R k × L, p L ) and ( µ i M i , p i ) → ( R k × M, p M ) for some bounded metric spaces L and M , then λ i µ i → diam( L )diam( M ) as i → ∞ and dim( L ) = dim( M ) . Proof.
Let a be any accumulation point of (cid:0) λ i µ i (cid:1) i ∈ N and (cid:0) λ ij µ ij (cid:1) j ∈ N be thecorresponding converging subsequence. Then ( λ i j M i j , p i j ) = (cid:16) λ i j µ i j · µ i j M i j , p i j (cid:17) → ( a ( R k × M ) , p M ) = ( R k × ( aM ) , p M ) as j → ∞ . Since this sequence converges to ( R k × L, p L ) as well, there is anisometry ( R k × ( aM ) , p M ) ∼ = ( R k × L, p L ) . Thus, dim( L ) = dim( M ) , diam( L ) = a · diam( M ) and all accumulationpoints of the bounded sequence ( λ i µ i ) i ∈ N equal diam( L )diam( M ) . In particular, (cid:0) λ i µ i (cid:1) i ∈ N is convergent. (cid:3) Next, the question will be answered under which condition the quotientof rescaling sequences belonging to two points in X r is bounded. The firstapproach in order to prove this is the special case of their good subsets tointersect. In the general case, the idea is to connect the points by a curvewhich itself is contained in X r and can be covered by finitely many ballssuch that subsequent subsets of good points intersect. In fact, this cannotbe expected to be possible for the same r > . However, it turns out that thequotient of the rescaling sequences is bounded if the points are connectedby a minimising geodesic contained in some X r ′ of a possibly different r ′ .Making all of this precise is the subject of the following lemma. Lemma 3.6.
Let ( M i , p i ) i ∈ N be a collapsing sequence of pointed completeconnected n -dimensional Riemannian manifolds which satisfy the uniformlower Ricci curvature bound Ric M i ≥ − ( n − and converge to a limit ( X, p ) of dimension k < n . XISTENCE OF TYPICAL SCALES 43 a) Let r m > , x m ∈ X r m and p x m i → x m , where ≤ m ≤ , such that G r ( p x i ) ∩ G r ( p x i ) = ∅ for all i ∈ N . Then ≤ λ x i ( r ) λ x i ( r ) ≤ for almost all i ∈ N and l x ω ( r ) = l x ω ( r ) . b) Let x ∈ X gen , p xi → x and r x ≥ R > r > . Then there is a naturalnumber m = m ( n, ˆ ε, r, R ) ∈ N such that − m ≤ λ xi ( r ) λ xi ( R ) ≤ m for almost all i ∈ N and l xω ( r ) = l xω ( R ) . c) Let γ : [0 , l ] → X be a minimising geodesic with image im( γ ) ⊆ X r forsome < r ≤ ˆ δ . Let x = γ (0) and y = γ ( l ) . Then there is a naturalnumber m = m ( n, ˆ ε, l, r ) ∈ N such that − m ≤ λ xi ( r ) λ yi ( r ) ≤ m for almost all i ∈ N and l xω ( r ) = l yω ( r ) . d) Let x, y ∈ X r and γ : [0 , l ] → X be a minimising geodesic with endpoints x = γ (0) , y = γ ( l ) and image im( γ ) ⊆ X r ′ for some r ′ ≤ r ≤ ˆ δ .Then there is a natural number m = m ( n, ˆ ε, l, r, r ′ ) ∈ N such that − m ≤ λ xi ( r ) λ yi ( r ) ≤ m for almost all i ∈ N and l xω ( r ) = l yω ( r ) .Proof. a) Without loss of generality, for ≤ m ≤ , all λ x m i ( r m ) are positive.Define a i := λ x i ( r ) λ x i ( r ) > and let q i ∈ G r ( p x i ) ∩ G r ( p x i ) be arbitrary.By Lemma 3.4, ( a i ) i ∈ N is bounded. Let a be an arbitrary accumulationpoint and ( a i j ) j ∈ N be the subsequence converging to a . Since q i j ∈ G r ( p x i j ) ,after passing to a subsequence, ( λ x i j ( r ) M i j , q i j ) → ( R k × K , · ) as j → ∞ for some compact metric space K with ≤ diam( K ) ≤ . Because of q i j ∈ G r ( p x i j ) , after passing to a further subsequence, ( λ x i j ( r ) M i j , q i j ) → ( R k × K , · ) as j → ∞ and K satisfies ≤ diam( K ) ≤ . By Lemma 3.5, a = lim j →∞ λ x i j ( r ) λ x i j ( r ) = diam( K )diam( K ) ∈ h , i and l x ω ( r ) = l x ω ( r ) .Since ( a i ) i ∈ N is a bounded sequence and all accumulation points are con-tained in [ , , only finitely many a i are not contained in [ , . b) Since C BG ( n, − , βR, R ) is monotonically increasing for decreasing β < ,there exists β = β ( n, ˆ ε, R ) < with C BG ( n, − , βR, R ) < ε − . Fix this β . Because C BG ( n, − , βρ, ρ ) is monotonically increasing for increasing ρ ,all ρ ≤ R satisfy C BG ( n, − , βρ, ρ ) < ε − as well.Let m = m ( n, ˆ ε, r, R ) ∈ N be maximal with r ≤ β m · R . Define r j := β j · R for ≤ j < m and r m := r ≤ β · r m − . Then C BG ( n, − , r j +1 , r j ) < ε − for all ≤ j < m . Moreover, x ∈ X r j for all ≤ j ≤ m due to r j ≤ R ≤ r x .Assume G r j ( p xi ) ∩ G r j +1 ( p xi ) = ∅ for some ≤ j < m and i ∈ N . Thisimplies G r j +1 ( p xi ) ⊆ B r j ( p xi ) \ G r j ( p xi ) , in particular, (1 − ˆ ε ) · vol( B r j ( p xi )) ≤ vol( G r j +1 ( p xi )) ≤ vol( B r j ( p xi ) \ G r j ( p xi )) ≤ ˆ ε · vol( B r j ( p xi )) ≤ ˆ ε · C BG ( n, − , r j +1 , r j ) · vol( B r j ( p xi )) . Hence, − ˆ ε ≤ ˆ ε · C BG ( n, − , r j +1 , r j ) < − ˆ ε , and this is a contradiction.Thus, G r j ( p xi ) ∩ G r j +1 ( p xi ) = ∅ for all ≤ j < m and i ∈ N . By a), ≤ λ xi ( r j ) λ xi ( r j +1 ) ≤ for almost all i and l xω ( r j ) = l xω ( r j +1 ) . Inductively, l xω ( r ) = l xω ( r m ) = l xω ( r ) = l xω ( R ) . Then λ xi ( R ) λ xi ( r ) = λ xi ( r ) λ xi ( r m ) = m − Y j =0 λ xi ( r j ) λ xi ( r j +1 ) proves the claim.c) Let d = d ( n, ˆ ε, r ) be as in Lemma 2.19 and m = m ( n, ˆ ε, l, r ) ∈ N bethe minimal natural number with l ≤ m · d .Define a sequence t < t < . . . < t m = l with pairwise t j +1 − t j ≤ d by t j := j · d for ≤ j ≤ m − and t m := l . For ≤ j ≤ m , define y j := γ ( t j ) ∈ X r . Now fix ≤ j < m . By Lemma 2.19 and Lemma 2.17, G r ( p y j i ) ∩ G r ( p y j +1 i ) = ∅ . The rest of the proof can be done analogously to the one of b).d) Let m x := m y := m ( n, ˆ ε, r ′ , r ) be as in b) and m = m ( n, ˆ ε, l, r ′ ) as inc). Then m = m ( n, ˆ ε, l, r, r ′ ) := m x · m · m y proves the claim. (cid:3) XISTENCE OF TYPICAL SCALES 45
Generic points and geodesics.
Throughout this subsection, fix acollapsing sequence ( M i , p i ) i ∈ N of pointed complete connected n -dimension-al Riemannian manifolds which satisfy the uniform lower Ricci curvaturebound Ric M i ≥ − ( n − and converge to a limit ( X, p ) of dimension k < n and use the notation introduced in subsection 3.1. Moreover, minimisinggeodesics are assumed to be parametrised by arc length.By Lemma 3.6, rescaling sequences corresponding to two different pointscan be compared if those points are connected by a geodesic lying in some X r . It remains to check for which points this is the case. It will turn outthat, if the strict interior of a minimising geodesic (i.e. the interior boundedaway from the endpoints) is generic, then it is already contained in X r forsufficiently small r > . In fact, nearly all pairs of points lie in the interiorof such a geodesic such that the part of the geodesic connecting these pointsis generic. Notation.
Define G := { ( x, y ) ∈ X gen × X gen | ∃ minim. geod. γ : [0 , l ] → X, < t x < t y < l ::= { ( x, y ) ∈ X gen × X gen |} x = γ ( t x ) , y = γ ( t y ) , im( γ | [ t x ,t y ] ) ⊆ X gen } , and for x ∈ X gen denote the image under the projection to the second factorby G x := { y ∈ X gen | ( x, y ) ∈ G} . Finally, define G ′ := { x ∈ X gen | G x has full measure in X } . Lemma 3.7.
The set G ′ has full measure in X .Proof. First, prove that G has full measure in X × X . Let S := { ( x, y ) ∈ X gen × X gen | ∃ minim. geodesic c : [0 , d ] → X ::= { ( x, y ) ∈ X gen × X gen |} x = c (0) , y = c ( d ) , im( c ) ⊆ X gen } S := { ( x, y ) ∈ X × X | ∃ minim. geodesic γ : [0 , l ] → X, < t x < t y < l ::= { ( x, y ) ∈ X × X |} x = γ ( t x ) , y = γ ( t y ) } and define S := S ∩ S . By [CN12, Theorem 1.20 (1), Theorem A.4 (3)], vol X × vol X ( X × X \ S ) = 0 and vol X × vol X ( X gen × X gen \ S ) = 0 . In particular, using that vol X ( X \ X gen ) = 0 , this proves vol X × vol X ( X × X \ S ) = 0 . Next, prove S ⊆ G : Let ( x, y ) ∈ S , c : [0 , d ] → X gen and γ : [0 , l ] → X begeodesics and < t x < t y < l with x = c (0) = γ ( t x ) and y = c ( d ) = γ ( t y ) .In particular, d = d ( x, y ) = t y − t x ≤ l . γ ( t x ) = c (0) = x y = c ( d ) = γ ( t y )˜ γ ( τ ) = γ ( τ ) γ ( τ )˜ γ ( τ ) = c ( τ − t x ) Figure 5.
Construction of ˜ γ .Define ˜ γ : [0 , l ] → X by ˜ γ ( τ ) := ( γ ( τ ) if τ ∈ [0 , t x ] ∪ [ t y , l ] ,c ( τ − t x ) if τ ∈ [ t x , t y ] , cf. Figure 5. A straightforward computation proves that ˜ γ is a minimisinggeodesic. Then ˜ γ verifies ( x, y ) ∈ G , and this proves vol X × vol X ( X × X \ G ) = 0 . Using X × X \ G = S x ∈ X { x } × ( X \ G x ) , X × X ( X × X \ G )= Z X vol X ( X \ G x ) dV ( x )= Z X \G ′ vol X ( X \ G x ) dV ( x ) . Since vol X ( X \ G x ) > for all x ∈ X \ G ′ , this proves vol X ( X \ G ′ ) = 0 . (cid:3) So far it was seen that almost all points can be connected by a geodesiclying in X gen which can be extended at both ends. By applying the followingtheorem of Colding and Naber, which describes the Hölder continuity of thegeometry of small balls with the same radius, to this situation, one obtainsthat the interior of the regarded geodesics not only lies in X gen , but in X r for some r > . Theorem 3.8 ([CN12, Theorem 1.1, Theorem 1.2]) . For n ∈ N there are α ( n ) , C ( n ) and r ( n ) such that the following holds: Let M be a complete n -dimensional Riemannian manifold with Ric M ≥ − ( n − or the limit spaceof a sequence of such manifolds, let γ : [0 , l ] → M a minimising geodesic(parametrised by arc-length) and fix β ∈ (0 , . For < r < r βl and βl < s < t < (1 − β ) l , d GH ( B Mr ( γ ( s )) , B Mr ( γ ( t ))) < Cβl · r · | s − t | α ( n ) . Lemma 3.9.
Let ˆ ε ∈ (0 , ) , γ : [0 , l ] → X be a minimising geodesic andassume < s < t < l to be times such that γ | [ s,t ] is contained in X gen . Thenthere is < r ′ = r ′ (ˆ ε, l, s, t ; n, k ) ≤ ˆ δ such that for all < r ≤ r ′ , im( γ | [ s,t ] ) ⊆ X r . XISTENCE OF TYPICAL SCALES 47
Proof.
Define β = β ( l, s, t ) := l · min { s, l − t } > . Then t, s ∈ ( βl, (1 − β ) l ) .Furthermore, let α ( n ) , C ( n ) , r ( n ) be as in Theorem 3.8 and define d = d (ˆ ε, l, s, t ; n, k ) := α ( n ) s βl · ˆ δ C ( n ) . Let m = m ( s, t ) be the maximal natural number with ( m − d ≤ t − s anddefine τ j := s + jd for ≤ j ≤ m . By definition, τ = s and τ m = s + md > t .Therefore, [ s, t ] ⊆ S mj =0 ( τ j − d, τ j + d ) .For every ≤ j ≤ m , choose λ j = λ j (ˆ ε, l, s, t ; n, k ) > as in Lemma 3.2and define r ′ = r ′ (ˆ ε, l, s, t ; n, k ) := min { ˆ δ, λ , . . . , λ m , ˆ δ · r ( n ) · βl } . Let < r ≤ r ′ and τ ∈ [ s, t ] be arbitrary. Choose ≤ j ≤ m with | τ − τ j | < d .By definition of d , | τ − τ j | α ( n ) < d α ( n ) = βl · ˆ δ C ( n ) , and so, using Theorem 3.8, d GH ( B r − X / ˆ δ ( γ ( τ )) , B R k / ˆ δ (0)) ≤ r · d GH ( B Xr/ ˆ δ ( γ ( τ )) , B Xr/ ˆ δ ( γ ( τ j ))) + d GH ( B r − X / ˆ δ ( γ ( τ j )) , B R k / ˆ δ (0)) ≤ r · C ( n ) βl · r ˆ δ · | τ j − τ | α ( n ) + ˆ δ < ˆ δ. (cid:3) Proof of the main theorem.
In order to prove Theorem 3.1, thefollowing technical result is needed which estimates the number of balls apoint can be contained in if the base points of these balls form an ε -net. Lemma 3.10.
Let X be an n -dimensional Riemannian manifold with lowerRicci curvature bound Ric ≥ ( n − · κ or the limit of a sequence of suchmanifolds. Then each point is contained in maximal C BG ( n, κ, r, r +2 R ) ballswith radii R whose base points have pairwise distance at least r .Proof. This result is an immediate consequence of the Bishop-Gromov The-orem: Let p , . . . , p m ∈ X be points with pairwise distance at least r and q ∈ T mi =1 B R ( p i ) .On the one hand, since d ( p i , p j ) ≥ r , one has B r ( p i ) ∩ B r ( p j ) = ∅ for any i = j . On the other hand, B r ( p i ) ⊆ B r + R ( q ) , hence, ` mi =1 B r ( p i ) ⊆ B r + R ( q ) .Furthermore, B r + R ( q ) ⊆ B r +2 R ( p i ) for any ≤ i ≤ m . Together, ≥ vol( ` mi =1 B r ( p i ))vol( B r + R ( q )) = m X i =1 vol( B r ( p i ))vol( B r + R ( q )) ≥ m X i =1 vol( B r ( p i ))vol( B r +2 R ( p i )) ≥ m C BG ( n, κ, r, r + 2 R ) . Thus, m ≤ C BG ( n, κ, r, r + 2 R ) . (cid:3) It remains to prove the main theorem. Again, the notation introduced insubsection 3.1 and subsection 3.3 is used.
Proof of Theorem 3.1.
The idea of the proof is the following: First, fix abound ˆ ε ∈ (0 , ) and choose a radius R such that X R (ˆ ε ; n, k ) has sufficientlylarge volume. Inside of this set of points, choose a point x and a finite R -net of points x j such that ( x , x j ) ∈ G , and take the union of the subsets G R ( p x j i ) . This has the required properties.Let ε ∈ (0 , be arbitrary and define ˆ ε = ˆ ε ( n, ε ) := ε · C BG (cid:0) n, − , , (cid:1) ∈ (cid:16) , (cid:17) . For arbitrary r > , define X ′ ( r ) := { x ∈ B − r ( p ) ∩ X gen | r x ≥ r } . For r ≤ r , obviously X ′ ( r ) ⊆ X ′ ( r ) . Further, S r> X ′ ( r ) = B ( p ) ∩ X gen .Thus, there exists a radius < R = R ( ε, X, p ; n ) ≤ such that vol X ( X ′ ( r )) ≥ (cid:16) − ε (cid:17) · vol X ( B ( p ) ∩ X gen ) = 1 − ε for all r ≤ R . Fix this < R ≤ .Since G ′ has full measure by Lemma 3.7, X ′ ( R ) ∩ G ′ is non-empty. Fix anarbitrary point x ∈ X ′ ( R ) ∩ G ′ , define X ′ := X ′ ( R ) ∩ G x and choose a maximal number of points x , . . . , x l ∈ X ′ with pairwise dis-tance at least R . By the maximality of the choice, X ′ ⊆ S lj =1 B R ( x j ) . Since G x has full measure by choice of x , vol X (cid:0) l [ j =1 B R ( x j ) (cid:1) ≥ vol X ( X ′ ) = vol X ( X ′ ( R )) ≥ − ε . On the other hand, by choice, B R ( x j ) ⊆ B ( p ) . Thus, vol X ( B ( p ) \ l [ j =1 B R ( x j )) ≤ ε . Let p x j i → x j and i ∈ N be large enough such that for all i ≥ i and all ≤ j < j ′ ≤ l , d ( p x j i , p x j ′ i ) ≥ · d ( x j , x j ′ ) ≥ R and vol M i ( B ( p i ) \ S lj =1 B R ( p x j i ))vol M i ( B ( p i )) ≤ · vol X ( B ( p ) \ S lj =1 B R ( x j ))vol X ( B ( p )) . XISTENCE OF TYPICAL SCALES 49
Fix i ≥ i . Then vol M i ( B ( p i ) \ l [ j =1 B R ( p x j i )) ≤ · vol X ( B ( p ) \ S lj =1 B R ( x j ))vol X ( B ( p )) · vol M i ( B ( p i )) ≤ ε · vol M i ( B ( p i )) . By Lemma 3.10, every point in the union S lj =1 B R ( p x j i ) is contained in atmost M different balls B R ( p x j i ) for M = M ( ε ; n, k ) := C BG (cid:16) n, − , R , R (cid:17) ≤ C BG (cid:16) n, − , , (cid:17) = ε ε . Therefore, l X j =1 vol M i ( B R ( p x j i )) ≤ M · vol M i (cid:0) l [ j =1 B R ( p x j i ) (cid:1) ≤ ε ε · vol M i ( B ( p i )) . Thus, vol M i (cid:0) l [ j =1 B R ( p x j i ) \ l [ j =1 G R ( p x j i ) (cid:1) ≤ vol M i (cid:0) l [ j =1 ( B R ( p x j i ) \ G R ( p x j i )) (cid:1) ≤ l X j =1 vol M i ( B R ( p x j i ) \ G R ( p x j i )) ≤ l X j =1 ˆ ε · vol M i ( B R ( p x j i )) ≤ ε · vol M i ( B ( p i )) . Hence, vol M i ( B ( p i ) \ l [ j =1 G R ( p x j i )) ≤ ε · vol M i ( B ( p i )) . Now define G ( p i ) := l [ j =1 G R ( p x j i ) and λ i := λ x i ( R ) . By construction, vol M i ( G ( p i )) ≥ (1 − ε ) · vol M i ( B ( p i )) . From now on, let λ x j i denote λ x j i ( R ) .Fix ≤ j ≤ l . By construction, ( x , x j ) ∈ G and x , x j ∈ X R . Thus, thereexists a minimising geodesic γ j : [0 , l j ] → X and < s j < t j < l j such that γ j | [ s j ,t j ] is contained in X gen , γ j ( s j ) = x and γ j ( t j ) = x j . By Lemma 3.9,there is r ′ j > such that for all < r ≤ r ′ j , γ j | [ s j ,t j ] is contained in X r . Let r j := min { r ′ j , R } . By Lemma 3.6 d), there is m j := m ( n, ˆ ε, d X ( x , x j ) , r j , R ) satisfying − m j ≤ λ x i λ x j i ≤ m j for almost all i ∈ N and l x ω ( R ) = l x j ω ( R ) . From now on, let i ≥ i be largeenough such that the above estimate holds for all ≤ j ≤ l .Given q i ∈ G ( p i ) , let ( Y, q ) be an arbitrary sublimit of ( λ i M i , q i ) , i.e. fora subsequence ( i s ) s ∈ N , ( λ i s M i s , q i s ) → ( Y, q ) as s → ∞ . For a further subsequence ( i s t ) t ∈ N there is ≤ j ≤ l with q i st ∈ G R ( p x j i st ) for all t and ( λ x j i st M i st , q i st ) → ( R k × ˜ K, · ) as t → ∞ for a compact metric space ˜ K satisfying diam( ˜ K ) ∈ [ , .On the other hand, (cid:18) λ i st λ x j i st · λ x j i st M i st , q i st (cid:19) = (cid:0) λ i st M i st , q i st (cid:1) → ( Y, q ) as t → ∞ , the sequence λ ist λ xjist converges to some α and Y is isometric to the product R k × K for K := α ˜ K . In particular, − m j ≤ α ≤ m j . So, diam( K ) ∈ [ D , D ] for D := 5 max { m j | ≤ j ≤ l } +1 . Moreover, for any non-principal ultrafilter ω , dim( K ) = dim( ˜ K ) = l x j ω ( R ) = l x ω ( R ) . In particular, for any two sublimits ( R k × K , · ) and ( R k × K , · ) comingfrom the same subsequence of indices, let ω be a non-principal ultrafiltersuch that these sublimits are ultralimits with respect to ω . Then dim( K ) = dim( K ) . (cid:3) References [And92] Michael T. Anderson,
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