# Collapsed Manifolds With Ricci Bounded Covering Geometry

We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows’ techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci bounded covering geometry.

## 0. Introduction

A complete manifold is called -collapsed, if the volume of any unit ball on is less than ; one may normalize with a bound on curvature. Collapsed manifolds with bounded sectional curvature, , has been extensively studied by Cheeger-Fukaya-Gromov ([CFG], [CG1, 2], [Fu1-3], [Gr]), and the basic discovery is a structure on , called a nilpotent structure, consisting of compatible local nilpotent almost symmetric structures whose orbits point to all collapsed directions of the underlying metric i.e., any short geodesic loop at a point is locally homotopy non-trivial and locally homotopic to a short loop in the orbit at the point. The existence of such a structure has found important applications in Riemannian geometry (cf. [Ro1]).

As showed by examples, structures on collapsed manifolds with bounded Ricci curvature are much more complicated; e.g., sectional curvature may blow up at some points ([An], [GW], [HSVZ]) or even everywhere ([Li]). Hence, a realistic goal is to first restrict to certain interesting class. Under various additional conditions, via smoothing techniques one may show the existence of a nearby metric of bounded sectional curvature, so as to conclude, by applying the above results, nilpotent structures on collapsed manifolds with Ricci curvature bounded below or in absolute value ([DWY], [PWY], [NZ], etc).

In [Ro2], nilpotent structures were (directly) constructed on collapsed manifolds with Ricci bounded local covering geometry. A compact -manifold is said to have Ricci bounded (resp. Ricci bounded below) local -covering geometry, if (resp. ), and local rewinding volume of satisfies

or local -covering geometry, if for any ,

where denotes the (incomplete) Riemannian universal cover, , denotes the metric -ball centered at , and is -Reifenberg point, if for all , the Gromov-Hausdorff distance,

where denotes an -ball in (cf. [CC2]). Clearly, (II) implies (I) and the converse does not hold. We point it out that for maximally collapsed manifolds with Ricci bounded below i.e. whose diameters are small, local covering geometry is equivalent to (global) covering geometry.

Note that local bounded covering geometry (I) is a necessary condition for collapsed manifolds with Ricci curvature bounded below to have nilpotent structures ([Ro2]). Indeed, if , then there exist constants such that

where denotes the infimum of the injectivity radii on . This crucial property was pointed in [CFG] based on the Gromov’s theorem on almost flat manifold [Gr], and a simple proof independent [Gr] is given in [Ro2].

The local construction of nilpotent structures in [Ro2] is independent of [CFG], and thus gives an alternative and relatively simple construction when restricting to collapsed manifolds with bounded sectional curvature. This seems to be the only alternative local construction since [Gr].

Our goal is to show the existence of nilpotent structures on collapsed manifolds with Ricci bounded below and local covering geometry (I) or (II). We point it out that without an upper bound on Ricci curvature, it seems to be difficult in carrying out the local construction from [Ro2].

In the present paper, we will restrict ourself in the case that the Riemannian universal cover of is not collapsed or all points on are -Reifenberg points. Our main tools are Ricci flows ([Ha]), the Perelman’s pseudo-locality ([Pe]), the structures on Ricci limit spaces ([CC1, 2]) and equivariant GH convergence (cf. [FY]); we will obtain a nearby metric with bounded sectional curvature, to which we are able to apply [Gr] and [CFG] to get nilpotent structures.

We now begin to state main results of this paper.

###### Theorem A

(Maximally collapsed manifolds with Ricci bounded blow local covering geometry) Given , there exist constants, , such that if a complete -manifold satisfies

then is diffeomorphic to an infra-nilmanifold , where is a point in the Riemannian universal cover of , is a simply connected nilpotent Lie group, and is a discrete subgroup of such that .

Theorem A implies Gromov’s theorem on almost flat manifolds i.e., and ; because which satisfies local bounded covering geometry.

Theorem A does not hold when removing the non-collapsing condition on ; any compact simply connected manifold of non-negative Ricci curvature serves as a counterexample.

###### Remark Remark 0.1

Theorem A substantially improves previous results on maximally collapsed manifolds with bounded Ricci or Ricci bounded below, and various additional conditions to apply smoothing techniques ([DWY], [PWY], [BW]); e.g., the conclusion of Theorem A was asserted in [BW] under the condition that and is close to in the Gromov-Hausdorff distance.

A map between two metric spaces, , is called an -Gromov-Hausdorff approaximation, briefly an -GHA, if is an -isometry and is -dense in . A sequence of Riemannian -manifolds converges to a metric space in the Gromov-Hausdorff topology, , if and only if there is an -GHA from to , .

###### Theorem B

(Nilpotent fibrations) Let be a sequence of complete -manifolds such that

where is a compact Riemannian manifold. Then for large, there is a smooth fibration, , such that

(B1) An -fiber is diffeomorphic to an infra-nilmanifold.

(B2) is an -GHA, and a -Hölder map, , where denotes a function in such that as , while is fixed.

Theorem B can be viewed as a parametrized version of Theorem A; as Theorem 1.2 a parametrized version of Theorem 1.1 (see Section 1). Note that Theorem B will be false if one removes the non-collapsed condition on (cf. [An]).

By (B2), the extrinsic diameter of any -fiber is less than , which may not imply that the intrinsic diameter of an -fiber is small (that holds in case sectional curvature is bounded in absolute value, see (1.2.4)). This is because in our approach to Theorem B, we use Ricci flows to obtain a nearby collapsed metric with bounded sectional curvature, by which we are able to apply the nilpotent structure result in [CFG]. Unfortunately, the Ricci flow metric is only weakly close to the original metric i.e., their distance functions are bi-Hölder close. Nevertheless, the following property holds (which is not used in the present paper): the normal subgroup, , preserves each component of , in which the -orbit in is -dense.

The bundle projection map in Theorem B defines a pure nilpotent structure on ([CFG]). A pure nilpotent structure on an open connected subset of Riemannian manifold is defined by an -invariant smooth affine fiber bundle on the orthogonal frame bundle over , , with fiber a nilpotent manifold (thus is smooth) i.e., the -action preserves both fibers and the structure group. The -invariance implies that descends to a map, , such that the following diagram commutes:

Note that a pure nilpotent structure, , is in general a singular fibration i.e. a singular fiber occurs when an -fiber and -orbit meets on a subset of positive dimension.

###### Theorem C

(Singular nilpotent fibrations) Given , there exists a small constant , such that if is a sequence of complete -manifolds converging to a compact space satisfying

then, for large, there is a singular fibration map, , such that

(C1) An -fiber is diffeomorphic to an infra-nilmanifold.

(C2) is an -GHA, and -Hölder map.

A (mixed) nilpotent structure on a manifold , , consists of a locally finite open cover for , , each admits a pure nilpotent structure such that the following compatible condition holds: if , then are both -invariant and -invariant such that -orbits sit in an -orbit or vice versa. By the compatibility, we define the -orbit at a point by the -orbit at the point of largest dimension.

###### Theorem D

(Mixed nilpotent structures) Given , there exist constants, , such that if an -collapsed compact -manifold, , satisfies

then admits a mixed nilpotent structure whose orbits at a point has extrinsic diameter .

###### Remark Remark 0.2

The existence of nilpotent structures on collapsed manifolds with has found several important applications in Riemannian geometry, cf. [Ro1]. By Theorems A-D, most of the applications, if not all, should hold on manifolds of Ricci bounded below and bounded covering geometry.

We now briefly describe our approach to Theorems A-C: we will use Ricci flows ([Ha]) and Perelman’s pseudo-locality ([Pe], [CTY]). Let denote the solution of

If exists for a definite time such that and if is much small than when either distance is less than (see Lemma 1.11), then we can apply nilpotent structural results in [CFG] to obtain the desired fibration structures on for large.

We first claim that the assumptions in Theorems A and B guarantee that points on the Riemannian universal cover are uniform Reifenberg points (see Lemma 2.1), which in turn, implies that satisfies the isoperimetric inequality condition in Perelman’s pseudo-locality theorem on Ricci flows ([Pe], [CM]). The verification of the claim is via equivariant convergence ([FY]) and structures on Ricci limit spaces ([CC1, 2]). Consequently, the Ricci flow solution on exists for a definite time and its curvature tensor, . Because the deck transformations on are also isometries with respect to , descends to on which is indeed the solution of Ricci flows on .

In Theorem A, based on the local estimate on (Lemma 1.11, cf. [CRX]) we see that is small so that we may apply [Gr] to conclude Theorem A.

In the proof of Theorem B, if is small, then we may directly apply the fibration theorem in [CFG] (cf. [Fu1]) to conclude the desired result. Unfortunately, the estimate on local distance functions is inadequate for the desired GH-closeness. Instead, we will show that in our circumstances, the construction of fibration in [Fu1] (cf. [Ro1]) can be modified according to the local closeness of distance functions (which does not require that is small), see Theorem 2.2.

In the proof of Theorem C, we show that the limit space of the Ricci flow metrics by a uniform definite time is bi-Hölder homeomorphic to , which is based on the bi-Hölder closeness of the initial metric and its Ricci flow metric in [BW] (cf. [CRX]). Note that this also implies a weak version of Theorem B.

To extend the approach of the present work to collapsed manifolds with Ricci bounded blow and local -bounded covering geometry, the key is to have a type of Perelman’s curvature estimate. We point it out that such an estimate, if holds, would be essentially different from Perelman’s pseudo-locality which relies on local geometry. Precisely, around a point whose universal cover satisfies the isopermetric inequality, there may not be a curvature bound if there is a far away point whose local cover does not satisfy the isoperimetric inequality (see below).

###### \examplename Example 0.3

(Topping, cf. [Cho]) Let be a topological sphere obtained by capping an -thin flat cylinder, , with two round -hemispheres. With a slightly smoothing, we may assume that is rotationally symmetric Riemannian manifold of nonnegative sectional curvature. For fixing , points that are -away from the centers of two hemispheres will satisfy local rewinding Reifenberg points condition. However, the Ricci flow time is proportional to area of which is less than 2. This may suggest that if one wants to have a type of Perelman’s pseuo-locality estimate on curvature of flowed metric, one has to use information on local rewinding volume at points far away.

The rest of the paper is organized as follows:

In Section 1, we will supply notions and basic properties that will be used in the proofs of Theorems A-D.

In Section 2, we will prove Theorems A-D.

In Appendix, for convenience of readers, we will outline a proof of Theorem 1.2 via embedding method in [Ro1] with minor modifications, and we will present a proof for the bi-hölder estimate of distance functions of the original metric and the Ricci flows with pseudo-locality ([BW]).

## 1. Preliminaries

In this section, we will supply notions and basic results that will be used through the rest of the paper.

### a. N-structures and collapsing with bounded sectional curvature

We will briefly recall main structural results on collapsed manifolds with bounded sectional curvature ([CFG], [CG1, 2], [Fu1-3], [Gr]).

Let be an open subset of an -manifold and let denote the orthogonal frame bundle on . A pure N-structure, , on refers to an -invariant fiber bundle, , such that an -fiber is a nilmanifold, , with a simply connected nilpotent Lie group, a co-compact lattice of , and the structural group is a subgroup of . The -invariance implies that the -action on descends to an -action on , and descends to a map, , such that the following diagram commutes,

Note that each -fiber is an infra-nilmanifold which may not have constant dimension; when -orbits and -fiber intersect more than isolated points. The minimal dimension of all -fibers is called the rank of the pure N-structure . If , we say that admits a pure N-structure. A subset of is called invariant if is the disjoint union of -fibers (or -orbits), and a metric on is called invariant, if the induced metric on every -fiber is left invariant.

A mixed N-structure on is defined by a locally finite open cover, , for , together with a pure N-structure on , which satisfies the following compatibility condition: whenever , are both invariant with respect to and , and on , every -fiber is a union of -fibers or vice versa.

An important case of a mixed N-structure is that all -fibers are tori, which is called an F-structure on . In [CG1], Cheeger-Gromov showed that if a complete manifold admits a (mixed) F-structure of positive rank, then admits a one-parameter family of invariant metrics, , which are -collapsed metrics such that and every orbit collapse to a point. Cheeger-Gromov proposed a similar construction with respect to a given mixed N-structure of positive rank, this was carried out in [CR].

We now begin to recall main structural results on collapsed manifolds with bounded sectional curvature.

###### Theorem 1.1

([Gr], [Ru]) There exists a constant such that if a compact -manifold satisfies

then is diffeomorphic to an infra-nilmanifold.

The converse of Theorem 1.1 holds ([Gr]). The following can be viewed as a bundle version of maximal collapsed manifolds.

###### Theorem 1.2

([Fu1], [CFG]) Given , there exist constants, , such that if a compact -manifold and an -manifold satisfy

then there is a fibration map, , such that

(1.2.1) is a -GHA, where is a constant.

(1.2.2) An -fiber is diffeomorphic to an infra-nilmanifold.

(1.2.3) is -Riemannian submersion, , where is orthogonal to the -fiber at .

(1.2.4) The second fundamental form of -fibers, .

###### Remark Remark 1.3

In [CFG], , and (1.2.1) may not been seen; due to the fact that averaging operation is performed on a -ball. In [Fu], [Ya] (cf. [Ro1]), is constructed via embedding and into an Euclidean space via (averaging) distance functions of two -close -nets on and . The embedding method works with condition, , where is -smooth (so (1.2.4) may not be seen). We point it out that with a minor modification of the construction of in [Ro1], plus the (additional) condition , one gets that is smooth and (1.2.4). For convenience of readers, we give a brief proof in Appendix.

For a collapsing sequence, , with and a compact metric space, the associate sequence of frame bundles equipped with canonical metrics, , and is a manifold ([Fu2]). By extending Theorem 1.2 to an -equivariant version on , one gets a singular fibration on .

###### Theorem 1.4

([Fu2], [CFG]) Let be a sequence of compact -manifolds with a compact length space and . Then for large, there is a singular fibration map, , such that

(1.4.1) An -fiber is diffeomorphic to an infra-nilmanifold.

(1.4.2) is an -GHA, .

(1.4.3) is a -submetry i.e., for all and ,

Note that the above construction of can be made local, and thus a bound on diameter can be removed for the singular fibration structure.

###### Theorem 1.5

([CG1, 2], [CFG]) There exists a constant such that if a complete -manifold satisfies

then admits a (mixed) N-structure of positive rank and at any , the orbit points to all collapsed directions with respect to a fixed scale.

### b. Ricci flows and the Perelman’s pseudo-locality

Given a complete manifold , the Ricci flow is the solution of

If (e.g., is compact), then there is a short time complete smooth solution ([Ha], [Sh1, 2]) which, for , satisfies

By [CZ], smooth solutions with bounded curvature tensor is unique. Let denote the supremum of for exists. If , then as . If a group acts isometrically with respect to , then acts by isometries with respect to .

In Riemannian geometry, Ricci flow has been a powerful tool to raise the regularity of original metrics, which requires a definite flow time. A fundamental result for estimating a lower bound on is the following Perelman’s pseudo-locality theorem on Ricci flows ([Pe]).

###### Theorem 1.6

(Perelman’s pseudo-locality) Given , there exist and satisfying the following. Let be a complete solution of the Ricci flow with bounded curvature, where , , and let be a point such that scalar curvature

and is -almost isoperimetrically Euclidean:

for every regular domain where is the Euclidean isoperimetric constant. Then

for all such that and , where is a constant depending on .

###### Remark Remark 1.7

Theorem 1.6 implies that the complete solution of the Ricci flow exists at least for , provided that the curvature conditions and the isoperimetric inequality hold everywhere.

The original statement of Theorem 1.6 in [Pe] restricts to compact . For a complete noncompact , see [CTY] (cf. Chapter 21 in [Cho]).

In case that , the almost isoperimetrically Euclidean condition is equivalent to the uniformly Reifenberg condition. For our purpose, we will formulate a special case of Corollary 1.3 in [CM] as follows:

###### Theorem 1.8

([CM]) Given , there exist such that, if is an -manifold with and where is relative compact, then given any , for every regular domain , the following almost Euclidean isoperimetric inequality holds,

where is given in Theorem 1.6.

We will apply Theorem 1.6 and 1.8 in the following situation: a compact -manifold satisfies that

Scaling by , in Theorem 1.8, we obtain

By Theorem 1.8, we conclude that any in a unit ball on satisfies

Because the above inequality is scaling invariant, also satisfies the above isoperimetric inequality condition. In short, we are able to apply Theorem 1.6 under condition (1.9).

An important consequence of the pseudo-locality is a local distance estimate (see below). Let the assumption be as in Theorem 1.6 which holds everywhere (see Remark 1.7), and for the sake of simple notation, let . Then by Hamilton’s integral version of Myer’s theorem (cf. Lemma 8.3 in [Pe]), for all and ,

Conversely, as observed in [BW] (cf. see Lemma 2.10 [CRX]), is also controlled by in the scale of . For our purpose, we need the following modification.

###### Lemma 1.11

Let the assumptions be as in Theorem 1.6. For any with or , we have

where as .

###### Demonstration Proof

By (1.10), it suffices to show . This holds if , see Lemma 2.10 in [CRX] where after normalizing (), but it actually holds without a normalization (see below).

Assume that , and we will show that implies that .

Arguing by contradiction, let be a segment from to at time such that . Then in time we can find a point on such that . By (1.12) for the case ,

So for sufficient small such that ,

a contradiction.

Finally, we point that with or implies that it holds with or ; dividing by and on each subinterval applying that . ∎

###### Remark Remark 1.13

Let be a compact -manifoild, and let denote the unique short time Ricci flow solution. Then the pullback on is also a Ricci flow solution of bounded curvature. If satisfies that and all points are -Reifenberg, then Theorems 1.6 applies to , and thus satisfies the same curvature estimate. Therefore, Ricci flow solution on exists in , and .

Moreover, the local distance estimate (1.12) also holds on .

### c. Ricci limit spaces

A pointed metric space is called a Ricci limit space, if there is a sequence of complete -manifolds of such that . A Ricci limit space is called -non-collapsed, if . Let (resp. ) denote the collection of Ricci limit space (resp. -non-collapsed Ricci limit spaces). We will briefly recall some basic results in the Cheeger-Colding theory on Ricci limit spaces that will be used in our proof of Theorem A-D.

For any , and any , by Gromov’s compactness it is easy to see, passing to a subsequence, , which is called a tangent cone of at . A tangent cone may depend on the choice of a subsequence. A point is called regular, if its tangent cone is unique and is isometric to an Euclidean space. A point is called singular, if it is not regular.

###### Theorem 1.14

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