Intrinsic Flat and Gromov-Hausdorff Convergence of Manifolds with Ricci Curvature Bounded Below

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature bounded below and diameter bounded above, Gromov-Hausdorff convergence agrees with intrinsic flat convergence. In particular, the limiting current is essentially unique, has multiplicity one, and mass equal to the Hausdorff measure. Moreover, the limit spaces satisfy a constancy theorem.

From the fact that the mass measure T equals the Hausdorff measure H n , it follows that it is the limit (in a weak sense) of the Riemannian volume measures of the manifolds in the approximating sequence. Indeed, Cheeger and Colding [3,Theorem 5.9] have shown that under Gromov-Hausdorff convergence, the Riemannian volume measures on the manifolds converge to the Hausdorff measure on the limit space.
We believe that the idea of the original proof by Sormani and Wenger in [12] can be used to extend their result to the case of Ricci curvature bounded below by an arbitrary constant. The important ingredients in their proof, such as the application of a volume estimate by Colding [6,Corollary 2.19] and Perelman's Main Lemma in [9], are applicable to manifolds of almost nonnegative curvature, and therefore they can be applied after scaling. To our knowledge, this was so far unknown. Our proof will differ from the one by Sormani and Wenger and does not use Perelman's Main Lemma.
Our techniques also allow to prove the uniqueness of the limiting current and the following local constancy theorem. Theorem B Let (X, d X , T ) ∈ M I F (n, , v, D). For all q ∈ X , and every integral current S on X such that ∂ S (B t (q)) = 0 for some t > 0, there exists an integer k ∈ Z such that S = kT on B t (q). Alternatively, Theorem B may be interpreted as stating that the local top-dimensional homology of the space is isomorphic to Z.
The structure of the manuscript is as follows. Section 1 gives a coarse background on integral currents in the sense of Ambrosio and Kirchheim [1], integral current spaces and intrinsic flat converge as introduced by Sormani and Wenger [13] and some of the elements we need from the theory on the structure of spaces with Ricci curvature bounded below by Cheeger and Colding [3,4]. In our notation, we generally try to stick to the notation in these articles.
Theorems A and B are direct consequences of Theorem 4.1 in the text. A crucial ingredient is a link between zero-dimensional slices and the degree, which we will explain in Sect. 2. Earlier work by Sormani and the second author [10] showed that integrals of the flat distance between lower-dimensional slices of currents can be controlled by the flat distances between the original currents. This will imply that the L 1 -distance between their degrees can be controlled locally. In Sect. 3, we show that Colding's volume estimate easily translates into an estimate on the degree for manifolds M ∈ M(n, , v, D). In Sect. 4 we combine these two ingredients to give a proof of the main theorem.

Background
In this section, we review integral currents on metric spaces as introduced by Ambrosio and Kirchheim [1], the intrinsic flat distance introduced by Sormani and Wenger [13] and some theory on the structure of spaces with Ricci curvature bounded below by Cheeger and Colding [3,4]. The prime purpose of this review is to fix notation. We adhere closely to the notation used in these articles, and therefore the reader familiar with these works could probably understand the rest of the manuscript without reading this section.

Currents
Let X be a complete metric space. For n ≥ 1, we define the set D n (X ) of all (n + 1)tuples ( f, π 1 , . . . , π n ) of Lipschitz functions on X , where additionally f is required to be bounded. For n = 0, we define D 0 (X ) as the set of bounded Lipschitz functions. It can be helpful to think of an element ( f, π 1 , . . . , π n ) ∈ D n+1 as an n-form f dπ 1 ∧ · · · ∧ dπ n .
An n-dimensional metric functional is a function T : D n (X ) → R such that the map is subadditive and positively 1-homogeneous with respect to the functions f and π 1 , . . . , π n . We denote the vector space of n-dimensional metric functionals on X by M F n (X ).
For n ≥ 1, the boundary of T ∈ M F n (X ) is the (n − 1)-dimensional metric functional denoted by ∂ T defined by If Y is another complete metric space and : X → Y is Lipschitz, we define the pullback operator that maps D n (Y ) to D n (X ) by We define the pushforward # T ∈ M F n (Y ) of T ∈ M F n (X ) by For T ∈ M F n (X ) and ω = (g, We say that T ∈ M F n (X ) has finite mass if there exists a finite Borel measure μ on X such that for all ( f, π 1 , . . . , π n ) ∈ D n (X ), where Lip(π i ) denotes the Lipschitz constant of π i . Moreover, the minimal measure satisfying this bound is called the mass of T and is denoted by T . When T has finite mass, it can be uniquely extended to a function on (n + 1)-tuples ( f, π 1 , . . . , π n ) for which f is merely bounded Borel, and π 1 , . . . , π n are Lipschitz.
An n-dimensional current T is an n-dimensional metric functional with additional properties. From the definition by Ambrosio and Kirchheim [1, Definition 3.1], immediately stronger properties may be derived. We choose to only phrase the stronger properties. The space of n-dimensional currents forms a Banach space, with respect to the mass norm M(T ) = T (X ). We denote the Banach space by M n (X ). Every T ∈ M n (X ) satisfies (i) T is multilinear in ( f, π 1 , . . . , π n ), and whenever f and π 1 are both bounded and Lipschitz, it holds that and where ψ = (ψ 1 , . . . , ψ n ) ∈ [C 1 (R n )] n and ∇ψ is bounded; (ii) The following continuity property is satisfied whenever f i − f → 0 in L 1 (X, T ) and π i j → π j pointwise in X with uniformly bounded Lipschitz constant Lip(π i j ) ≤ C; (iii) The following locality property holds: where B i are Borel and π i is constant on B i .
We say that a sequence of currents T i ∈ M n (X ) converges weakly to T ∈ M n (X ) if for all ω ∈ D n (X ), The mass of open sets is lower-semicontinuous under weak convergence, that is for O ⊂ X open, and T i converging weakly to T , A very important example of an n-dimensional current on the Euclidean space R n is given by the current induced by a function g ∈ L 1 (R n ) which we denote by g and is defined by g ( f, π 1 , . . . , π n ) := R n g f dπ 1 ∧ · · · ∧ dπ n = R n g f det(∇π) dx. (10) We say that a current T ∈ M n (X ) is normal if ∂ T ∈ M n−1 (X ).
A subset S ⊂ X is called countably H n -rectifiable if there are compact sets K i ⊂ R n and Lipschitz functions f i : We say T ∈ M n (X ) is rectifiable if T is concentrated on a countably H n -rectifiable set and vanishes on H n -negligible Borel sets. We call T integer rectifiable if for all φ ∈ Lip(X, R n ) and all open O ⊂ X it holds that φ # (T O) = θ for some θ ∈ L 1 (R n , Z). Finally, the collection of integral currents will consist of all integer rectifiable currents that are also normal. We will denote this collection by I n (X ) and in this manuscript, we will only deal with integral currents. We denote by ω n the (Lebesgue) volume of the unit ball in R n . For a Borel measure μ on X , we define respectively the n-dimensional lower and upper density of μ in If these values coincide, we call the common value the n-dimensional density of μ in x and we denote it by n (μ, x). We define set(T ) ⊂ X as For an integer rectifiable current T , the mass T is always concentrated on set(T ) and set(T ) is rectifiable. Integer rectifiable currents allow for a parametric representation. It is a simple consequence of Lusin's theorem and [1, Theorem 4.5] that if T is an n-dimensional integer rectifiable current, there exist a sequence of compact sets K i , numbers θ i ∈ N and bi-Lipschitz functions f i : An n-dimensional oriented Riemannian manifold M naturally induces a current (on its geodesic metric space), that we will also denote by M , given by integration of ω ∈ D n over M, where τ is a (unit) orienting n-vector field. In this case, the mass of M equals the Riemannian volume. The intrinsic representation of rectifiable currents by Ambrosio and Kirchheim [1, Theorem 9.1] shows that at least in some sense, this formula holds for any integer rectifiable current. More precisely, if Z is a w * -separable dual space (i.e. Z = G * for a separable Banach space G), and T is an integer rectifiable current on Z , then there exists a countably H n -rectifiable set Y ⊂ Z , a Borel function θ T : Y → N (which we call the multiplicity of T ) with Y θ T dH n < ∞ and an orientation τ of Y such that for ( f, π 1 , . . . , π n ) ∈ D n (Z ). We sometimes write T = Y, θ T , τ . The multiplicity θ T corresponds to the θ i in the parametric representation (14), in the sense that for Moreover, the mass of T satisfies for a Borel function λ : Y → [c(n), C(n)] (the area factor) that is bounded away from zero and infinity by constants that only depend on the dimension. It is true that we have not defined the objects appearing in (16). For a precise definition and formulation see [1]. For the purpose of the paper we just would like to stress the analogy with the formula for a current induced by a Riemannian manifold.
Additionally, the representation formula makes clear that if there is another integral current S supported on a subset of Y , it holds that while on the other hand for every Borel function g : A very useful technique in dealing with currents on metric spaces is called slicing. For the purpose of this manuscript, we only need zero-dimensional slices. For T ∈ I n (X ), a Lipschitz map : X → R n and points x ∈ R n , the slices T, , x ∈ I 0 (X ) are characterized by the property from which it follows that for any bounded Borel function f on X , By [1,Theorem 9.7], if X is in addition a w * -separable dual space, and T = Y, θ T , τ , for a rectifiable set Y ⊂ X , the slices are very easy to interpret. Indeed, for L n -a.e.
x ∈ R n , −1 (x) ∩ Y contains at most finitely many points and for some choice of a p ∈ {−1, 1}. Finally, for instance by [1,Theorem 5.7] Finally, we note that a separable space Y can always be isometrically embedded (24)

Integral Current Spaces and Intrinsic Flat Convergence
Let S, T ∈ I n (X ). The flat distance between S and T in X is defined as As briefly mentioned in the introduction, an n-dimensional integral current space (X, d X , T ) is a pair of a metric space (X, d X ), which is not necessarily complete, and a current T ∈ I n (X ) on the completion of X . Additionally, by convention, it is assumed that the current is completely settled, that is X = set(T ).
The intrinsic flat distance between two integral current spaces (X, d X , T ) and )) = 0, this implies that there exists a current-preserving isometry φ : X → Y , that is an isometry such that φ # T = S. Note that without the convention X = set(T ) (or a similar condition), this would certainly not be the case in general. We will denote the metric space of (equivalence classes of) n-dimensional integral current spaces with the intrinsic flat distance by M I F n . There is an involution ι acting on M I F n , given by We may endow the quotient space M I F n /ι by the quotient distance: In general, such a definition only yields a pseudometric. However, in this special case, the quotient distance is indeed a distance, and is in fact given by We denote the metric space M I F n /ι endowed with this quotient metric by M I F/ι n .

The Structure of Spaces with Ricci Curvature Bounded Below
In [3][4][5], Cheeger and Colding study the structure of metric spaces that arise as the Gromov-Hausdorff limits of manifolds with Ricci curvature uniformly bounded from below. Cheeger and Colding consider both noncollapsed and collapsed limit spaces. From the point of view of intrinsic flat convergence, the collapsed case is trivial as in that case the approximating sequence of Riemannian manifolds converges in the intrinsic flat distance to the zero integral current space. We therefore consider only noncollapsed limit spaces, for which the results by Cheeger and Colding are much stronger.
We borrow the following definitions. If X ∈ M G H (n, , v, D), and p ∈ X , we say p ∈ R ,δ if and only if d G H (B r ( p), B r (0)) < r, for all r < δ.
We further define the regular set R by In words, p ∈ X is regular if for all > 0 there exists a δ > 0 such that B r ( p) is -close to a Euclidean ball at every scale smaller than δ.
Cheeger and Colding show that in the noncollapsed case, the Hausdorff codimension of the complement X \R is at least 2. They use this to prove in [4, Corollary 3.9 and 3.10] a (local) connectedness result. More precisely, for all q 1 , q 2 ∈ R, and every , σ > 0, there is a δ > 0 such that there is a path in R ,δ of length smaller than d(q 1 , q 2 ) + σ connecting q 1 and q 2 . It follows immediately that when q ∈ X , q 1 , q 2 ∈ B t (q) ∩ R, for every > 0 there is a path in R ,δ connecting q 1 and q 2 that remains inside B t (q). Indeed, there is a σ > 0 such that q 1 , q 2 ∈ B t−3σ (q). Next, there exists a q 3 ∈ R with d(q 3 , q) < σ by the Bishop-Gromov estimate and the fact that R has full measure. By the above, there exists a δ > 0 such that there are paths in R ,δ from q 1 to q 3 and from q 3 to q 2 respectively, both of length smaller than t − σ , as for k = 1, 2, These paths are contained in B t (q) and by concatenating them we obtain a path from q 1 to q 2 in B t (q) ∩ R ,δ .

Degree Estimate
For an integral current T ∈ I n (X ) on a complete metric space X and a Lipschitz : X → R n the pushforward # T ∈ I n (R n ) is represented by a unique BV function, by a representation theorem due to Ambrosio and Kirchheim [1, Theorem 3.7]. The theorem also ensures that the mass measure of the boundary of # T equals the total variation of the distributional derivative of the representing function. We call such a function the degree of with respect to T .

Definition 2.1
Let X be a complete metric space and let T ∈ I n (X ). Let : X → R n be a Lipschitz map. We define the degree of with respect to T as the (unique) function deg(T, , .) ∈ BV(R n ), taking values in Z, that satisfies If T is a current induced by an oriented Riemannian manifold, then the degree defined above indeed corresponds to the usual topological degree. By This observation is very useful in light of the estimates by Sormani and the second author that show that integrals of flat distances between slices of two currents can be controlled by the flat distance between the full currents. This translates to an L 1 estimate on the difference between the degrees.

Lemma 2.2
Let Z be a complete metric space and let T k ∈ I n (Z ), k = 1, 2. Let k : Z → R n be such that every component k j is 1-Lipschitz. Then Proof We use the estimate by Sormani and the second author [10, Proposition 4.17] From this inequality, we conclude the estimate (34) by using the link between slices and the degree explained in (33). Indeed, for L n -a.e. x ∈ R n ,

A Consequence of Colding's Volume Estimate
Throughout this section we use the notation B r (0) to denote the open ball of radius r centered at 0 in the Euclidean space R n . Let X be a metric space, x ∈ X , and a radius R > 0, such that Let {x 1 , . . . , x n } ⊂ X and consider the map x,x 1 ,...,x n given by We call x,x 1 ,...,x n an ( , R)-chart around x if there exist a metric space Z and isometric embeddings φ : and where {e i } is the standard orthonormal frame in R n , and d Z H denotes the Hausdorff distance in Z .
Note that every coordinate function of x,x 1 ,...,x n is Lipschitz with Lipschitz constant 1.
In [6,Sect. 2], Colding shows the following result, that we phrase as a lemma. and From the above lemma, we can derive that on a large set, the ( , R)-chart is locally one-to-one.
Proof We choose the constants , R and as in Lemma 3.1, so that Since every component of the map is 1-Lipschitz, the Jacobian J of is bounded, |J | ≤ 1. We apply the coarea formula and find Therefore, Proof We may without loss of generality assume that η n ≤ ω n /4 =:η n .
Note that is a σ -Gromov-Hausdorff approximation, for R large enough and small enough, only depending on n.

Proof of the Main Theorems
Theorems A and B in the introduction are implied by the following theorem. (i) a subsequence of the associated integral current spaces M i converges in the flat distance to (X, d X , T ), with set(T ) = X, (ii) the mass T equals H n , the Hausdorff measure on X , (iii) T has multiplicity one H n -a.e., (iv) for every q ∈ X , every t > 0, and every S ∈ I n (X ) with ∂ S (B t (q)) = 0, there exists an integer k ∈ Z such that S = kT on B t (q).
Before we prove Theorem 4.1, we explain how it implies the theorems in the introduction. Certainly, Theorem B is a direct consequence of part (iv) in the above theorem.
The map F : considered in Theorem A is well-defined as Sormani and Wenger have shown that if the intrinsic flat distance between two integral current spaces is zero, there is a (current-preserving) isometry between the spaces [13]. Therefore, the map does not depend on the choice of representative. Moreover, by item (i) of Theorem 4.1, X is indeed a compact metric space. The map F is one-to-one by (iv). A one-to-one continuous map from a compact to a Hausdorff space is automatically a homeomorphism. So the only non-trivial part of Theorem A left to show is the continuity of F.
If (X i , d X i , T i ) ∈ M I F/ι (n, , v, D) converge in the intrinsic flat distance to an integral current space (X, d X , T ), there are Riemannian manifolds M i ∈ M I F/ι (n, , v, D) such that the associated integral current spaces [M i ] converge in the intrinsic flat distance to (X, d X , T ) while and thus Y is isometric to X . We will now prove Theorem 4.1.
Proof By the Ambrosio-Kirchheim compactness theorem [1, Theorem 5.2], a subsequence of the associated currents T i converges in the weak sense to an integral current T without boundary. A theorem by Wenger [14,Theorem 1.4] implies that the T i converge to T in the flat distance in Z as well. Let p ∈ X be a regular point, that is p ∈ R ⊂ X . (The definition of the sets R and R ,δ are as in [4], see also Sect. 1.3). Let η > 0. Let (n), (η, n) and R(η, n) be as in Lemma 3.3. Since p ∈ R, there exists a number 0 < δ < √ /K such that p ∈ R /R,δ R . In other words, for all r < δ, We will now show that for a subsequence, we may localize the flat convergence to balls. Since M i → X in the Hausdorff distance in Z , there exists a sequence p i ∈ M i such that p i → p. By the proof of [11,Lemma 4.1], see also [10,Theorem 4.16], for yet another subsequence, for L 1 -a.e. radius 0 < r < δ, the currents restricted to balls of radius r converge, that is We choose p 1 , . . . , p n ∈ X such that is an ( r, r R)-chart around p.
In order to prove (ii), we will lift this ( , R)-chart to charts on the manifolds in the approximating sequence. By the results in the previous section, we have good control of the degree of these charts, and Lemma 2.2 allows us to pass this control to the limit. Estimates on the degree will immediately imply density estimates.
We argue as follows. Since In particular, ≥ ω n r n − η n r n .
Moreover, by lower-semicontinuity of the mass measure under weak convergence, ≤ ω n r n + η n r n /3.
In particular, p ∈ set(T ), and since η > 0 was arbitrary, the density n ( T , p) of T in p is 1. Since H n (X \R) = 0, in fact H n = T , which shows (ii).
By the results by Cheeger and Colding however, we know that H n satisfies the Bishop-Gromov estimate, so that in particular, at every point in X the density of H n (= T ) is (strictly) positive. Consequently, set(T ) = X , which finishes the proof of (i).
Our next objective is to show (iii), namely that T has multiplicity one. To this end, we use that the measure of the set {| deg (T B r ( p), , .)| = 1} is very close to T (B r ( p)). This can only happen if most points in the image of have exactly one pre-image, where the multiplicity is one.
For L n -a.e. x ∈ G, there is a unique p x ∈ set(T )∩ B r ( p) such that ( p x ) = x and the multiplicity θ T ( p x ) = 1. Therefore, again using that T (B r ( p)) ≤ ω n r n + η n r n /3, we find This shows that in every p ∈ R, the density of the set where the multiplicity of T is larger than 1 equals zero. Since H n (X \R) = 0, θ T ≡ 1. This shows (iii). Finally, we need to show (iv). Let therefore q ∈ X and S ∈ I n (X ) such that ∂ S (B t (q)) = 0, for some t > 0. Let p ∈ R ∩ B t (q) and r > 0 be as above, with the additional assumption that r < d( p, X \B t (q)). The representation theorem for integer rectifiable currents [1, Theorem 9.1] implies that there is a Borel function S/ T : X → Z such that We will prove (iv) by showing that on a large set S/ T is equal to the ratio of two degrees, which in turn are constant. The next few lines will be devoted to the choice of a good representative for S/ T . We first introduce an "average" version, based on a majority vote: where, if the maximum is not unique, we give preference to the smallest absolute value of k, and if this gives no decision, the positive value. However, this is quite irrelevant. By the Lebesgue differentiation theorem, for H n -a.e. q 1 ∈ X , For convenience, we will from now on only work with the precise representative of S/ T which we define to be the right-hand-side of (63) if this limit exists, and 0 otherwise.
By the characterization of slices [1, Theorems 5.7 and 9.7], for L n -a.e. x ∈ G, the slice T, , x is just a signed delta-measure, while We apply both sides to the function identically equal to 1, and conclude that If we inspect the proof of Lemma 3.3, we see that the constants , R, and were chosen such that is a σ r -Gromov-Hausdorff approximation. Hence, both deg(T B r ( p), , .) and deg(S B r ( p), , .) are constant on the ball B (1−σ )r (0) and we can find a compact subsetG ⊂ G, L n (G) > (ω n − 3η n )r n such that every x ∈G has exactly one pre-image p x under in set(T ) ∩ B r ( p), and for this p x It follows that ≤ ω n r n + η n r n /3 − ω n r n + 3η n ≤ 4η n r n .
Therefore, for η small enough, only depending on the dimension, the average ( S/ T ) p,r is jointly continuous in ( p, r ) on the domain p ∈ R /R,δ R ∩ B t (q), 0 < r < min(δ, d X ( p, X \B t (q))). In particular, S/ T is continuous on R /R,δ R ∩ B t (q). However, by a result by Cheeger and Colding [4, Corollary 3.9 and 3.10] (see also Sect. 1.3), for all q 1 , q 2 ∈ B t (q) ∩ X , there is a δ such that q 1 and q 2 lie in the same component of R /R,δ R ∩ B t (q). Therefore, S/ T is constant on B t ( p). This shows (iv).

Remark 4.2
To obtain the necessary control on the degree, we use that an ( , R)chart is a Gromov-Hausdorff approximation. It is possible to conclude such control under weaker assumptions. A lot of information can be extracted from the existence of a map : X → R n on the limit space X , for which every component is 1-Lipschitz, and such that the excess T (B r ( p)) − T (χ B r ( p) d ) is small. We have decided to not present this argument in the manuscript, as it is considerably longer, and at this point, it is unclear whether there are applications in which the easier argument by a Gromov-Hausdorff approximation cannot be applied.