Collapsed Ricci limit spaces as non-collapsed RCD spaces

In this short note we provide several conjectures on the regularity of measured Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature bounded below, from the point of view of the synthetic treatment of lower bounds on Ricci curvature for metric measure spaces. Convergence theory of Riemannian manifolds Let (M i , gi) be a sequence of n-dimensional complete Riemannian manifolds with the pointed Gromov-Hausdorff (pGH) convergence: (M i , dgi , xi) pGH −→ (X, d, x) (0.1) to some pointed metric space (X, d, x). Then the convergence theory of Riemannian manifolds states that under suitable curvature restriction: (⋆) Establish regularity results on (X, d). (⋆⋆) Find relationships between M i and X. In this short note we provide several conjectures related to the problem (⋆) under lower Ricci curvature bounds. Before introducing them, let us start to discuss the case of sectional curvature shortly in order to clarify the difference from the case of Ricci curvature. Lower bound on sectional curvature Let us consider (0.1) in the case when the sectional curvature of (M i , gi) is bounded below by a constant K ∈ R: Seci M i ≥ K. (0.2) Then (X, d) is a k-dimensional Alexandrov space of curvature bounded below by K for some k ∈ [0, n] ∩ N, which is a direct consequence of the stability of Alexandrov spaces with respect to the pGH convergence proved in [BurGrPer92]. In particular nice geometric properties of (X, d) are carried from Alexandrov geometry which gives the best framework on the synthetic treatment of lower bounds on sectional curvature for metric spaces. For example any point p of X has a neighbourhood Up of p which is homeomorphic to the tangent cone at p, which is proved in [BurGrPer92, Per91] (see also [Kap07]). This gives a geometric answer to the problem (⋆). The fibration theorem proved in [Yam91] also gives a geometric answer to the problem (⋆⋆). ∗Tohoku University, shouhei.honda.e4@tohoku.ac.jp. The author acknowledges supports of the Grantin-Aid for Young Scientists (B) 16K17585 and Grant-in-Aid for Scientific Research (B) of 18H01118. This paper is submitted to a special issue of SIGMA on scalar and Ricci curvature in honor of Misha Gromov’s 75th birthday.

to some pointed metric space (X, d, x). Then the convergence theory of Riemannian manifolds states that under suitable curvature restriction: (⋆) Establish regularity results on (X, d).
(⋆⋆) Find relationships between M n i and X. In this short note we provide several conjectures related to the problem (⋆) under lower Ricci curvature bounds. Before introducing them, let us start to discuss the case of sectional curvature shortly in order to clarify the difference from the case of Ricci curvature.

Lower bound on sectional curvature
Let us consider (0.1) in the case when the sectional curvature of (M n i , g i ) is bounded below by a constant K ∈ R: Then (X, d) is a k-dimensional Alexandrov space of curvature bounded below by K for some k ∈ [0, n] ∩ N, which is a direct consequence of the stability of Alexandrov spaces with respect to the pGH convergence proved in [BurGrPer92]. In particular nice geometric properties of (X, d) are carried from Alexandrov geometry which gives the best framework on the synthetic treatment of lower bounds on sectional curvature for metric spaces. For example any point p of X has a neighbourhood U p of p which is homeomorphic to the tangent cone at p, which is proved in [BurGrPer92,Per91] (see also [Kap07]). This gives a geometric answer to the problem (⋆). The fibration theorem proved in [Yam91] also gives a geometric answer to the problem (⋆⋆).

Lower bound on Ricci curvature
Next let us consider (0.1) in the case: It is trivial that (0.2) implies (0.3). After passing to a subsequence by [ChCol97,Fuk87], with no loss of generality we can assume that the pointed measured Gromov-Hausdorff (pmGH) convergence holds: for some Borel measure m on X. Then (X, d, m) is so-called a Ricci limit space. The structure theory of Ricci limit spaces is established in [ChCol97,ChCol00a,ChCol00b]. For example, the Laplacian ∆ on (X, d, m) is well-defined via the rectifiablity as a metric measure space. Moreover the spectrums behave continuously with respect to the convergence (0.4) if (X, d) is compact, which confirms a conjecture raised in [Fuk87]. This gives an analytic answer to the problem (⋆⋆).
On the geometric side, in general, similar fibration result as in [Yam91] is not satisfied in this setting. A counter example can be found in [And92]. However we know that if (X, d) is compact with no singular set, and (0.4) is non-collapsed (as explained below), then X is homeomorphic to M n i for any sufficiently large i, which is proved in [ChCol97]. This gives a geometric answer to the problem (⋆⋆).
Let us consider the problem (⋆). It is proved in [ChCol97] that the same geometric property as in the previous section also holds for regular points if the sequence (0.4) is non-collapsed whose definition is to satisfy m = aH n for some a ∈ (0, ∞), where H n is the n-dimensional Hausdorff measure. That is, if m = aH n , then any n-dimensional regular point p of X has a neighbourhood U p of p which is homeomorphic to R n . 1 This gives a geometric answer to the problem (⋆). See also [ChJianNab00b] for a recent development along this direction.
However if the sequence (0.4) is collapsed, then such a nice geometric property is unknown. Although one of the central topics in this story is to develop the local structure theory in general situation, it is still very hard.
In connection with these observations, it is also interesting to ask: ( * ) When can we find other non-collapsed sequence of k-dimensional complete Riemannian manifolds (M k i ,ĝ i ) with Ricci curvature bounded below by a constant such that holds as a non-collapsed sequence (even if the sequence (0.4) is collapsed)?
In general this question ( * ) has a negative answer. A counter example can be found in [ChCol97] as a "metric horn" which will be discussed later. See also [Hat17,Men00b,1 We say that a point p is n-dimensional regular if any tangent cone at p is isometric to (R n , d R n , 0n). In general tangent cones at a point are not unique even in the non-collapsed setting. More strongly, there exists a 5-dimensional non-collapsed Ricci limit space with a base point p such that there exist two tangent cones at p which are not homeomorphic to each other. See [ColNab13]. Thus it is hard to find nice topological results around singular points, which is very different from the case of sectional curvature (0.2). See also [Men00a]. Men01] for other examples along this direction. Let us emphasize that if (0.5) holds (as a non-collapsed sequence), then m = bH k holds for some b ∈ (0, ∞).
We are now in a position to provide the first conjecture: The synthetic condition of lower bounds on Ricci curvature for metric measure spaces, RCD(K, N ) condition, is explained in the next section.
Let us compare Conjecture 0.1 with the following conjecture raised in [ChCol97]: Conjecture 0.2 (Cheeger-Colding). It holds that for all y ∈ X, Vol m (r) denotes the volume of a ball of radius r in the m-dimensional space form whose sectional curvature is equal to −1.
In connection with Conjecture 0.2, it is natural to ask: Because if this question ( * * ) has a positive answer for some suitable K 1 , N 1 , then Conjecture 0.2 holds by the Bishop-Gromov inequality in the RCD theory. However this question ( * * ) has a negative answer for a metric horn. See the next section.

Synthetic treatment of lower bound on Ricci curvature
In order to keep the short presentation, we skip the precise definition of RCD(K, N ) space. Roughly speaking, a metric measure space (X, d, m) 2 is said to be a RCD(K, N ) space (or RCD space for short) if the Ricci curvature is bounded below by K ∈ R, the dimension is bounded above by N ∈ [1, ∞], and it has the Riemannian structure in some sense. See [AmbGigSav14b,ErKuwSt15,Gig13,LottVill09,St06a,St06b] for the precise definitions 3 (see also [AmbMonSav19,CavMil16]). It is worth pointing out that any Ricci limit space as obtained by (0.4) is a RCD(K(n − 1), n) space by the stability of RCD(K, N ) spaces with respect to the pmGH convergence proved in [GigMonSav13], and that any n-dimensional Alexandrov space of curvature bounded below by K with H n is also a RCD(K(n − 1), n) space, which is proved in [Pet11,ZhanZh10]. From now on we fix K ∈ R and a finite N < ∞. Thanks to recent quick developments on the study of RCD(K, N ) spaces, most of well-known properties on Ricci limit spaces can be covered by the RCD theory. For example, it is proved in [BrSem18] that the essential dimension, denoted by dim (X, d, m), 4 whose definition is the unique k such that the kdimensional regular set has positive m-measure, is well-defined. This gives a generalization of a result proved in [ColNab12] to RCD(K, N ) spaces.
On the other hand, a special class of RCD(K, N ) spaces, so-called non-collapsed RCD(K, N ) spaces, is proposed in [DePhGig18] as the synthetic counterpart of noncollapsed Ricci limit spaces. A RCD(K, N ) space is said to be non-collapsed if m = H N holds. Let us emphasize that it is essential that the upper bound N of dimension (as RCD spaces) coincides with the dimension N of the Hausdorff measure H N in the definition of the non-collapsed condition. 5 Then non-collapsed RCD(K, N ) spaces have nicer properties, including that for non-collapsed Ricci limit spaces, rather than general RCD(K, N ) spaces. For instance, any N -dimensional regular point p has a neighbourhood U p of p which is homeomorphic to R N (see also [KapMon19]).
Let us mention that dim(X, d, m) is equal to N if (X, d, m) is a non-collapsed RCD(K, N ) space. It is conjectured in [DePhGig18] that the converse implication is also true up to multiplication by a positive constant to the measure, that is: This conjecture is true if (X, d) is compact, which is proved in [Hon19]. Note that since the RCD(K, N )-condition is unchanged under multiplication by a positive constant to the measure, if (X, d, aH N ) is a RCD(K, N ) space, then (X, d, H N ) is a non-collapsed RCD(K, N ) space.
Conjecture 0.1 can be formulated in the RCD setting as follows: Note that the implication from Conjectures 0.4 to 0.1 is trivial by letting n = N . Let us recall some structure results on the reference measure m. Based on results obtained in [MonNab19,DePhMarRin17,GigPas16,KellMon18], it is proved in [AmbHonTew18] that for any RCD(K, N ) space (X, d, m) with k = dim(X, d, m), the limit exists in (0, ∞) for m-a.e. x ∈ X (we denote by R * k the set of all such points x ∈ X) and coincides with the Radon-Nikodym derivative of the restricition of m to R * k with respect to H k , where ω k = H k (B 1 (0 k )). In particular we see that (0.7) implies k = dim(X, d, m).
5 In general, the optimal dimension of (X, d, m) as RCD spaces is diferent from the Hausdorff dimension. See also [HonSunZhang19].
One implication, from (a) to (b), is equivalent to Conjecture 0.3. That is, if (a) holds, then it follows from a result of [BrSem18], which confirms a conjecture raised in [DePhGig18], that (X, d, m) is a RCD(K, k) space, where k = dim(X, d, m), in particular, Conjecture 0.3 yields (b).
Conversely, thanks to a result of [Han18], the implication from (b) to (a) is true if Conjecture 0.4 holds. It is worth pointing out that under assuming that (X, d) is compact with (b), it is proved in [Hon19] that (X, d, H k ) is a non-collapsed RCD(K, k) space if and only if inf x∈X,r∈(0,1) holds. Thus in the case when (X, d) is compact, the remaining issue for Conjecture 0.5 is only to prove (0.10) under assuming (0.7). Finally let us go back to the question ( * * ) appeared in the previous section. It is constructed in [ChCol97] that for any sufficiently small ǫ > 0 there exists a sequence of complete Riemannian metrics g i on R 8 with positive Ricci curvature such that holds for some Borel measure ν ǫ , where p is the cusp and g S 4 is the standard Riemannian metric on S 4 . This limit space is called a metric horn. Let us denote it by (X ǫ , d ǫ , p, m ǫ ). From now on we will check that for any N 1 ) space). Before proving it, let us remark by definition of h ǫ that we have Assume that (X ǫ , d ǫ , H 5 ) is a RCD(K 1 , ∞) space for some K 1 ∈ R. Note that it is easy to see that an open set B R (p) \ B r (p) for all r, R ∈ (0, ∞) with r < R is isometric as Riemannian manifolds to an open subset of a closed Riemannian manifold (by gluing "two caps" along the boundary). In particular the localities of the minimal relaxed slope, of the Laplacian and of the Hessian (see [Gig18]) yield Xǫ ∇f, ∇ϕ dH 5 = − Xǫ tr(Hess f )ϕ dH 5 , ∀f ∈ D(∆), ∀ϕ ∈ C ∞ c (X ǫ \ {p}). (0.13) Thus we see that ∆f (x) = tr(Hess f )(x) for H 5 -a.e. x ∈ X ǫ . Then it follows from the Bochner inequality proved in [Gig18] that in the weak sense: This shows that (X ǫ , d ǫ , H 5 ) is a RCD(K 1 , 5) space, that is, it is non-collapsed. In particular the Bishop-Gromov inequality yields lim inf r→0 + H 5 (B r (p)) r 5 > 0 (0.15) which contradicts (0.12).

Regularity on reference measure
Let (X, d, m) be a RCD(K, N ) space and let f be a function defined on a Borel subset A of X. We say that f is differentiable for m-a.e. x ∈ A if there exists a family of Borel subset {A i } i∈N of A such that m(A \ i A i ) = 0 holds and that f | A i is Lipschitz for any i. See [AmbBrTr18,Hon14]. Typical examples can be found in Sobolev functions, that is, wee see that any g ∈ H 1,p (U ) for some open subset U of X and some p ∈ [1, ∞] is differentiable for m-a.e. x ∈ U via the standard telescopic argument. 6 It is trivial from [Gig18] that if f is differentiable for m-a.e. x ∈ A with m(X \ A) = 0, then ∇f is well-defined in L 0 (T X), that is, ∇f is a Borel measurable vector field on X.
Let us remark that the implication from Conjecture 0.6 to (0.9) is trivial. A partial contribution to Conjecture 0.6 can be found in [Hon19].
The technique provided in [Hon19] is useful for all conjectures above in the case when (X, d) is compact. This is to apply a geometric flow defined by embedding maps in L 2 via the global heat kernel. 8 Such embedding maps are introduced and studied first in [BerBesGall94] for closed Riemannian manifolds. Recently in [AmbHonPorTew18] this observation is generalized to RCD(K, N ) spaces by using stability results of Sobolev functions with repect to the pmGH convergence proved in [AmbHon17, AmbHon18]. It seems to the author that this technique, using a geometric flow, is useful for all conjectures proposed in this paper even in the case when (X, d) is non-compact.