Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing
aa r X i v : . [ m a t h . DG ] A ug COLLAPSING GEOMETRY WITH RICCI CURVATURE BOUNDED BELOWAND RICCI FLOW SMOOTHING
SHAOSAI HUANG, XIAOCHUN RONG, AND BING WANG
Dedicated to Misha Gromov on the occasion of his th birthday. A bstract . We survey some recent developments in the study of collapsing Riemannian manifoldswith Ricci curvature bounded below, especially the locally bounded Ricci covering geometry andthe Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold M is su ffi cientlyvolume collapsed with bounded diameter and sectional curvature, then there is a nearby Ricci-flatK¨ahler metric on M together with a compatible pure nilpotent Killing structure — this is related toan open question of Cheeger, Fukaya and Gromov.
1. I ntroduction
In the seminal work [39], Gromov discovered a gap phenomenon for the sectional curvature(denoted by K g for a smooth Riemannian metric g ) to detect the infranil manifold structure (seealso [7, 78]): Theorem 1.1 (Gromov’s almost flat manifold theorem, 1978) . There is a dimensional constant ε ( m ) ∈ (0 , such that if a closed m-dimensional Riemannian manifold ( M , g ) satisfies diam( M , g ) max ∧ T M (cid:12)(cid:12)(cid:12) K g (cid:12)(cid:12)(cid:12) ≤ ε , (1.1) then M is di ff eomorphic to an infranil manifold. Here we say that M is an infranil manifold if on the universal covering ˜ M of M there is a flatconnection with parallel torsion, defining a simply connected nilpotent Lie group structure N on ˜ M such that π ( M ) is a sub-group of N ⋊ Aut ( N ) with rank π ( M ) = m and [ π ( M ) : π ( M ) ∩ N ] < ∞ — in the case of Gromov’s almost flat manifold theorem, it is also shown that such index has auniform dimensional upper bound C ( m ).Ever since its birth, Gromov’s almost flat manifold theorem has inspired the research of Rie-mannian geometers by two themes of generalizations. One theme is to find parametrized versionsof Theorem 1.1, as indicated by Fukaya’s fiber bundle theorem [33]: if a Riemannian manifold withbounded diameter and sectional curvature is su ffi ciently Gromov-Hausdor ff close to a lower dimen-sional one with bounded geometry, then it is di ff eomorphic to the total space of a smooth familyof almost flat manifolds parametrized over the lower dimensional one (see Theorem 2.1). Thegeneralized version of this theorem (see Theorem 2.2), combined with the more intrinsic approachof Cheeger and Gromov [16, 17] on F -structures of positive rank, whose existence is equivalent tothe existence of a one-parameter family of Riemannian metrics collapsing with bounded sectional Date : September 1, 2020.
Key words and phrases.
Almost flat manifold, collapsing geomtry, locally bounded Ricci covering geometry, nilpo-tent Killing structure, Ricci flow. curvature, nurtured the rich and splendent theory of the collapsing geometry with bounded sec-tional curvature, notably the construction (in [14]) and various applications of the nilpotent Killingstructure (see Theorem 2.3). This theory, as we will briefly recall in § Ricci almost non-negatively curved manifold of unit diameter has its first Betti number equal toits dimension, then the manifold is di ff eomorphic to a flat torus (see Theorem 2.5). Obviously, theweaker curvature assumption alone is insu ffi cient to conclude the infranil manifold structure, andcertain extra assumptions are necessary — just as the case of the Colding-Gromov gap theorem;see also [30, 70, 49, 55] and § ffi culties for our understanding, and it isnatural to start with some extra assumptions. For instance, one could impose some extra conditionson topology such as the first Betti numbers: in [53], it is shown by the first- and third-namedauthors that if a Riemannian manifold with Ricci curvature bounded below is su ffi ciently Gromov-Hausdor ff close to a lower dimensional one with bounded geometry, and the di ff erence of their firstBetti numbers is equal to their dimensional di ff erence, then the higher dimensional manifold isdi ff eomorphic to a torus bundle over the lower dimensional one (see Theorem 2.10). This theoremgeneralizes Fukaya’s fiber bundle theorem and the Colding-Gromov gap theorem simultaneously,in setting of collapsing Riemannian manifolds with Ricci curvature bounded below — relatedresults will be surveyed in § §
3. In fact, the major e ff ort in proving Theorem 2.10 is devoted to decoding the topologicalinformation associated with the first Betti numbers and show that the higher dimensional manifoldhas locally bounded Ricci covering geometry.A key issue in the study of collapsing Riemannian manifolds with Ricci curvature boundedbelow is the low regularity of the metric due to the (weaker) Ricci curvature assumption. This con-fines our understanding on the finer structures of the collapsing geometry, and suitable smoothingof the given metric is usually inevitable — in §
4, we will survey the relevant Ricci flow smoothingtechniques for locally collapsing manifolds with Ricci curvature bounded below.In fact, the Ricci flow smoothing technique also enhances our understanding on the classicaltheory of collapsing with bounded sectional curvature. Cheeger, Fukaya and Gromov asked in[14] the following question which remains open today.
OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 3
Open Question 1.2.
Given a complete Riemannian manifold ( M , g ) with bounded sectional cur-vature, for any ε > there is a regular ( ρ, k ) -round metric g ε such that k g − g ε k C < ε and acompatible nilpotent Killing structure N . If the initial metric g is K¨ahler or Einstein, can we findg ε in the same category? In the last section of this note, we will prove, based on Ricci flow techniques, that if a Ricci-flatK¨ahler metric is very collapsed with bounded diameter and sectional curvature, then the approx-imating metric compatible with a nilpotent Killing structure may indeed be found as a nearbyRicci-flat K¨ahler metric. Notice that the approximating metric g ε obtained from [14, Theorem 1.7]is not necessarily K¨ahler or Ricci-flat, but we manage to evolve it along the Ricci flow to find adesirable one. Since the Ricci flow respects local isometries, the evolved metrics are compatiblewith the original nilpotent Killing structure N . We wish our result could cast some light on thegeneral case of Cheeger, Fukaya and Gromov’s open question.2. C ollapsing geometry with sectional or R icci curvature bounds In this section we give a short survey of the two directions generalizing Theorem 1.1 — thecollapsing geometry with bounded sectional curvature and the almost flatness characterized byweaker curvature conditions — as well as the study of collapsing geometry with only Ricci cur-vature bounded below. While there have been comprehensive surveys on the theory of collapsinggeometry [37, 74], we still briefly go through some classical theorems so as to put the study ofcollapsing geometry with Ricci curvaure bounded below in the historical context.2.1.
Collapsing with bounded sectional curvature.
Gromov’s almost flat manifold theorem,when embedded in the framework of the coarse geometry on the space of all Riemannian manifolds(see [40]), opened a new chapter in the study of Riemannian geometry: the collapsing geometry ofRiemannian manifolds with bounded curvature. In this note, we consider the following collectionsof Riemannian manifolds:(1) M Rm ( m , D ) denoting the collection of m -dimensional Riemannian manifolds with sectionalcurvature bounded by 1 in absolute value, and diameter bounded from above by D ≥ M Rm ( m , D , v ) denoting the sub-collection of M Rm ( m , D ) with volume bounded below by v > ff topology, the moduli space M Rm ( m , D ) is pre-compact [40].Based on the work of Cheeger (see [12, 38]), the sub-collection M Rm ( m , D , v ) is not just compactin the Gromov-Hausdor ff topology, but also has only finitely many di ff eomorphism classes. Onthe contrary, if we consider a sequence { ( M i , g i ) } in M Rm ( m , D ), then under the Gromov-Hausdor ff topology, it is possible that M i GH −−→ N , for some lower dimensional manifold ( N , h ) in M Rm ( k , D , v )with k < m . In this case, we say that { ( M i , g i ) } collapses to N with bounded curvature and diameter .In [33] it is shown that such situation can only occur when M i are infranil fiber bundles over N . Theorem 2.1 (Fukaya’s fiber bundle theorem, 1987) . Given D ≥ and v > , there is a uniformconstant ε ( m , v ) ∈ (0 , such that if ( M , g ) ∈ M Rm ( m , D ) and ( N , h ) ∈ M Rm ( k , D , v ) with k ≤ msatisfy d GH ( M , N ) < δ for some δ < ε ( m , v ) , then there is a C submersion f : M → N such that (1) f is an almost Riemannian submersion, i.e. ∀ ξ ⊥ ker f ∗ , e − Ψ F ( δ | m , v ) | ξ | g ≤ | f ∗ ξ | h ≤ e Ψ F ( δ | m , v ) | ξ | g ,where Ψ F ( δ | m , v ) ∈ (0 , satisfies lim δ →∞ Ψ F ( δ | m , v ) = ; and SHAOSAI HUANG, XIAOCHUN RONG, AND BING WANG (2) the fiber of f is di ff eomorphic to an infranil manifold. This theorem describes the di ff eomorphism type of those su ffi ciently collapsed manifolds in M Rm ( m , D ) by those “minimal models” in M Rm ( k , D , v ), as long as we can find such a lower di-mensional model space. In general however, we cannot expect a sequence in M Rm ( m , D ) to col-lapse to an element in M Rm ( k , D , v ). We will refer to those Hausdor ff k -dimensional ( k < m ) metricspaces arising as the Gromov-Hausdor ff limits of sequences in M Rm ( m , D ) as the collapsing limitspaces . The local structure of such spaces is descrited in [35, Theorem 0.5]: for any point x in acollapsing limit space ( X , d X ), there is an open neighborhood V of x , a Lie group G x admiting afaithful representation to O ( n ) (for some n ≤ m ), and a G x -invariant Riemannian metric ˜ g on anopen neighborhood U of ~ o ∈ R n , such that the identity component of G x is isomorphic to a torusand ( V , d X ) ≡ ( U , ˜ g ) / G x . It is consequently shown in [35, 36, 14] that a manifold M ∈ M Rm ( m , D )su ffi ciently Gromov-Hausdor ff close to a collapsing limit space X exhibits a singular fiberationover X : Theorem 2.2 (Singular fibration, 1988-1992) . Given D ≥ , there are uniform constants ε ( m , D ) and c ( m ) > to the following e ff ect: if ( M , g ) ∈ M Rm ( m , D ) satisfies | M | g < ε ≤ ε ( m , D ) , thenthe frame bundle F ( M ) of M, equipped with the canonical metric ¯ g, is Gromov-Hausdor ff close tosome ( Y , h ) ∈ M Rm ( m ′ , D ′ , v ′ ) with m ′ < m + m ( m − and D ′ , v ′ > determined by m and D, suchthat O ( m ) acts isometrically on ( Y , h ) and there is a fiber bundle ¯ f : ( F ( M ) , O ( m )) → ( Y , O ( m )) which is O ( m ) -equivariant; the fiber of ¯ f is di ff eomorphic to a compact nilmanifold N / Γ (withN simply connected and Γ ≤ N a co-compact lattice), and the structure group is contained in ( C ( N ) / ( C ( N ) ∩ Γ )) ⋊ Aut ( Γ ) ; moreover, ¯ f induces a singular fibration f : M → X = Y / O ( m ) thatfits into the following commutative diagram: ( F ( M ) , O ( m )) N / Γ M ( Y , O ( m )) X = Y / O ( m ) . / / (cid:15) (cid:15) / O ( m ) / / f / / ¯ f (cid:15) (cid:15) / O ( m ) (2.1) Finally, ¯ f is an Ψ ( ε | m , D ) -Gromov-Hausdor ff approximation satisfying Item (1) of Theorem 2.1,and the second fundamental form of each ¯ f fiber is uniformly bounded by c ( m ) in magnitude. Here the key observation is that two isometries of a Riemannian manifold are identical if their1-jets agree at some point, and the total space of 1-jets of isometries is conveniently represented byself-maps of the frame bundle; see [35]. Notice that the above mentioned fiber bundle theorems arefor manifolds collapsing also with bounded diameter, and they are in the di ff erentiable category.However, due to the existence of abundant local symmetries for those very collapsed manifoldswith bounded sectional curvature, it is natural to wonder if the extra symmetry provided by theinfranil fibers can be reflected on the level of Riemannian metrics, locally around a given fiber.This direction has been studied by Cheeger and Gromov [16, 17] for the central part of the infranilfibers (constructing the F -structure), and is thoroughly investigated in the fundational work ofCheeger, Fukaya and Gromov [14]: on the very collapsed part of a complete Riemannian manifoldwith bounded sectional curvature, a nilpotent Killing structure of positive rank is constructed,providing the finest description of the collapsing geomtry.We now define the nilpotent structure (a.k.a. N -structure) on a given complete Riemannianmanifold ( M , g ). Roughly speaking, it is the local singular fiber bundle as in Theorem 2.2 patched OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 5 together. Let { U j } be a locally finite open covering of M , then for each U j we can associate an elementary N-structure N j , which is nothing but a singular fiber bundle f j : U j → X j satisfyingthe description in Theorem 2.2 — this can be seen as a localization of that theorem, and all theinformation is encoded in the following commutative diagram( F ( U j ) , O ( m )) N j / Γ j U j ( Y j , O ( m )) X j = Y j / O ( m ) , / / (cid:15) (cid:15) pr j / / f j / / ¯ f j (cid:15) (cid:15) pr j (2.2)where pr j and pr j denote the natural projections onto the space of O ( m ) orbits. By the commu-tativity of this diagram, we see for any x ∈ X j that f − j ( x ) = pr j (cid:16) ¯ f − j (cid:16) pr − j ( x ) (cid:17)(cid:17) is an infranilmanifold, called the N j - orbit passing through any point of f − j ( x ); we let O j ( p ) denote such anorbit passing through a given p ∈ f − j ( x ). An open set V ⊂ U j is said to be N j - invariant if itis the union of N j -orbits of points in V . An N -structure N on M is then a collection of elemen-tary N -structure { N j } satisfying the compatibility condition: there is an ordering of { j } such thatif U j ∩ U j ′ , ∅ with j < j ′ , then it is both N j - and N j ′ -invariant; moreover, there is an O ( m )-equivariant fiber bundle ¯ f j j ′ : ¯ f j ′ ( F ( U j ∩ U j ′ )) → ¯ f j ( F ( U j ∩ U j ′ )) such that ¯ f j j ′ ◦ ¯ f j ′ = ¯ f j . Noticethat this implies O j ′ ( p ) ⊂ O j ( p ) for any p ∈ U j ∩ U j ′ , and the N -orbit passing through p ∈ M isthen defined as O ( p ) : = ∪ j O j ( p ). Clearly, we have a partition M = ∪O ( p ) into disjoint unions of N -orbits. We also define the rank of N j as rank N j : = min p ∈ U j dim O j ( p ), and the rank of N as rank N : = min j rank N j . If N j = N for all j , we say that N is a pure nilpotent structure, which isalways the case when the collapsing sequence has uniformly bounded diameter, as shown in [36].If each N j is abelian, we say that N defines an F -structure.Notice that on each F ( U j ), the fiber of ¯ f j is di ff eomorphic to the symmetric space N j / Γ j , onwhich the simply connected nilpotent Lie group N j acts (or equivalently, the sheaf of its Lie al-gebra maps homomorphically into the sheaf of vector fields tangent to the ¯ f j fibers). We say thatthe N -structure N is a nilpotent Killing structure compatible with a Riemannian metric g ′ on M ,if the canonically induced metric ¯ g ′ on F ( M ) has its restriction ¯ g ′ | F ( U j ) for each j being left invari-ant under the actions of N j (or equivalently, the sheaf of its Lie algebra maps homomorphicallyinto the sheaf of ¯ g ′ - Killing vector fields tangent to the ¯ f j fibers). Notice that the existence of thenilpotent Killing structure makes each fiber bundle map ¯ f j : F ( U j ) → Y j a Riemannian submer-sion. Moreover, the ¯ g ′ -Killing vector fields induced by each N j descends to U j , defining g ′ -Killingvector fields tangent to the N j -orbits. When N is pure, the partition of M into the N -orbits thendefines a (singular) Riemannian foliation, with leaves being the N -orbits; see [63] and § Theorem 2.3 (Nilpotent Killing structure, 1992) . There is a δ ( m ) > such that if ( M , g ) is acomplete Riemannian manifold with sup M (cid:12)(cid:12)(cid:12) K g (cid:12)(cid:12)(cid:12) ≤ , and for any x ∈ M, (cid:12)(cid:12)(cid:12) B g ( x , (cid:12)(cid:12)(cid:12) < δ ( m ) , then thefollowing hold: (1) ∀ ε ∈ (0 , , there exists a regular ( ρ, k ) -round Riemannian metric g ε such that k g − g ε k C ( M , g ) < ε ;(2.3)(2) there is a nilpotent Killing structure N of positive rank, compatible with g ε . SHAOSAI HUANG, XIAOCHUN RONG, AND BING WANG
Here we say that a Riemannian metric g is ( ρ, k )- round , with ρ and k determined by M and ε ,if for any p ∈ M there is an open neighborhood U ⊂ M of B g ( p , ρ ), such that U has a normalRiemannian covering ˜ U with deck transformation group Λ , whose injectivity radius is uniformlybounded below by ρ , and there is an isometric action on ˜ U by a Lie group N with nilpotent identitycomponent N , extending the deck transformations by Λ and h N : N i = h Λ : Λ ∩ N i ≤ k . Inthe presence of a compatible N -structure N = { N j } , for any p ∈ M there is some U j such that B g ( p , ρ ) ⋐ U j , and N (cid:27) N j / R d as Lie groups, with d = dim G f j ( p ) . Here f j : U j → X j is the localsingular fibration over the collapsing limit space X j , and G f j ( p ) is the isotropy group of f j ( p ) ∈ X j asdiscussed previously. Since the identity component of G f j ( p ) is a d -torus, its Lie algebra is indeed R d . Moreover, we have Λ ∩ N (cid:27) q ( Γ j ) with q : N j → N being the natural quotient map.Besides serving as a fundamental theorem of collapsing geometry with bounded sectional cur-vature, Theorem 2.2 and Theorem 2.3 have stimulated many exciting discoveries in Riemanniangeometry. To name a few, low dimensional collapsed manifolds were investigated, for which somerationality conjectures of Cheeger and Gromov on the geometric invariants associated to collaps-ing were verified [71, 73], and the 4-dimensional case of Gromov’s gap conjecture on the minimalvolume was confirmed [72]; in [19, 20], the singular structures described in Theorem 2.2 were in-vestigated, and the existence of a mixed polarized sub-structure (or a pure polarization when thereis bounded covering geometry) was proven; in [9, 10], an Abelian structure has been constructedon very collapsed manifolds with bounded non-positive sectional curvature, verifying the Buyuloconjecture [3, 4] on Cr -structure in general dimensions; in [31, 32, 68, 69], di ff eomorphism sta-bility and finiteness results have been established for very collapsed 2-connected manifolds; andin [8], one-parameter families of collapsing metrics with bounded sectional curvature have beenconstructed under the presence of the nilpotent Killing structures of positive rank, extending theprevious works of Cheeger and Gromov [16].2.2. Almost flatness by weaker curvature assumptions.
Besides the far-reaching generalizationof Gromov’s almost flat manifold theorem to parametrized versions in the collapsing geometrywith bounded sectional curvature, another direction of generalization is to weaken the curvatureassumption (1.1) in Theorem 1.1.In [30], Dai, Wei and Ye obtained a generalization for almost Ricci-flat manifolds whose conju-gate radii are uniformly bounded below.
Theorem 2.4 (Almost Ricci-flat manifolds, 1996) . There is a uniform constant ε ( m ) ∈ (0 , suchthat if a closed m-dimensional Riemannian manifold ( M , g ) with conjugate radii bounded belowby satisfies diam( M , g ) max M (cid:12)(cid:12)(cid:12) Rc g (cid:12)(cid:12)(cid:12) g ≤ ε , then M is di ff eomorphic to an infranil manifold. While this theorem seems to be expected directly from Gromov’s almost flat manifold theorem,the proof actually requires the Ricci flow smoothing technique — to the authors’ knowledge, thisis the first instance where such a technique is employed for manifolds satisfying certain Riccicurvature bounds. This theorem is further generalized by Petersen, Wei and Ye in [70], where theregularity of the metrics is characterized by the harmonic C , norm of the manifolds; see [70,Theorem 1.4], which we will not restate in this note for the sake of brevity. OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 7
A more striking conjecture of Gromov [40] predicted that even if only assuming almost non-negative Ricci curvature, when the first Betti number of the manifold is equal to its dimension,then it has to be a flat m -torus. This conjecture is an e ff ective version of the Bochner technique[5, 6] which asserts that a Ricci non-negatively curved manifold with maximal first Betti number(never exceeding the dimension) must be a flat torus, and is confirmed by Colding based on hisvolume continuity theorem [27]: Theorem 2.5 (Colding-Gromov gap theorem, 1997) . There is a dimensional constant ε ( m ) ∈ (0 , such that if a closed m-dimensional Riemannian manifold ( M , g ) satisfies diam( M , g ) Rc g ≥ − ε g and b ( M ) = m , then M is di ff eomorphic to a flat torus. This theorem can also be viewed as a quantitative version of the Cheeger-Gromoll splittingtheorem [15] when we consider the universal covering space of the given manifold.
Remark . In [27], Colding proved that M is homotopic to the torus when dim M =
3, and ishomeomorphic to the torus when dim M >
3. The di ff eomorphism statement was later proven inthe joint work of Cheeger and Colding using the Reifenberg method; see [13, Appendix A].Obviously, the weaker assumption of almost non-negative Ricci curvature by itself is not enoughto detect the infranil manifold structure, and certain extra assumptions should be expected. Thesecond-named author proposed in 2014 to study Riemannian manifolds with locally bounded Riccicovering geometry , considering those manifolds with Ricci curvature bounded below and non-collapsing local universal covering spaces. Much progress has been made since then (see § Theorem 2.6 (Bounded Ricci covering geometry, 2018) . Given m , v > , there is a uniform con-stant ε ( m , v ) ∈ (0 , such that if a closed m-dimensional Riemannian manifold ( M , g ) satisfies diam( M , g ) Rc g ≥ − ε g and (cid:12)(cid:12)(cid:12) B ˜ g ( ˜ p , (cid:12)(cid:12)(cid:12) ≥ v , then M is di ff eomorphic to an infranil manifold. Here ˜ p is any point in ˜ M, the universal coveringspace of M equipped with the covering metric ˜ g. Other types of weaker curvature assumptions include certain mixed curvature conditions consid-ered by Kapovitch: in [55], the sectional curvature lower bound in Gromov’s almost flat manifoldtheorem is weakened to a Bakry- ´Emery Ricci tensor lower bound.
Theorem 2.7 (Mixed curvature conditions, 2019) . There exists a uniform constant ε ( N ) ∈ (0 , such that if an m-dimensional (m ≤ N) weighted closed Riemannian manifold ( M , g , e − f H m ) sat-isfies diam( M , g ) max ∧ T M K g ≤ ε ( N ) and diam( M , g ) (cid:16) Rc g + Hess f (cid:17) ≥ − ε ( N ) g , then M is di ff eomorphic to an infranil manifold. We notice here that the lower bound of the Bakry- ´Emery Ricci tensor Rc g + Hess f is essen-tially weaker than the corresponding Ricci curvature lower bound, which would imply almostnon-negative sectional curvature in the context: as shown in [55, Lemma 8.1], on the 2-torus there SHAOSAI HUANG, XIAOCHUN RONG, AND BING WANG is a sequence of Riemannian metrics satisfying the assumptions of the theorem, but the minimumof their sectional curvature (in ∧ T T ) have no finite lower bound. We also point out that it is morenatural to consider the Bakry- ´Emery Ricci curvature lower bound in the collapsing setting: see thework of Lott [60].2.3. Collapsing manifolds with Ricci curvature bounded below.
More generally, one may con-sider the Gromov-Hausdor ff limits of manifolds in the collection M Rc ( m ) of complete Riemannian m -manfiolds with the lowest eigenvalue of the Ricci tensor uniformly bounded below by − ( m − ff limits of a sequence in M Rc ( m ) with an extra uniform volume lowerbound (a non-collapsing sequence ) has been thoroughly investigated through the works [13, 18],our understanding of a possibly collapsing sequence in M Rc ( m ) (i.e. a sequence without volumelower bound) is very limited. The ideal here is to develop a parallel theory as the collapsing geome-try with bounded sectional curvature, and describe the geometry around points where the sectionalcurvature becomes unbounded.Recall that in the classical theory, the collapsing with bounded sectional curvature is causedby the extra symmetry of the infranil fibers. It is therefore natural to focus on the local isome-tries of manifolds in M Rc ( m ). The understanding of such local isometry is encoded in the fiberedfundamental group Γ δ ( p ), defined for any p ∈ M and δ ∈ (0 ,
1) as Γ δ ( p ) : = Image h π ( B g ( p , δ ) , p ) → π ( B g ( p , , p ) i . This group collects all loops contained in B g ( p , δ ) and based at p , but that are allowed to deformwithin B g ( p , Theorem 2.8 (Generalized Margulis lemma, 2011) . There are uniform constants C ( m ) ≥ and ε ( m ) ∈ (0 , such that for any ( M , g ) ∈ M Rc ( m ) and any p ∈ M, the fibered fundamental group Γ ε ( m ) ( p ) contains a nilpotent sub-group of nilpotency rank ≤ m and index ≤ C ( m ) . Based on a rescaling and contradiction argument, this theorem is strengthened by Naber andZhang [65] when a geodesic ball is Gromov-Hausdor ff close to a lower dimensional Euclideanball: if for some k ≤ m it holds that d GH (cid:16) B g ( p , , B k (2) (cid:17) < ε ( m ), then rank Γ ε ( m ) ( p ) ≤ m − k . Moresignificantly, they discovered certain topological conditions that guarantee a very strong regularityof the metric. Theorem 2.9 ( ε -Regularity for Ricci curvature, 2018) . For any ε > there is a uniform constant δ ( m , ε ) ∈ (0 , such that if ( M , g , p ) is a pointed Riemannian m-manifold with Rc g ≥ − ( m − g,and B g ( p , ⋐ B g ( p , , then for any normal covering π : ( W , ˆ p ) → ( B g ( p , , p ) with π ( ˆ p ) = p,covering metric ˆ g and deck transformation group G, if (1) d GH (cid:16) B g ( p , , B k (2) (cid:17) < δ , and (2) and the almost nilpotent group satisfies rank b G δ ( p ) = m − k,then for some r ∈ ( δ, it holds thatd GH (cid:16) B ˆ g ( ˆ p , r ) , B m ( r ) (cid:17) < ε r . OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 9
If we impose the extra assumption of a uniform Ricci curvature upper bound, this theoremdirectly proves a local fiber bundle theorem for domains collapsing to lower dimensional Euclideanballs; see [65, Proposition 6.6]. The same conclusion actually holds without the Ricci curvatureupper bound, and this has been proven very recently by the authors based on the Ricci flow localsmoothing techniques ([54, Theorem 1.4]); see Theorem 4.9 in the next section. Notice that theAssumption (1) in Theorem 2.9 is crucial in the original blow up argument, and the case of orbifoldcollapsing limit in [54, Theorem 1.4] is considerably more di ffi cult; see Remark 3. Based on theseresults, the following theorem is recently proven in [53]. Theorem 2.10 (Rigidity of the first Betti number, 2020) . Given D ≥ and v > , there is auniform constant ε ( m , D , v ) ∈ (0 , such that if ( M , g ) ∈ M Rc ( m ) and ( N , h ) ∈ M Rm ( k , D , v ) satisfyd GH ( M , N ) < ε ( m , D , v ) with k ≤ m, then b ( M ) − b ( N ) ≤ m − k. Moreover, if the equality holds,then M is di ff eomorphic to an ( m − k ) -torus bundle over N. Here we notice that the assumptions of Theorem 2.9 are purely local, and a key di ffi culty inthe proof was to localize the topological information encoded in the first Betti number, which isglobal in nature. Here we introduced the so-called pseudo-local fundamental group ˜ Γ δ ( p ), whichis defined for any p ∈ M and δ ∈ (0 ,
1) as ˜ Γ δ ( p ) : = Image h π ( B g ( p , δ ) , p ) → π ( M , p ) i . Thisconcept provides a bridge linking π ( M , p ) with Γ δ ( p ). We also considered H δ ( M ; Z ), generated bysingular homology classes with a representation by a geodesic loop of length not exceeding 10 δ .Under the assumption d GH ( M , N ) < δ , it is then shown that b ( M ) − b ( N ) = rank H δ ( M ; Z ); andfor δ < ε ( m ) su ffi ciently small, generalizing the work of Colding and Naber [28] it is shown that rank H δ ( M ; Z ) ≤ rank ˜ Γ ε ( m ) ( p ) ≤ m − k for any p ∈ M . Therefore, by Theorem 2.9 the assumption b ( M ) − b ( N ) = m − k ensures the universal covering of M to locally resemble the m -Euclideanspace. We could then run the Ricci flow to obtain a regular metric that still collapses, and applyingTheorem 2.1 we established the theorem — discussions on such Ricci flow smoothing techniquewill be in §
4. 3. L ocally bounded R icci covering geometry In this section, we discuss the program initiated by the second-named author around 2014 toinvestigate Riemannian manifolds with Ricci curvature bounded below and non-collapsing localuniversal covering spaces: its current status and its goal; see also the previous survey [75] by thesecond-named author for related discussions.More precisely, we let f M Rc ( m , ρ, v ) denote the collection of complete m -dimensional Riemann-ian manifolds ( M , g ) satisfying Rc g ≥ − ( m − g and for any x ∈ M , (cid:12)(cid:12)(cid:12) B ˜ g ( ˜ x , ρ ) (cid:12)(cid:12)(cid:12) ≥ v , where ˜ g is thecovering metric of the (incomplete) Riemannian universal covering space of B g ( x , ρ ), and ˜ x is anypoint covering x . We call the quantity (cid:12)(cid:12)(cid:12) B ˜ g ( ˜ x , ρ ) (cid:12)(cid:12)(cid:12) the local rewinding volume of x at scale ρ , denotedby g Vol g ( x , ρ ). So roughly speaking, f M Rc ( m , ρ, v ) consists of manifolds with Ricci curvature andlocal rewinding volume (at scale ρ ) uniformly bounded below, and we say that such manifolds have locally ( ρ, v ) -bounded Ricci covering geometry , or just locally bounded Ricci covering geometry.The goal of studying locally bounded Ricci covering geomtry is mainly to establish an ana-logue, for manifolds in f M Rc ( m , ρ, v ) and their (pointed) Gromov-Hausdor ff limits, of the nilpotentstructure theorey of Cheeger, Fukaya and Gromov (see § As pointed out in the introduction, understanding the collapsing behaviors of manifolds of thistype is a very natural and immediate step in our study of general collapsing phenomena of man-ifolds with Ricci curvature bounded below. If a Riemannian m -manifold has sectional curvatureuniformly bounded by 1 in absolute value, then it has locally ( ρ ( m ) , v ( m ))-bounded Ricci coveringgeometry, where the constants only depend on the dimension m ; see [14, 77].3.1. Singular infranil fiber bundles.
Recall that our ideal of study collapsing geometry withRicci curvature bounded below is to recover, at least over most parts of the collapsing limit, theinfranil fiber bundle structure. In the setting of collapsing with locally bounded Ricci coveringgeometry, this is indeed the case over the regular part of the collapsing limit.
Theorem 3.1 (Infranil fiber bundle, 2020) . Given m , k ∈ N with k < m, D ≥ and ρ, v , v > there is a uniform constant δ ( m , D , ρ, v , v ) > such that if ( M , g ) ∈ f M Rc ( m , ρ, v ) is δ -Gromov-Hausdor ff close to a manifold ( N , h ) ∈ M Rm ( k , D , v ) , then there is a fiber bundle f : M → N whichis also a Ψ ( δ | m , D , ρ, v , v ) -Gromov-Hausdor ff approximation, with whose fibers di ff eomorphic toan ( m − k ) -dimensional infranil manifold, and whose structure group reduced to a generalizedtorus group as described in Theorem 2.2. This theorem summarizes the contributions from [48] and [77]. In [48], the existence of thetopological fiber bundle map f is obtained using the canonical Reifenberg method in [18]. How-ever, the infranil manifold structure of the fiber obtained in [77] is not a direct application ofTheorem 2.6, since an f -fiber may have no uniform Ricci curvature lower bound. Considerationson the ambient geometry is instead carried out in [77] — this work, when restricting to the caseof almost flat manifolds, provides for the first time an approach entirely di ff erent from the originalone in [39, 78]; see also [76]. Notice that the arguments in [48, 77] are local and our statementshere are valid even if N is an open set in an k -dimensional manifold.Besides substaintially generalizing the condition of collapsing with bounded sectional curvature,the condition of locally bounded Ricci covering geometry is also more general than the assumptionof maximal nilpotency rank discussed in Theorem 2.9, which plays a key role in the proof ofTheorem 2.10. In fact, even when collapsing with bounded sectional curvature occurs, the fiberedfundamental group (or the pseudo-local fundamental group) based at the fiber over a corner pointcannot have maximal nilpotency rank. Here we recall that for a collapsing limit space X , we havea singular fiber bundle f : M → X as described in Theorem 2.2, and we say x ∈ X is a corner point if dim G x >
0. We make, however, the following remark.
Remark . When the collapsing limit X is a manifold, these three concepts of collapsing coincide:both (a) collapsing with locally bounded Ricci covering geometry, and (b) collapsing with Riccicurvature bounded below and maximal nilpotency rank at every point, imply the same collapsinginfranil fiber bundle structure arising from collapsing with bounded sectional curvature, whichclearly implies the cases (a) and (b).For a general metric spaces arising as the Gromov-Hausdor ff limits of manifolds in f M Rc ( m , ρ, v ),we propose the following conjecture; compare also Theorem 2.2. Conjecture 3.2 (Singular nilpotent fibration) . Suppose a sequence { ( M i , g i ) } ⊂ f M Rc ( m , ρ, v ) col-lapses to a lower dimensional compact metric space X, i.e. M mi GH −−→ X k where the dimension isthe sense of Colding-Naber [28] . Then for each i su ffi ciently large, there is a singular fibration OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 11 f i : M i → X, such that a regular fiber is an infranil manifold, and a singular fiber is a finitequotient of an infranil manifold.
Some progress on understanding the structure of a Gromov-Hausdor ff limit of a sequence in f M Rc ( m , ρ, v ) has already been made: see e.g. [50]. Notice that if { ( M i , g i ) } ⊂ M Rc ( m ) and M i GH −−→ X for some metric space ( X , d ) with diam( X , d ) ≤ D , then there is a renormalized measure ν on X ,and by the work of Colding and Naber [28], there is a unique k ≤ m such that at ν -a.e. point of X any tangent cone is isometric to R k : this k will be denoted as dim CN X , the dimension of X inthe sense of Colding-Naber. On the other hand, as a metric space one can talk about the Hausdor ff dimension dim H X . When k = m we know that dim CN X = dim H X , but when k < m this is notnecessarily the case. However, by [75] we have the following Proposition 3.3. If { ( M i , g i , p i ) } ⊂ f M Rc ( m , ρ, v ) and M i pGH −−−→ X with ( X , d , p ) a pointed metricspace, then there is some k ≤ m such that any tangent cone at any point of X is a k-dimensionalmetric cone. Moreover, dim H X = k = dim CN X. Almost maximal local rewinding volume.
In this sub-section, we consider an “extremal”case of locally bounded Ricci covering geometry: those manifolds in f M Rc ( m , ρ, v ) with almostmaximal local rewinding everywhere; compare also [26]. Given ( M , g ) ∈ e M Rc ( m , ρ, v ), since forany x ∈ M , the local covering metric has the same Ricci curvature lower bound as the originalmetric, the Bishop-Gromov volume comparison is in e ff ect, implying that g Vol g ( ˜ x , ρ ) ≤ Λ m λ ( x ) ( ρ ),which denotes the volume of a geodesic ρ -ball in the space form of sectional curvature equal to λ ( x ), with λ ( x ) denoting the lowest eigenvalue of Rc g on B g ( x , ρ ). When M is compact, we let λ ( M , g ) : = min x ∈ M λ ( x ). If, however, we know that the local rewinding volume is almost maximal,then strong structural results have been obtained by the second-named author with his collaboratorsin [21, 22]. Theorem 3.4 (Quantitative space form rigidity, 2019) . Given m , ρ, v > with ρ < and a closedmanifold ( M , g ) ∈ M Rc ( m ) whose Riemannian universal covering ( ˜ M , ˜ g ) has some ˜ p ∈ ˜ M satisfy-ing (cid:12)(cid:12)(cid:12) B ˜ g ( ˜ p , (cid:12)(cid:12)(cid:12) ˜ g ≥ v, then we have the following: (1) if λ ( M , g ) = and inf x ∈ M g Vol g ( x , ρ ) ≥ (1 − ε ) Λ m ( ρ ) for some uniform ε ∈ (0 , determinedby m, ρ and v, then M is di ff eomorphic to a spherical space form by a Ψ ( ε | m , ρ, v ) -isometry; (2) if λ ( M , g ) = , diam( M , g ) ≤ and inf x ∈ M g Vol g ( x , ρ ) ≥ (1 − ε ) Λ m ( ρ ) for some uniform ε ∈ (0 , determined by m, ρ and v, then M is isometric to a flat manifold; and (3) if λ ( M , g ) = − , diam( M , g ) ≤ d and inf x ∈ M g Vol g ( x , ρ ) ≥ (1 − ε ) Λ m − ( ρ ) for some uniform ε ∈ (0 , determined by d, m, ρ and v, then M is di ff eomorphic to a hyperbolic manifoldby a Ψ ( ε | m , ρ, v , d ) -isometry. Note that manifolds satisfying Items (1) or (2) in this theorem may be arbitrarily collapsed.In [21, Theorem D], a quantitative rigidity theorem for hyperbolic spaces (compare Item (3) inTheorem 3.4) has been obtained for manifolds in M Rc ( m ) in terms of the volume entropy [58]. Infact, by the assumed uniform lower bound of the local rewinding volume, it is natural to ask if wecan drop the non-collapsing assumption of the universl covering spaces: Conjecture 3.5.
Theorem 3.4 still holds for ( M , g ) ∈ M Rc ( m ) even if we do not assume the exis-tence of ˜ p ∈ ˜ M so that (cid:12)(cid:12)(cid:12) B ˜ g ( ˜ p , (cid:12)(cid:12)(cid:12) ˜ g ≥ v > . In the case when a uniform Ricci curvature upper bound is additionally assumed, this conjecturehas been verified by the second-named author and his collaborators in [22].4. S moothing the locally collapsing metrics with R icci curvature bounded below As mentioned in the introduction, the proofs of Theorems 2.1, 2.6, 2.7 and 2.10 all rely on thesmoothing e ff ect by globally running the Ricci flow. The Ricci flow with initial data ( M , g ), firstintroduced by Hamilton [44] on closed 3-manifolds to deform a given Riemannian metric g withpositive Ricci curvature to a positive Einstein metric, is a smooth family of Riemannian metrics g ( t ) on M solving the following initial value problem for t ≥ ∂ t g ( t ) = − Rc g ( t ) ; g (0) = g . (4.1)In harmonic coordinates, the Ricci flow becomes a non-linear heat-type equation for the metrictensor, and by the nature of the heat flows, notably Shi’s estimates [79], a key e ff ect of runningRicci flow is that the evolved metric has much improved regularity: ∀ l ∈ N , ∃ C l > , sup M (cid:12)(cid:12)(cid:12) ∇ l Rm g ( t ) (cid:12)(cid:12)(cid:12) g ( t ) ≤ C l t − − l . (4.2)Here the constants C l depend on the dimension of M , as well as (cid:13)(cid:13)(cid:13) Rm g (cid:13)(cid:13)(cid:13) C ( M , g ) . In fact, the finitenessof (cid:13)(cid:13)(cid:13) Rm g (cid:13)(cid:13)(cid:13) C ( M , g ) guarantees the Ricci flow solution to (4.1) to exist for a definite amount of timedetermined by its value, even if ( M , g ) is complete but non-compact.In view of Shi’s estimates, the Ricci flow also becomes a useful tool to smooth a given Riemann-ian metric by replacing the initially given metric g with the evolved metric g ( t ), whose regularityis controlled by (4.2) — in order to take advantage of such an estimate, a uniform lower boundof the Ricci flow existence time then becomes crucial. This method has been investigated in [30]for closed mainfolds, producing fruitful applications, such as the proofs of Theorem 2.4 and [65,Proposition 6.6] (see also [70]); but as the collapsing phenomenon may be observed locally ona geodesic ball, the localization of the Ricci flow existence results is usually necessary for thesmoothing purpose. In this section, we will discuss the recent developments on the Ricci flowlocal smoothing techniques for collapsing initial data with Ricci curvature bounnded below.4.1. Local existence of the Ricci flow.
As shown in [54, Lemma 2.2], we could in fact start theRicci flow locally on any Riemannian manifold with Ricci curvature bounded below.
Lemma 4.1.
Given a complete Riemannian manifold ( M m , g ) with Rc g ≥ − ( m − g and let K bea compact subset. For any R > there is a smooth family of Riemannian metrics g ( t ) on B g ( K , R ) satisfying ∂ t g ( t ) = − Rc g ( t ) on B g ( K , R ) × (0 , T ] , g (0) = g on B g ( K , R ) , for some T > , such that ∀ t ∈ (0 , T ] , sup B g ( K , R ) (cid:12)(cid:12)(cid:12) Rm g ( t ) (cid:12)(cid:12)(cid:12) g ( t ) ≤ Ct − , where the positive constants C and T depend on g, K and R. OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 13
This lemma is proven using Hochard’s conformal transformation technique in [47, § ∂ B g ( K , R ) to infinity, obtaining a complete Riemannian metric h defined on B g ( K , R ). Since B g ( K , R ) is compact, the sectional curvature of g is bounded, and thusso is the sectional curvature of the complete metric h by making a good choice of the conformalfactor. One could then rely on Shi’s short time existence theorem to start a Ricci flow solution h ( t )with initial data h (0) = h . But since the conformal factor can be designed to be 1 on B g ( K , R ),one can view the Ricci flow solution h ( t ) | B g ( K , R ) as the local Ricci flow starting from the initial data( B g ( K , R ) , g ). We point out that the the conformal factor could also be designed so that the scalarcurvature and local isoperimetric constant lower bounds for h are comparable to the ones for g .As mentioned in the introduction, in order to use Ricci flow as a smoothing tool one needs adefinite lower bound of the existence time. If the initial data has only Ricci curvature lower bound,then the existence time lower bound relies on certain non-collapsing condition — even if the actual m -dimensional initial data may collapse to a lower dimensional space, the local covering spacesare usually assumed to resemble the local m -dimensional Euclidean space. In this setting, our mostrecent result ([54, Theorem 1.2]) gives: Theorem 4.2.
Given any α ∈ (0 , − ) , any positive m , l ∈ N and any R ∈ (0 , , there areuniform constants δ O ( m , l , R , α ) , ε O ( m , α ) ∈ (0 , to the following e ff ect: let K be a compact andconnected subset of ( M m , g ) , an m-dimensional Riemannian manifold with Rc g ≥ − ( m − g,suppose for some k ≤ m and δ ≤ δ O it satisfies for any p ∈ B g ( K , R ) the following assumptions: (1) there are a finite group G p < O ( k ) with (cid:12)(cid:12)(cid:12) G p (cid:12)(cid:12)(cid:12) ≤ l and a φ p ∈ Hom (cid:16) π ( B g ( K , R ) , p ) , G p (cid:17) which is surjective, (2) d GH (cid:16) B g ( p , − R ) , B k (4 − R ) / G p (cid:17) < δ , and (3) rank ˜ Γ δ ( p ) = m − k,then there is a Ricci flow solution with initial data ( B g ( K , R ) , g ) , existing for a period no shorterthan ε O , and with curvature control ∀ t ∈ (0 , ε O ] , sup B g ( K , R ) (cid:12)(cid:12)(cid:12) Rm g ( t ) (cid:12)(cid:12)(cid:12) g ( t ) ≤ α t − + ε − O . (4.3)Here the notation ˜ Γ δ ( p ) : = Image h π ( B g ( p , δ ) , p ) → π ( B g ( K , R ) , p ) i is the pseudo-local fun-damental group for B g ( K , R ), containing all geodesic loops in B g ( p , δ ) with base point p , and areallowed to be deformed within the entire B g ( K , R ).Theorem 4.2 is proven roughly as following: by Conditions (1) and (2), for each p ∈ B g ( K , R )we can find a finite normal covering of B g ( p , R ), so that it is Ψ ( δ )-Gromov-Hausdor ff close to B k ( R ); this condition, together with the nilpotency rank assumption in Condition (3), enable us toshow that the isoperimetric constant in a fix-sized geodesic ball around any point of the univer-sal covering space of B g ( K , R ) is very close to the m -Euclidean isoperimetric constant. By thedesign of the conformal factor, such almost locally Eucliean property is almost preserved underthe conformal transformation, and together with the (relaxed) scalar curvature lower bound of theconformally transformed metric, it enables us to apply Perelman’s pseudo-locality theorem (seethe next sub-section) to bound the Ricci flow existence time from below.Theorem 4.2 characterizes the “almost locally Euclidean covering space” assumption via alge-braic conditions, i.e. the maximality of the rank of the pseudo-local fundamental groups and the existence of a surjective homomorphism of the local fundamental group onto the orbifold groups.One could also directly assume that the local universal covering space resembles the m -Euclideanspace up to a fixed scale, defining the so-called ( δ, ρ )-Reifenberg points. For any p ∈ M m , we sayit is a ( δ, ρ ) -Reifenberg point , if for any lift ˜ p of p in the Riemannian universal covering space of B g ( p , ρ ), ∀ r ∈ (0 , ρ ] , r − d GH (cid:16) B ˜ g ( ˜ p , r ) , B m ( r ) (cid:17) ≤ δ. This definition essentially appears in the work [49] of the second-named author and his collabo-rators, and is for the purpose of defining the concept of Ricci bounded local covering geometry.Notice that with bounded Ricci curvature, if p is a ( δ, ρ )-Reifenberg point for δ su ffi ciently small,then B ˜ g ( ˜ p , ρ ) has a uniform lower bound on the C , harmonic radius. On the other hand, one couldalways run the Ricci flow locally around a ( δ, ρ )-Reifenberg point for a definite amount of time. Theorem 4.3.
For any α, ρ ∈ (0 , there are uniform δ ( m , α, ρ ) ∈ (0 , and ε ( m , α, ρ ) ∈ (0 , such that if ( M , g ) is a complete Riemannian manifold with Rc g ≥ − ( m − g, and p ∈ M is a ( δ, ρ ) -Reifenberg point, then there is a Ricci flow solution with initial data ( B g ( p , ρ ) , g ) , that existsup to time ε and for any t ∈ [0 , ε ] , the curvature satisfies sup B g ( p ,ρ ) (cid:12)(cid:12)(cid:12) Rm g ( t ) (cid:12)(cid:12)(cid:12) ≤ α t − . Sketch of proof.
We could always start the Ricci flow h ( t ) by Lemma 4.1 on B g ( p , ρ ). Moreover,we could make sure that the initial data satisfies h (0) | B g ( p ,ρ ) ≡ g | B g ( p ,ρ ) . We only need to bound theexistence time of the Ricci flow from below, which in turn relies on showing that the isoperimetricconstant at any point of the covering space is almost Euclidean on a fixed scale. But this is straight-forward by the definition of the ( δ, ρ )-Reifenberg property, as long as δ is su ffi ciently small. Onethen relies on Perelman’s pseudo-locality theorem to prove that the flow exists for a definite periodof time. (cid:3) Let us also mention that in the case of Kapovitch’s mixed curvature condition, it can be shownthat the assumptions of Theorem 2.7 guarantees the local universal covering space at a point of themanifold to be almost locally Euclidean, via the Aspherical Theorem ([55, Theorem 5.3]).4.2.
The pseudo-locality theorem.
In all the results discussed above, once the almost locallyEuclidean condition for the local covering space is verified, the lower bound of the existence timeof the Ricci flow is guaranteed by Perelman’s pseudo-locality theorem, stated in its various formsas following:
Theorem 4.4 (Perelman’s pseudo-locality theorem) . For any α ∈ (0 , , there are uniform positiveconstants ε P = ε P ( m , α ) and δ P = δ P ( m , α ) such that if ( M , g ) is a Ricci flow solution define fort ∈ [0 , T ] with each time slice ( M , g ( t )) being a complete Riemannian manifold, and if one of theconditions holds for p ∈ M: (1) R g (0) ≥ − on B g (0) ( p , and I B g (0) ( p , ≥ (1 − δ P ) I m , or (2) Rc g (0) ≥ − δ P g (0) on B g (0) ( p , and (cid:12)(cid:12)(cid:12) B g (0) ( p , (cid:12)(cid:12)(cid:12) g (0) ≥ (1 − δ P ) ω m , OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 15 where I m and ω m stands for the isoperimetric constant and volume of the m-Euclidean unit ball,respectively, and I Ω denotes the isoperimetric constant for the domain Ω ⊂ M, then ∀ t ∈ (0 , ε P ] , sup B g ( t ) ( p ,ε P ) (cid:12)(cid:12)(cid:12) Rm g ( t ) (cid:12)(cid:12)(cid:12) g ( t ) ≤ α t − + ε − P . (4.4)The theorem is originated from Perelman for closed manifolds satisfying Condition (1) above;see [67, Theorem 10.1]. Later a version for complete non-compact manifolds was obtained byChau, Tam and Yu; see [11, Theorem 8.1]. The theorem with Condition (2) was proven by Tianand the third-named author for closed manifolds in [81, Proposition 3.1], and its counterpart forcomplete non-compact data appears in the recent work of the authors’ in [53, Proposition 6.1]. Wepoint out that all the later works essentially follow Perelman’s original idea and arguments.In the proofs of Theorem 4.2, Theorem 4.3 and [55, Theorem 7.2], the almost locally Euclideanproperty for the local covering spaces checked before allows us to apply the pseudo-locality theo-rem to the covering flow and obtain a uniform lower bound on the existence time of the Ricci flowstarted via Lemma 4.1: if the existence time T of the Ricci flow were shorter than ε P , then for somesequence t i ր T we could observe points x i ∈ M such that lim t i → T (cid:12)(cid:12)(cid:12) Rm h ( t i ) (cid:12)(cid:12)(cid:12) h ( t i ) ( x i ) = ∞ ; especially,we will get (cid:12)(cid:12)(cid:12) Rm h ( t i ) (cid:12)(cid:12)(cid:12) h ( t i ) ( x i ) > α T − + ε − P for all i large enough, contradicting the conclusion (4.4)since T > “A natural question is whether the assumption on the volume of the ball is superfluous. We notice however, that there are examples (see e.g. [62, 49]) that show the direct removal ofthe initial local volume non-collapsing assumption is fatal:
Example 4.5 (Topping) . Let M δ denote the smooth manifold obtained from capping o ff the δ -thincylinder δ S × [ − , ( S is identified with the unit circle in C with base point ∈ C ) by two discs ofradius approximately π δ and slightly smoothing near the ends of the cylinder. The natural metricg δ can be easily made to have non-negative sectional curvature. It is also obvious that around thebase point p δ = ( δ, of M δ , the geodesic ball B g δ ( p δ , ) is flat and is δ -Gromov-Hausdor ff closeto ( − , ) . However, the Ricci flow starting from M δ exits only for a period determined by the areaof M δ , which is proportional to δ . Therefore, as δ ց , a curvature bound of the form (4.4) cannotbe obtained for any uniform ε > . Fortunately, in many natural settings, the scalar curvature is indeed uniformly bounded alongthe Ricci flow, and here we raise the following
Conjecture 4.6.
Given α ∈ (0 , , there are positive constants δ = δ ( m , α ) and ε = ε ( m , α ) suchthat if ( M , g ) is an m-dimensional Ricci flow solution on [0 , T ] with each of whose time slices beingcomplete, and for some p ∈ M it satisfies sup B g (0) ( p , (cid:12)(cid:12)(cid:12) Rm g (0) (cid:12)(cid:12)(cid:12) g (0) ≤ , d GH (cid:16) B g (0) ( p , , B k (1) (cid:17) ≤ δ, and sup M × [0 , T ] (cid:12)(cid:12)(cid:12) R g ( t ) (cid:12)(cid:12)(cid:12) ≤ , then we have for any t ∈ [0 , ε ] , the curvature bound sup B g ( t ) ( p ,ε ) (cid:12)(cid:12)(cid:12) Rm g ( t ) (cid:12)(cid:12)(cid:12) g ( t ) ≤ α t − + ε − . (4.5)4.3. Distance distortion estimates.
Once the Ricci flow exists for a definite amount of time, forthe purpose of smoothing, it is of key importance to compare the initial metric with the evolvedmetric. In general, the distance distortion estimate for Ricci flows is of central importance in theunderstanding of the geometry along the Ricci flows, and we refer the readers to [45, 23, 81, 24, 1,2, 25, 52] for previous works on this topic in various settings. Very recently, based on the previouscontributions, especially the local entropy theory developed in [83], the distance distortion estimatefor collapsing initial data [52], and the H¨older distance estimate for non-collapsing initial data in[49], we obtain the following H¨older distance estimate for collapsing initial data [54, TheoremA.1]:
Theorem 4.7.
Given a positive integer m, positive constants ¯ C , C R , T ≤ and α ∈ (0 , , there areconstants C D ( ¯ C , C R , m ) ≥ and T D ( ¯ C , C R , m ) ∈ (0 , T ] such that for an m-dimensional completeRicci flow ( M , g ( t )) defined for t ∈ [0 , T ] , if for some x ∈ M and any t ∈ [0 , T ] we have R g (0) ≥ − C R in B g (0) ( x , , (4.6) (cid:12)(cid:12)(cid:12) Rc g ( t ) (cid:12)(cid:12)(cid:12) g ( t ) ≤ α t − in B g ( t ) (cid:16) x , + √ t (cid:17) , (4.7) and the initial metric has a uniform bound ¯ C on the doubling and Poincar´e constant for thegeodesic ball B g (0) ( x , , then for any x , y ∈ B g (0) ( x , √ T D ) and t ∈ [0 , T D ] , we haveC − D d g (0) ( x , y ) + α d g (0) ( x , y ) ≤ d g ( t ) ( x , y ) ≤ C D d g (0) ( x , y ) − α . (4.8)Notice that the curvature assumption is natural (in view of the pseudo-locality theorem) andthe comparison with the initial time slice is the key di ffi culty — for positive time slices the Riccicurvature bound makes the estimate trivial. Another handy distance distortion estimate for theapplication of smoothing the collapsing initial is the following Lemma 4.8.
For any α ∈ (0 , , there is a positive quantity Ψ D ( α | m ) with lim α → Ψ D ( α | m ) = ,such that under the assumption of Theorem 4.2 or Theorem 4.3, for any x , y ∈ B g ( p , and anyt ∈ (0 , ε P ( m , α )] , if d g ( x , y ) ≤ √ t, then we have (cid:12)(cid:12)(cid:12) d g ( t ) ( x , y ) − d g ( x , y ) (cid:12)(cid:12)(cid:12) ≤ Ψ D ( α | m ) √ t . (4.9)This lemma is a slight re-wording of [49, Lemma 1.11], which concerns non-collapsing initialdata. See also [54, Lemma 4.1] for a proof.4.4. Applications of the Ricci flow local smoothing technique.
With the Ricci flow smooth-ing tool kit at hand (the flow existence time lower bounds and the distance distortion estimates),we could in many cases reduce our consideration of collapsing manifolds with Ricci curvaturebounded below to the classical collapsing geometry with bounded sectional curvature.Locally, one could obtain infranil fiber bundle structure around points where the Ricci flowsmoothing results (Theorem 4.2 and Theorem 4.3) apply:
OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 17
Theorem 4.9.
There is a positive constant δ = δ ( m ) such that if ( M , g ) is an m-dimensional com-plete Riemannian manifold with Rc g ≥ − ( m − g, then for any p ∈ M which has a geodesic ballsatisfying d GH (cid:16) B g ( p , , B k (2) (cid:17) < δ and one of the following conditions: (1) p is a ( δ, -Reifenberg point [49] , or (2) rank Γ δ ( p ) = m − k [54] ,there is an open neighborhood U of p such that B g ( p , − δ ) ⋐ U ⋐ B g ( p , + δ ) , and Uis di ff eomorphic to an infranil fiber bundle over B k (1) , with the extrinsic diameter of the fibersbounded above by δ .Remark . In fact, as shown in [54, Theorem 1.4], there is a positive constant δ = δ ( m , l ) < p ∈ M has a geodesic ball satisfying d GH (cid:16) B g ( p , , B k (2) / G (cid:17) < δ for some G < O ( k ) with | G | ≤ l , rank Γ δ ( p ) = m − k , and there exists a surjective φ ∈ Hom ( Γ δ ( p ) , G ), then the same infranilfiber bundle structure over the orbifold neighborhood B k (1) / G can be obtained. One can of coursereplace the assumption on the nilpotency rank with the Reifenberg property as in Item (1). Seealso [36, §
7] for related concepts.Theorem 4.9 generalizes Naber and Zhang’s ε -regularity theorem (Theorem 2.9) from the caseof manifolds with bounded Ricci curvature to manifolds with Ricci curvature only bounded frombelow. It is also a localization of [49, Theorem B]. To prove this theorem, we first notice thatfor any α ∈ (0 ,
1) su ffi ciently small the assumptions enable us to run a Ricci flow with the localinitial data for a definite period of time, and obtain a smoothing metric g ( ε ) which is regular;by the distance distortion estimate Lemma 4.8, we know that up to scale ε , the original metricstructure defined by g is Ψ ( α )-Gromov-Hausdor ff close to the metric structure defined by g ( ε );therefore, since the domain ( B g ( p , , g ) is δ -Gromov-Hausdor ff close to B k (1), we know that thedomain ( B g ( p , ) , g ( ε )) is δ + Ψ ( α ) ε -Gromov-Hausdor ff close to B k (1) on scales up to ε ; but thenthe regularity of the metric g ( ε ) allows us to appeal to the classical theory of collapsing geometry([14, Theorem 2.6]) with bounded sectional curvature to obtain the infranil fiber bundle structureover B k (1).Here we would like to emphasize that the classical theorems (e.g. Theorems 2.1 and 2.3, aswell as [14, Theorem 2.6]) on collapsing with bounded sectional curvature essentially describe agap phenomenon, rooted back in Gromov’s almost flat manifold theorem (Theorem 1.1): when themanifold is su ffi ciently Gromov-Hausdor ff close to a lower dimensional space, then the manifolditself already acquires some non-trivial symmetry. Such a gap phenomenon allows us to slightlyperturb the given metric locally to one with much better regularity, but remains to be su ffi cientlycollapsed (in the metric sense) so that the symmetry structure could still be observed.The Ricci flow local smoothing results can also help with proving global results when the col-lapsing limit is singular. In particular, we make the following Conjecture 4.10.
Given D ≥ , m , l ∈ N and ι > there is an ε ( m , l , ι ) > such that if ( M , g ) ∈ M Rc ( m , D ) and an ( l , ι ) -controlled k-dimensional Riemannian orbifold ( X , d X ) satisfythe conditions d GH ( M , X ) < ε and b ( M ) − b ( X ) = m − k, then M is a torus bundle over X. Here by saying the Riemannian orbifold ( X , d X ) is ( l , ι )- controlled we mean that for any x ∈ X , B d X ( x , ι ) ≡ B k / G x with the order of the orbifold group G x < O ( k ) bounded above by l .5. C ollapsing R icci - flat K¨ ahler metrics with bounded curvature Combining the classical theory of collapsing geometry with bounded curvature and the Ricciflow smoothing technique, in this section we prove the following
Theorem 5.1.
Given a closed Calabi-Yau manifold ( M , ¯ g , ¯ J ) , there is a constant β ( M , ¯ g , ¯ J ) > such that if β ( M , ¯ g , ¯ J ) < , then (1) there is a Ricci-flat K¨ahler metric g (together with a compatible complex structure J), suchthat k g − ¯ g k C k ( M , ¯ g ) < ¯ η for some ¯ η ∈ (0 , solely determined by ¯ g; (2) there are a Ricci-flat orbifold X and a Riemannian submersion f : M → X with respect tog, such that the fibers are totally geodesic tori (see [36, § for related definitions), and gis invariant under the trous action. The idea here is that for the given data ( M , ¯ g , ¯ J ) which is su ffi ciently volume collapsed withbounded sectional curvature, by the work of Cheeger, Fukaya and Gromov (see [74, Theorem5.1]) there is a C approximating metric. On the other hand, if the perturbation is su ffi cientlysmall, then by the work of Dai, Wang and Wei [29] the Ricci flow starting from the perturbedmetric converges back to a Ricci-flat K¨ahler metric. The issue here is that Cheeger, Fukaya andGromov’s approximating metric is only in a C neighborhood of ¯ g , but Dai, Wang and Wei’s sta-bility result requires much higher regularity for the neighborhood. The new input here is thereforethe regularity improvement (Theorem 5.2), which finds a C neighborhood of ¯ g where the Ricciflow is stable. The quantity β ( M , ¯ g , ¯ J ) is then defined as ¯ η − | M | ¯ g diam( M , ¯ g ) − m for some ¯ η to bespecified later. If β ( M , ¯ g , ¯ J ) <
1, then the manifold is very (volume) collapsed with respect to the C stability radius ¯ η of ¯ g .While the assumption on the volume collapsing is rather strong, in that β ( M , ¯ g , ¯ J ) depends onthe specific K¨ahler manifold, the existence of an invariant critical metric drastically reduces thetopological complexity of the manifold: it is a torus bundle over a Ricci-flat orbifold. The invariantmetric allows us to apply the O’Neill’s formula [66], together with the central density (see [51, § corner singularities of the collapsing limitspace, and following the arguments in [61, 64] we can show that the collapsing fibers must be toriand the fibration must locally be a Riemannian product, implying the Ricc-flatness of the collapsinglimit.5.1. Existence of invariant Ricci-flat K ¨ahler metric.
In this sub-section we prove the first claimin Theorem 5.1. By the stability result [29] of Dai, Wang and Wei for Ricci-flat K¨ahler metrics,we know that for the Calabi-Yau manifold ( M , ¯ g , ¯ J ), there are some positive constants ¯ η < ¯ η < g , such that if g is another smooth Riemannian metric with k g − ¯ g k C k ( M , ¯ g ) < ¯ η ,then the Ricci flow with initial data g exists for all time and converges to a Ricci-flat K¨ahler metricin B C k ( M , ¯ g ) ( ¯ g , ¯ η ). Here k : = l ¯ η − m is solely determined by ¯ g . Since k may be a very large number,our first priority is to prove the following regularity improvement result. Theorem 5.2.
There is a positive constant ¯ η < ¯ η determined by ¯ g, such that if k g − ¯ g k C ( M , ¯ g ) < ¯ η ,then the Ricci flow starting from g exists for all time and converges to a Ricci-flat K¨ahler metric inB C k ( M , ¯ g ) ( ¯ g , ¯ η ) . OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 19
Before proving Theorem 5.2, we state the following slight variant of Lemma 4.8:
Lemma 5.3.
There is a positive constant ¯ η ′ < determined by ¯ g such that if g ∈ B C ( M , ¯ g ) ( ¯ g , ¯ η ′ ) ,and the Ricci flow g ( t ) with initial data g satisfies max M (cid:12)(cid:12)(cid:12) Rc g ( t ) (cid:12)(cid:12)(cid:12) g ( t ) < α t − for t ∈ [0 , T ) and some α ∈ (0 , , then there is a Ψ ′ D ( α | ¯ g ) > with lim α → Ψ ′ D ( α | ¯ g ) = such that for any x , y ∈ M withd g ( x , y ) ≤ √ t, we have (cid:12)(cid:12)(cid:12) d g ( x , y ) − d g ( t ) ( x , y ) (cid:12)(cid:12)(cid:12) ≤ Ψ ′ D ( α | ¯ g ) √ t . (5.1)The proof of this lemma is the same as that of [21, Lemma 2.10]: in the contradiction argumentinvolved, the rescaling limit will be exactly R n as the quantity Ψ ′ D ( α | ¯ g ) depends on ¯ g . With suchdistance distortion estimate at our disposal, we now prove the C stability around a stable Ricci-flatK¨ahler metric. Proof of Theorem 5.2.
We notice that since M is a closed manifold, for any smooth Riemannianmetric g there is a Ricci flow solution with initial data g . Moreover, for any α >
0, there is alwaysome ε > α and ( M , g )) such that the following curvature bound is valid for t ∈ (0 , ε ]: (cid:13)(cid:13)(cid:13) Rm g ( t ) (cid:13)(cid:13)(cid:13) C ( M , g ( t )) ≤ α t − . (5.2)We now make the following claim regarding the Ricci flow solution: Claim 5.4.
There are some α ∈ (0 , and δ ∈ (0 , ¯ η ′ ) depending on ( M , ¯ g ) such that if k g − ¯ g k C ( M , ¯ g ) < δ and ∀ t ≤ ε , (cid:13)(cid:13)(cid:13) Rm g ( t ) (cid:13)(cid:13)(cid:13) C ( M , g ( t )) ≤ α t − , then (cid:13)(cid:13)(cid:13) g ( ε ) − ¯ g (cid:13)(cid:13)(cid:13) C k ( M , ¯ g ) < ¯ η .Proof of the claim. We now prove this claim via a contradiction argument: if the theorem fails, wemay find a sequence of smooth Riemannian metrics g i on M and sequences of positive numbers α i → δ i →
0, such that k g i − ¯ g k C ( M , ¯ g ) = δ i → i → ∞ , the Ricci flow solutions g i ( t )satisfy ∀ t ≤ ε i , (cid:13)(cid:13)(cid:13) Rm g i ( t ) (cid:13)(cid:13)(cid:13) C ( M , g i ( t )) ≤ α i t − . (5.3)for some ε i ∈ (0 , (cid:13)(cid:13)(cid:13) g i ( ε i ) − ¯ g (cid:13)(cid:13)(cid:13) C k ( M , ¯ g ) ≥ ¯ η for all i su ffi ciently large.From the contradiction hypothesis we may find points p i ∈ M such that for all i large enough, (cid:12)(cid:12)(cid:12) g i ( ε i ) − ¯ g (cid:12)(cid:12)(cid:12) C k ( p i ) ≥ ¯ η . (5.4)On the other hand, by the curvature control (5.3) and Shi’s estimates we have for any l ∈ N that (cid:13)(cid:13)(cid:13)(cid:13) ∇ l Rm g i ( ε i ) (cid:13)(cid:13)(cid:13)(cid:13) C ( M , g i ( ε i )) ≤ C l ε − − li , (5.5)with C l > m for l >
1, and C = α i . Moreover, applying Lemma 5.3 with(5.3) we have the distance distortion estimate for any x , y ∈ M with d g i ( x , y ) ≤ ε i : (cid:12)(cid:12)(cid:12)(cid:12) d g i ( x , y ) − d g i ( ε i ) ( x , y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ψ ′ D ( α i | ¯ g ) ε i . (5.6) We now rescale the metrics so that ε i
2, setting h i ( t ) : = ε − i g i ( ε − i t ) for t ∈ [0 , ε i ] and¯ h i : = ε − i ¯ g , and estimate for all x , y ∈ M with d h i (0) ( x , y ) ≤
2, that (cid:12)(cid:12)(cid:12) d ¯ h i ( x , y ) − d h i (1) ( x , y ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) d ¯ h i ( x , y ) − d h i (0) ( x , y ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) d h i (0) ( x , y ) − d h i (1) ( x , y ) (cid:12)(cid:12)(cid:12) ≤ Ψ ( δ i ) + Ψ ′ D ( α i | ¯ g ) , which approaches 0 as i → ∞ .Consequently, the we have the pointed Gromov-Hausdor ff distance estimate d pGH (cid:16) B h i (1) ( p i , , B ¯ h i ( p i , (cid:17) < Ψ ( δ i ) + Ψ ′ D ( α i | ¯ g ) , (5.7)and such Gromov-Hausdor ff distance bounds are realized by the identity map.Since { ε i } ⊂ [0 ,
1] and { p i } ⊂ M , by the compactness we may pass to convergent sub-sequences,still denoted by { ε i } and { p i } respecitvely. We denote lim i ε i = ε and lim i p i = p ∈ M . There aretwo possibilities: either ε i → ε > ε = ε >
0, we notice that ¯ h i = ε − i ¯ g → ε − ¯ g smoothly as i → ∞ . Moreover, since M is compact,the collection of geodesic balls n B ¯ h i ( p i , o has uniformly bounded geometry — the geometry isultimately bounded by that of B ε − ¯ g ( p , ⊂ M . Consequently, there is a smooth limit metric ¯ h ∞ such that as i → ∞ , ( B ¯ h i ( p i , , ¯ h i ) pCG −−−→ ( B ε − ¯ g ( p , , ¯ h ∞ ) . Here pCG means pointed Cheeger-Gromov (smooth) convergence. On the other hand, by (5.7), itis clear that as i → ∞ , ( B h i (1) ( p i , , h i (1)) pGH −−−→ ( B ε − ¯ g ( p , , ¯ h ∞ ) . However, since the metrics { h i (1) } has uniformly controlled regularity (5.5), this last conver-gence must also be in the pointed Cheeger-Gromov sense. Say, the limit metric h ∞ is definedon B ε − ¯ g ( p ,
1) and we have h ∞ ≡ ¯ h ∞ . But the pointed Cheeger-Gromov convergence also implies,by (5.4), that (cid:12)(cid:12)(cid:12) h ∞ − ¯ h ∞ (cid:12)(cid:12)(cid:12) C k ( p ) = lim i →∞ (cid:12)(cid:12)(cid:12) h i (1) − ¯ h i (cid:12)(cid:12)(cid:12) C k ( p i ) ≥ ¯ η , and this contradicts the conclusion h ∞ ≡ ¯ h ∞ .If ε =
0, as ¯ h i is nothing but the metric ¯ g scaled up — by the compactness of M , and that ε i → i → ∞ , B ¯ h i ( p i , pCG −−−→ B m (1). By (5.7) we then have B h i (1) ( p i , pGH −−−→ B m (1) as i → ∞ . However, the uniform regularity control (5.5) improves the convergence also to pointedsmooth Cheeger-Gromov convergence, i.e. as i → ∞ we in fact have( B h i (1) ( p i , , h i (1)) pCG −−−→ ( B m (1) , h ∞ ) , with the limit smooth metric h ∞ ≡ g Euc . But by (5.4) and (5.5) we have | h ∞ − g Euc | C k ( o ) = lim i →∞ (cid:12)(cid:12)(cid:12) h i (1) − ¯ h i (cid:12)(cid:12)(cid:12) C k ( p i ) ≥ ¯ η , which is a contradiction. (cid:3) We now return to the proof of the theorem. Fix the α and δ = : ¯ η determined by ¯ g through theclaim, then for any smooth Riemannian metric g with k g − ¯ g k C ( M , ¯ g ) < ¯ η , we see that the evolved OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 21 metric g ( ε ) ∈ B C k ( M , ¯ g ) ( ¯ g , ¯ η ), and thus the Ricci flow continuing from g ( ε ) exists for all time andconverges to a Ricci-flat K¨ahler metric in B C k ( M , ¯ g ) ( ¯ g , ¯ η ). (cid:3) The proof of the first claim in Theorem 5.1 is now a simple combination of Theorem 5.2 andTheorem 2.3.
Proof of Theorem 5.1 Claim (1).
By Theorem 2.3 we may find some positive constant ¯ η < ε ( m )determined by ¯ g with Ψ N ( ¯ η | m ) < ¯ η , and set β ( ¯ g ) : = ¯ η − | M | ¯ g diam( M , ¯ g ) − m . If β ( ¯ g ) <
1, sincemax M (cid:12)(cid:12)(cid:12) K ¯ g (cid:12)(cid:12)(cid:12) ¯ g ≤
1, the original work of Cheeger, Fukaya and Gromov enables us to find an approx-imating metric g ′ ∈ B C ( M , ¯ g ) ( ¯ g , ¯ η ) which is ( ρ, k )-round and compatible with a nilpotent Killingstructure N whose orbits are of diameter less than ¯ η . Now by Theorem 5.2 and the fact that g ′ ∈ B C ( M , ¯ g ) ( ¯ g , ¯ η ), the Ricci flow starting from g ′ exists for all time and converges to a Ricci-flatK¨ahler metic g : = g ′ ( ∞ ) ∈ B C k ( M , ¯ g ) ( ¯ g , ¯ η ). Since the Ricci flow is intrinsic, the infinitesimal isome-tries are preserved (i.e. it preserves the Killing vector fields). Consequently, N is still a nilpotentKilling structure compatible with g , which is our desired metric. (cid:3) Reduction of the di ff eomorphism type. In this sub-section we prove the second claim ofTheorem 5.1. Under the assumption that β ( M , ¯ g , ¯ J ) < η , we see that M admits a nilpotent Killing structure N . Notice that N is compatible with g in that the local nilpotentgroup actions on M are isometric with respect to g . In fact, since M is very collapsed with boundeddiameter, the structure is pure , i.e. there is a single nilpotent Lie algebra n such that the germ ofthe acting Lie group at every point has its identity component generated by n . Moreover, this givesus a singular Riemannian submersion f : M → X over some collapsing limit space ( X , d X ) whosetopologcial structure is described by Theorem 2.2.In fact, the nilpotent Killing structure N determines a (singular) Riemannian foliation N (see[63]), defined by the distribution of Killing vector fields tangent to the orbits of the structure N .Clearly, the leaf of N passing through p is O ( p ), the orbit of p under the nilpotent Killing structure n , and it is also a component of the fiber f − ( f ( p )). Within N we can define the central distribution F , consisting of the center C ( N p ) E N p at every p ∈ M . This defines another (singular) Riemann-ian foliation by the Frobinius theorem — in fact, this defines an F -structure F a la Cheeger andGromov [16, 17].The leaf space of N (or equivalently the orbit space of N ) is isometric to the collapsing limit( X , d X ), and the leaf space of F is isometric to a metric space ( W , d W ). Recalling our descriptionsin § X = ˜ R ⊔ ˜ S , where ˜ R is an open (incomplete)Riemannian orbifold, and ˜ S consists of corner points , i.e. those x ∈ X with dim G x >
0. Forsuch an x , G x has its non-trivial identity component as a torus, and more significantly, for any p ∈ f − ( x ), the infinitesimal action of G x is contained in F p = C ( N p ); see [35, Lemma 5.1]. Let g p denote the Lie algebra of G x for any p ∈ f − ( x ), then g p (cid:27) R d as Lie algebras, with d = x ∈ ˜ R and d ∈ { , . . . , m − dim X } when x ∈ ˜ S .Clearly, N p , the distribution N located at each p ∈ M , is isomorphic to n / g p as Lie algebras,and F p is isomorphic to C ( n ) / g p since g p E C ( n ). Consequently, the foliation N is a Riemannianfoliation (i.e. non-singular) if and only if ˜ S = ∅ (which is also equivalent to saying that thenilpotent Killing structure N is polarized ), and the same conclusion holds for F and F . In fact,by [35, Theorem 0.5], the possibly singular Riemannian foliations N and F are always linearized;see [63]. We will denote k ′ = dim C ( n ) ≤ m − dim X . The N (and thus F ) invariant metric g , when restricted to a leaf of F , defines a non-negativedefinite 2-tensor field G : for any two vector fields ξ, ζ ∈ Γ ( F , M ), G ( ξ, ζ ) : = g ( ξ, ζ ), which is leftinvariant along the leaves of F . Consequently, the central density — det G — is a non-negativebasic function for F , i.e. it descends to a non-negative function on W ; see also [51]. Clearly,det G is smooth around the regular leaves of F , i.e. those leaves whose tangents are isomorphicto C ( n ). By the previous discussion on F p for p ∈ f − ( ˜ S ), we see that det G vanishes exactly on f − ( ˜ S ), i.e. { p ∈ M : det G ( p ) = } = f − ( ˜ S ). This is because at p ∈ f − ( ˜ S ) we can extend G by 0 on F p ⊕ g p (cid:27) C ( n ); compare also [34, Theorem 0.6]. Notice that G induces a left invariantRiemannian metric on the leaves of F ; but since F consists of commuting vector fields, G isbi-invariant and actually flat along the leaves of F .In the formulas below, we will use the Roman letters i , j , k , l to index the coordinates alongthe leaf directions of F , and for directions perpendicular to a leaf of F , we use the Greek letters α, β as indicies. Moreover, we employ the Einstein summation convention, adding the repeatedindicies. As the basic function det G is constant along the leaves of F , if it were not a constantthroughout M , then max M ln det G = max W ln det G is attained at some p ∈ f − ( ˜ R ), since det G ≥ G | f − ( ˜ S ) ≡
0. By the O’Neill’s formula [66] applied to the (singular) Riemannian foliation F in a small enough open neighborhood around p ∈ M and the flatness of the leaves, we havesome locally defined basic 1-form A l αβ such that Rc i j = − (cid:16) G i j ; αα − G ik ,α G jk ,α (cid:17) − G kl G kl ,α G i j ,α + G ik G jl A k αβ A l αβ . (5.8)Now tracing by G i j on the leaf directions and by the Ricci-flatness of the metric on M , we see that12 ∆ ⊥ ln det G + (cid:12)(cid:12)(cid:12) ∇ ⊥ ln det G (cid:12)(cid:12)(cid:12) = | A | . (5.9)Here ∇ ⊥ and ∆ ⊥ denote the derivatives taken perpendicular to the leaf directions. However, sinceln det G ( p ) = max M ln det G , we must have ∆ ⊥ ln det G ( p ) < ∇ ⊥ ln det G ( p ) = G isa positive constant. This implies that ˜ S = ∅ , i.e. X = ˜ R is a Riemannian orbifold. Moreover, F isa Riemannian foliation — the orbit space W of F is also a Reimannian orbifold.Clearly, G i j descends to germs of smooth functions on the regular part of W ; if w ∈ W is asingularity, it can only be an orbifold point and we may pull the corresponding quantities back toits local orbifold covering — the di ff erentials of G i j are well-defined throughout W . Notice that ∇ W G i j = ∇ ⊥ G i j and ∆ W G i j = ∆ ⊥ G i j . Moreover, by (5.9) the constancy of ln det G ensures that | A | ≡ X , and thus by (5.8), ∆ W G i j = ∆ ⊥ G i j = G ik ,α G jk ,α . (5.10)If we further check the O’Neill’s formula for the regular part of W , the vanishing of (cid:12)(cid:12)(cid:12) ∇ ⊥ ln det G (cid:12)(cid:12)(cid:12) and | A | , together with the Ricci-flatness of M tell that( Rc W ) αβ = G i j ,α G i j ,β . (5.11)We notice that the quantity |∇ W G | is a globally defined non-negative smooth function on W . Bythe compactness of W , if |∇ W G | .
0, then we have ∆ W |∇ W G | ( w ) < w ∈ W . On the other hand, since det G is a constant, we may view the h G i j i as a matrix valued map G : U → S L ( k ′ , R ) / S O ( k ′ ), with U ⊂ W being an open neighborhood of w where we can write OLLAPSING WITH RICCI CURVATURE BOUNDED BELOW AND RICCI FLOWS 23 down G in coordinates. We now notice that the codomain S k ′ : = S L ( k ′ , R ) / S O ( k ′ ) is a negativelycurved symmetric space. Now by the Bochner formula, we can calculate at w ∈ U to see ∆ W |∇ W G | ( w ) = |∇ W ∇ W G | ( w ) + ( Rc W ) αβ G i j ,α G i j ,β ( w ) − (cid:16) G ∗ Rm S k ′ (cid:17) αββα ( w ) > |∇ W G | ( w ) > , which is impossible. Therefore, we see the tensor ∇ ⊥ G ≡ M . Consequently, we see that(5.11) reduces to Rc W ≡
0. Moreover, as ∇ ⊥ G stands for the second fundamental form of theleaves of F , its vanishing tells that the the leaves of F are totally geodesic. So the Riemannianmetric g locally splitts, and by the left invariance of g with respect to N , this implies the splittingof the nilpotent Killing structure, i.e. passing through each point p ∈ M , we have the splitting ofLie algebras N p = F p ⊕ N ′ p for some nilpotent Lie algebra N ′ p . This however leads to F = N :otherwise, N ′ p , p ∈ M , and by the nilpotency it has to have a non-trivial center C ( N ′ p ),but the above splitting shows that F p = C ( N p ) = F p ⊕ C ( N ′ p ), which is absurd. Consequently,we have F = N , X ≡ W as Ricci-flat orbifolds, and that M fibers over X by flat tori with totallygeodesic fibers (the leaves of N ). R eferences [1] Richard H. Bamler and Qi S. Zhang, Heat kernel and curvature bounds in Ricci flows with bounded scalarcurvature. Adv. Math.
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Camb. J. Math. haosai H uang , D epartment of M athematics , U niversity of W isconsin - M adison , 480 L incoln D rive , M adison ,WI, 53706, U.S.A. E-mail address : [email protected] X iaochun R ong , D epartment of M athematics , R utgers U niversity , 110 F relinghuysen R oad , P iscataway , NJ,08854, U.S.A. E-mail address : [email protected] B ing W ang , I nstitute of G eometry and P hysics , and S chool of M athematical S ciences , U niversity of S cience and T echnology of C hina , 96 J inzhai R oad , H efei , A nhui P rovince , 230026, C hina E-mail address ::