ON DIMENSIONS OF TANGENT CONES IN LIMIT SPACES WITH LOWER RICCI CURVATURE BOUNDS

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1 ON DIMENSIONS OF TANGENT CONES IN LIMIT SPACES WITH LOWER RICCI CURVATURE BOUNDS VITALI KAPOVITCH AND NAN LI Abstract. We show that if X is a limit of n-dimensional Riemannian manifolds with Ricci curvature bounded below and γ is a limit geodesic in X then along the interior of γ same scale tangent cones T γ(t) X have the same dimension in the sense of Colding-Naber. 1. Introduction In this paper we obtain new restrictions on tangent cones along interiors of limit geodesics in Gromov-Hausdorff limits of manifolds with lower Ricci curvature bounds. Our main technical result is the following Theorem 1.1. For any H R and < δ < 1/3, there exist r (n, δ, H), ε(n, δ, H) > such that the following holds: Suppose that (M n, g) is a complete n-dimensional Riemannian manifold with Ric M (n 1)H and let γ: [, 1] M be a unit speed minimizing geodesic. Then for any t 1, t 2 (δ, 1 δ) with t 1 t 2 < ε and any r < r there are subsets C r i B r (γ(t i )) (i = 1, 2) such that vol C r i vol B r (γ(t i )).9 and there exists a bijective map f r : C r 1 Cr 2 which is 4-Bilipschitz (i.e both f r and fr 1 are 4-Lipschitz). d vol Let d vol i,r = vol B r (γ(t i )) (i = 1, 2) be the normalized volume measures at γ(t i). It s then obvious that under the assumptions of the theorem we have (1.1) C 1 (n) d vol 2,r ( f r ) # (d vol 1,r ) C(n) d vol 2,r for some universal constant C(n). Therefore, using precompactenss and a standard Arzela- Ascoli type argument Theorem 1.1 easily yields Corollary 1.2. Let M n j X where Ric M j (n 1)H. Let γ: [, 1] X be a unit speed geodesic which is a limit of geodesics in M i. Let t 1 (, 1). Then for any t 2 sufficiently close to t 1 there exist subsets C i (i = 1, 2) in the unit ball around the origin o i in the same scale tangent cones T γ(ti )X (i = 1, 2) such that vol i C i.9 and there exists a map f : C 1 C 2 satisfying (i) f is bijective and 4-Bilipschitz; 1991 Mathematics Subject Classification. 53C2. The first author was supported in part by a Discovery grant from NSERC. The second author was supported in part by CUNY PDAC Travel Award. 1

2 2 VITALI KAPOVITCH AND NAN LI (ii) There exists a universal constant C(n) 1 such that C 1 (n) d vol 2 f # (d vol 1 ) C(n) d vol 2. In particular, f # (d vol 1 ) ( f 1 # (d vol 2)) is absolutely continuous with respect to vol 2 (vol 1 ). Here d vol i denotes the renormalized limit volume measure on B 1 (o i ) T γ(ti )X. (In particular, vol i (B 1 (o i )) = 1.) Let X be a limit of n-manifolds with Ricci curvature bounded below. Recall that a point p X is called k-regular if every tangent cone T p X is isometric to R k. The collection of all k-regular points is denoted by R k (X). (When the space X in question is clear we will sometimes simply write R k ). The set of regular points of X is the union (1.2) R(X) k R k (X). The set of singular points S is the complement of the set of regular points. It was proved in [CC97] that vol(s) = with respect to any renormalized limit volume measure d vol on X. Moreover, by [CCb, Theorem 4.15], dim Haus R k k and d vol is absolutely continuous on R k (X) with respect to the k-dimensional Hausdorff measure. In particular, (1.3) dim Haus R k = k if vol(r k ) >. It was further shown in [CN12, Theorem 1.18] that there exists unique integer k such that (1.4) vol(r k ) >. Altogether this implies that there exists unique integer k such that (1.5) vol(x\r k ) =. Moreover, it can be shown (Theorem 1.7 below) that this k is equal to the largest integer m for which R m is non-empty. Following Colding and Naber we will call this k the dimension of X and denote it by dim X. (Note that it is not known to be equal to the Hausdorff dimension of X in the collapsed case). Corollary 1.2 immediately implies Theorem 1.3. Under the assumptions of Corollary 1.2 the dimension of same scale tangent cones T γ(t) X is constant for t (, 1). Proof. For t 2 sufficiently close to t 1 = t let C i B 1 (o i ) (i = 1, 2), f : C 1 C 2 be provided by Corollary 1.2. Let k i = dim T γ(ti )X. Suppose k 1 k 2, say k 1 < k 2. By using (1.5) and Corollary 1.2 (ii) we can assume that C i R ki (T γ(ti )X). By above this means that dim Haus C i = k i. Since f is Lipschitz we have dim Haus ( f (C 1 )) dim Haus C 1 = k 1. Since d vol 2 is absolutely continuous with respect to the k 2 -dim Hausdorff measure on R k2 and k 2 > k 1 this implies that vol 2 f (C 1 ) =. This is a contradiction since vol 2 f (C 1 ) = vol 2 (C 2 ) >. Note that a cusp can exist in the limit space of manifolds with lower Ricci curvature bound, for example, a horn [CC97, Example 8.77]. Theorem 1.3 indicates that a cusp cannot occur in the interior of limit geodesics. In particular, it provides a new way to rule the trumpet [CCa, Example 5.5] and its generalizations [CN12, Example 1.15]. Moreover, we show that the following example cannot arise as a Gromov-Hausdorff limit of manifolds with lower Ricci bound, even through the tangent cones are Hölder (in fact, Lipschitz) continuous along the interior of geodesics. This example cannot be ruled out by previously known results.

3 ON DIMENSIONS OF TANGENT CONES IN LIMIT SPACES WITH LOWER RICCI CURVATURE BOUNDS 3 Figure 1. Y = { (x, y, z) R 3 : z x 4 + y x 2} Example 1.4. Let Y = { (x, y, z) R 3 : z x 4 + y x 2}. Then T (x,,) Y = { } (y, z) R 2 : z y 2x R for x and T(,,) Y = R 2 + R. Let X be the double of Y along its boundary. Then all points not on the x-axis are in R 3 and along the x-axis we have that for x T (x,,) X = (double of { } (y, z) R 2 : z y 2x ) R (i.e. it s 2 a cone R) degenerating to T (,,) X = R + R. So dim T (,,) X = 2 but dim T (x,,) X = 3 for x. Lastly, any segment of the geodesic γ(t) = (t,, ) is unique shortest between its end points and hence it s a limit geodesic if Y is a limit of manifolds with Ric (n 1)H. Hence Theorem 1.3 is applicable to γ and therefore X is not a limit of n-manifolds with Ric (n 1)H. Note that one can further smooth out the metric on X along Y\{x axis} to obtain a space X 1 with similar properties but which in addition is a smooth Riemannian manifold away from the x-axis. In particular X 1 is non-branching. Next we want to mention several semicontinuity results about the Colding-Naber dimension which further suggest that this notion is a natural one. Let M n be the space of Gromov-Hausdorff limits of manifolds with Ric (n 1). Recall the following notions from [CC97] Definition 1.5. Let X M n. WE k (X) = {x X such that some tangent cone T x X splits off isometrically as R k Y}. E k (X) = {x X such that every tangent cone T x X splits off isometrically as R k Y}. (WE k ) ε (X) = {x X such that there exist < r 1, Y and q R k Y such that d G H (B r (x), B Rk Y r (q)) < εr}.

4 4 VITALI KAPOVITCH AND NAN LI By [CC97, Lemma 2.5] there exists ε(n) > such that if p (WE k ) ε (X) for some ε ε(n) then vol B r (p) E k > for all sufficiently small r. Suppose (X i, p i ) M n, (X i, p i ) (X, p) and p R k (X). Then p (WE k ) ε(n) (X) which obviously implies that p i (WE k ) ε(n) (X i ) for all large i as well. By above this implies that vol E k (X i ) > for all large i. This together with (1.5) yields the following result of Honda proved in [Hon13b, Prop 3.78] using very different tools. Theorem 1.6. [Hon13b, Prop 3.78] Let X i M n and dim X i = k. Let (X i, p i ) G H (X, p). Then dim X k. In other words, the dimension function is lower semicontinuous on M n with respect to the Gromov-Hausdorff topology. This theorem, applied to the convergence ( 1 r X, p) (T px, o) = (R k, ) for p R k, r immediately gives the following result which also directly follows from [Hon13a, Prop 3.1] and (1.5). Theorem 1.7. Let X M n. Then dim X is equal to the largest k for which R k (X). Another immediate consequence of Theorem 1.6 is the following Corollary 1.8. [Hon13b, Prop 3.78] Let X M n. Then for any x X and any tangent cone T x X it holds that dim T x X dim X. It is obvious from (1.3) that for any X M n we have dim X dim Haus X. However, as was mentioned earlier, the following natural question remains open. Question 1.9. Let X M n. Is it true that dim X = dim Haus X? 1.1. Idea of the proof of Theorem 1.1. Let γ: [, 1] M n be a unit speed shortest geodesic in an n-manifold with Ric (n 1)H. In [CN12] Colding and Naber constructed a parabolic approximation h τ to d(, p) given as the solution of the heat equation with initial conditions given by d(, p), appropriately cut off near the end points of γ and outside a large ball containing γ. They showed that h r 2 provides a good approximation to d = d(, p) on an r-neighborhood of γ [δ,1 δ]. In particular, they showed that (1 δ) ( ) (1.6) Hess hr 2 2 dt c(n, δ, H) δ B r (γ(t)) for all r r (n, δ, H). They used this to show that for any t (δ, 1 δ) most points in B r (γ(t)) remain r-close to γ under the reverse gradient flow of d for a definite time s ε = ε(n, δ, k). In section 3 we show that the same holds true for the reverse gradient flow φ s of h r 2. Next, the standard weak type 1-1 inequality for maximum function applied to the inequality (1.6) implies that (1 δ) ( ) (1.7) (Mx Hess hr 2 ) 2 dt c(n, δ, H) δ B r (γ(t)) as well. This implies that for every x in a subset C r (γ(t)) in B r (γ(t)) of almost full measure the integral ε Mx Hess h r 2 (φ s (x))ds is small (see estimate (4.4)). Using a small modification of a lemma from [KW11] this implies that for any such point x and any < r 1 r most points in B r1 (x) remain r 1 -close to φ s (x) for all s ε under the flow φ s. This then easily implies that φ s is Bilipschitz on C r (γ(t)) using Bishop-Gromov volume comparison and the triangle inequality.

5 ON DIMENSIONS OF TANGENT CONES IN LIMIT SPACES WITH LOWER RICCI CURVATURE BOUNDS Acknowledgements. We are very grateful to Aaron Naber for helpful conversations and to Shouhei Honda for bringing to our attention results of [Hon13a] and [Hon13b]. 2. Preliminaries In this section we will list most of the technical tools needed for the proof of Theorem Segment inequality. Throughout the rest of the paper unless indicated otherwise we will assume that all manifolds M n involved are n-dimensional complete Riemannian satisfying Ric M n (n 1). We will need the following result of Cheeger and Colding Theorem 2.1 (Segment inequality). [CC96, Theorem 2.11] Given n and r > there exists c = c(n, r ) such that the following holds. Let F : M n R + be a nonnegative measurable function. Then for any r r and A, B B r (p) it holds d(x,y) F(γ x,y (u)) du d vol x d vol y c r (vol A + vol B) F(z) d vol z, A B where γ z1,z 2 denotes a minimal geodesic from z 1 to z Generalized Abresch-Gromoll Inequality. Let γ: [, L] M be a minimizing unit speed geodesic with γ() = p, γ(l) = q where L = d(p, q). To simplify notations and exposition from now on we will assume that L = 1. Let d = d(, p), d + = d(, q), and let e = d + + d d(p, q) be the excess function. The following result is a direct consequence of [CN12, Theorem 2.8] and, as was observed in [CN12], using the fact that e 2 it immediately implies the Abresch-Gromoll estimate [AG9]. Theorem 2.2 (Generalized Abresh-Gromoll Inequality). [CN12, Theorem 2.8] There exist c(n, δ), r (n, δ) > such that for any < δ < t < 1 δ < 1, < r < r it holds e c(n, δ) r 2. B r (γ(t)) 2.3. Parabolic approximation for distance functions. Fix δ > and let h ± t be parabolic approximations to d ± constructed in [CN12]. They are given by the solutions to the heat equations d dt h± t = h ± t h ± (x) = λ(x) d± (x) for appropriately constructed cutoff function λ. We will need the following properties of h t established in [CN12]. Lemma 2.3. [CN12, Lemma 2.1] There exists c(n, δ) such that (2.1) h ± t c(n, δ). Theorem 2.4. [CN12, Theorem 2.19] There exist c(n, δ), r (n, δ) > such that for all r 1 r there exists r [ r 1 2, 2r 1 ] such that the following properties are satisfied B 2r (p) (i) h ± d ± (x) c r 2 for any x B r 2 2 (p)\(b δ (p) B δ (q)) with e(x) r 2 (ii) B r (x) h± 2 1 c r. r 2 (iii) (1 δ) δ B r (γ(t)) h± 2 1 c r 2. r 2 (iv) (1 δ) Hess δ B r (γ(t)) h ± 2 c. r 2

6 6 VITALI KAPOVITCH AND NAN LI 2.4. First Variation formula. We will need the following lemma (cf. [CN12, Lemma 3.4] ). Lemma 2.5. Let X be a smooth vector field on M and let σ 1 (t), σ 2 (t) be smooth curves. Let p = σ 1 (), q = σ 2 (). Then d + dt d(σ 1(t), σ 2 (t)) t= X(p) σ 1 () + X(q) σ 2 γ () + X, p,q where γ p,q : [, d(p, q)] M is a shortest geodesic from p to q. Here X means the norm of the full covariant derivative of X i.e. norm of the map v v X. In particular, if h: M R is smooth and X = h, then d + dt d(σ 1(t), σ 2 (t)) t= X(p) σ 1 () + X(q) σ 2 γ () + Hess h. p,q Proof. The lemma easily follows from the first variation formula for distance functions and the triangle inequality Maximum function. Let f : M R be a nonnegative function. Consider the maximum function Mx ρ f (p) := sup r ρ B r (p) f for ρ (, 4]. We ll set Mx f := Mx 1 f. The following lemma is well-known [Ste93, p. 12]. Lemma 2.6 (Weak type 1-1 inequality). Suppose (M n, g) has Ric (n 1) and let f : M R be a nonnegative function. Then the following holds. (i) If f L α (M) with α 1 then Mx ρ f is finite almost everywhere. (ii) If f L 1 (M) then vol { x M : Mx ρ f (x) > c } C(n) c f for any c >. M (iii) If f L α (M) with α > 1 then Mx ρ f L α (M) and Mx ρ f α C(n, α) f α. This lemma easily generalizes to functions defined on subsets as follows: Corollary 2.7. Let Ric M n (n 1) and f : M R + be measurable. Let A M be measurable such that f L α (U ρ (A)) where α > 1. Here U ρ (A) denotes the ρ-neighborhood of A. Then Mx ρ f L α (A) C(n, α) f L α (U ρ (A)). Proof. Let f = f χ Uρ (A). Obviously, Mx ρ f (x) = Mx ρ f (x) for any x A. The result follows by applying Lemma 2.6 (iii) to f. 3. Gradient flow of the parabolic approximation Let φ s be the reverse gradient flow of h = h r 2 (i.e. the gradient flow of h r 2 ) and let ψ s be the reverse gradient flow of d. We first want to show that for most points x B r (γ(t)) we have that φ s (x) B 2r (γ(t s)) for all t (δ, 1 δ) and s [, ε] for some uniform ε = ε(n, δ). Note that this (and more) is already known for ψ s by [CN12]. Following Colding-Naber we use the following Definition 3.1. For < s < t < 1 define the set A t s(r) {z B r (γ(t)) : ψ u (z) B 2r (γ(t u)) u s}. Similarly, we define B t s(r) {z B r (γ(t)) : φ u (z) B 2r (γ(t u)) u s}. An important technical tool used to prove the main results of [CN12] is the following

7 ON DIMENSIONS OF TANGENT CONES IN LIMIT SPACES WITH LOWER RICCI CURVATURE BOUNDS 7 Proposition 3.2. [CN12, Proposition 3.6] There exist r (n, δ) and ɛ (n, δ) such that if t (δ, 1 δ) and ε ɛ then r r as in Theorem 2.4 we have 1 2 vol(at ε(r)) vol(b r (γ(t))). Unlike Colding-Naber we prefer to work with the gradient flow of the parabolic approximation h rather than the gradient flows of d ±, because the gradient flow of h provides better distance distortion estimates since in that case the two terms outside the integral in Lemma 2.5 vanish and the resulting inequality scales better in the estimates involving maximum function (see Lemma 4.2 below). Therefore, our first order of business is to establish the following lemma which says that Proposition 3.2 holds for the gradient flow of h as well: Lemma 3.3. There exists r 1 (n, δ) and ɛ 1 (n, δ) such that if δ < t ε < t < 1 δ with ε ε 1 then r r 1 we have 1 2 vol(at ε(r)) vol(b r (γ(t))) and 1 2 vol(bt ε(r)) vol(b r (γ(t))). The proof of Proposition 3.2 uses bootstrapping in ε, r starting with infinitesimally small (depending on M!) r (cf. Lemma 4.2 below) for which the claim easily follows from Bochner s formula applied to d along γ. We don t utilize bootstarpping in r and instead use that the result has already been established for the gradient flow of d. Proof. Of course, we only need to prove the second inequality as the first one holds by Proposition 3.2 for some r (n, δ), ɛ (n, δ) >. By possibly making r smaller we can ensure that it satisfies Theorem 2.4. Let < ε < ε be small (how small it will be chosen later). Let { (3.1) S t s < t δ : 1 } 2 < vol(bt s(r)). vol(b r (γ(t))) We wish to show that S t contains [, ε] for some uniform ε = ε(n). Obviously S t is open in [, ε] so it s enough to show that it s also closed. To establish this it s enough to show that if ε ε and [, ε ) S t then ε S t. For any < s < t we define c t s to be the characteristic function of the set A t s(r) B t s(r). The same argument as in [CN12] shows that (3.2) c t s(x, y) Hess h d vol x d vol y B r (γ(t)) B r (γ(t)) γ ψs(x),φs(y) ( ) 2 vol(br (γ(t s))) C(n, δ) r Hess h. vol(b r (γ(t))) B 5r (γ(t s)) Indeed, we have (3.3) c t s(x, y) Hess h d vol x d vol y B r (γ(t)) B r (γ(t)) γ ψs(x),ψs(y) = Hess h d vol x d vol y A t s(r) B t s(r) γ ψs(x),ψs(y) C(n, δ) Hess h d vol x d volȳ, ψ s (A t s(r)) φ s (B t s(r)) γ x,y

8 8 VITALI KAPOVITCH AND NAN LI where the last inequality follows from the fact that h c(n, δ) by Lemma 2.3 and hence the Jacobian of φ s satisfies (3.4) J φs e C(n,δ)s. Similar inequality holds for ψ s by Bishop-Gromov volume comparison. Since ψ s (A t s(r)), φ s (B t s(r)) B 2r (γ(t s)) by definition, by the segment inequality (Theorem 2.1 ) we have (3.5) Hess h d vol x d volȳ ψ s (A t s(r)) φ s (B t s(r)) γ x,y C(n, δ) r [vol(ψ s (A t s(r))) + vol(φ s (B t s(r)))] Hess h B 5r (γ(t s)) C(n, δ) r vol(b 5r (γ(t s))) Hess h B 5r (γ(t s)) = C(n, δ) r vol(b 5r (γ(t s))) 2 Hess h B 5r (γ(t s)) C(n, δ) r vol(b r (γ(t s))) 2 Hess h, B 5r (γ(t s)) where the last inequality follows by Bishop-Gromov. Thus, (3.6) c t s(x, y) Hess h d vol x d vol y B r (γ(t)) B r (γ(t)) γ ψs(x),ψs(y) C(n, δ) r vol(b r (γ(t s))) 2 Hess h. B 5r (γ(t s)) Dividing by vol(b r (γ(t))) 2 we get (3.2). By [CN12, Cor 3.7] we have that (3.7) C 1 vol(b r(γ(t s))) vol(b r (γ(t))) C. for some universal C = C(n, δ) and therefore (3.8) c t s(x, y) Hess h d vol x d vol y B r (γ(t)) B r (γ(t)) γ ψs(x),φs(y) C(n, δ) r Hess h. Let (3.9) Ĩε r B r (γ(t)) B r (γ(t)) ε B 5r (γ(t s)) c t s(x, y) Hess h ds d vol x d vol y. γ ψs(x),φs(y) Then by (3.8) and Theorem 2.4 we have that ε (3.1) Ĩε r = c t u(x, y) Hess h d vol x d vol y ds B r (γ(t)) B r (γ(t)) γ ψu(x),φu(y) ε ( ) C(n, δ) r Hess h d vol ds B 5r (γ(t s)) C(n, δ)r ( 1 δ ) 1/2 ε Hess h 2 d vol ds C(n, δ) ε r. δ B 5r (γ(s))

9 ON DIMENSIONS OF TANGENT CONES IN LIMIT SPACES WITH LOWER RICCI CURVATURE BOUNDS 9 Let T η r x B r(γ(t)) : x A t ε(r) and {x} B r (γ(t)) ε c t s(x, y) Hess h η 1 Ĩε r, γ ψs(x),φs(y) and for x T η r let us define ε (3.11) T η(x) r y B r(γ(t)) : c t s(x, y) Hess h ds η 2 Ĩ r ε. γ ψs(x),φs(y) Here η = η(n, δ, d(p, q)) > is small and chosen first. Then ε is chosen later depending on η. By [CN12, (3.37)] we can assume that (3.12) Therefore, by construction we have that (3.13) and hence (3.14) vol(a t ε(r)) 1 C(n, δ)η. vol(b r (γ(t))) vol( T η) r 1 C(n, δ)η, vol(b r (γ(t))) vol( T r η(x)) vol(b r (γ(t))) 1 C(n, δ)η, x T r η. We choose η so that C(n, δ)η 1 in (3.13) and (3.14). Let x T η r T η r/1 (this intersection is non-empty for small η = η(n) by Bishop- Gromov) and let y T η(x). r We will fix η > satisfying the above conditions from now on. We claim that then y B t ε (r). Indeed d(ψ s (x), γ(t s)) r/5 for all s ε since x T η r/1 A t ε(r/1). So by the triangle inequality it s enough to show that d(ψ s (x), φ s (y)) 1.1r for any s ε. Let S = {s ε : y B t s(r)}. This set is obviously open and connected in [, ε ]. We claim that S = [, ε ]. Let s = sup{s: s S }. Note that for any < s < s we have that c t s(x, y) = 1. Therefore, by (3.1) and (3.11) for any < s < s we have s (3.15) Hess h du η 2 Ĩ r C(n, δ) t s εr.1r γ ψu(x),φu(y) η 2 if ε = ε(η) is chosen small enough. Next, recall that by [CN12, Lemma 2.2(3)] we have that for any x B r (γ(t)), (3.16) s Further, by Theorem 2.2 we know that (3.17) h(ψ u (x)) d (ψ u (x)) du c(n, δ) s ( e(x) + r). B r (γ(t)) e c(n, δ) r 2. Therefore, without losing generality by making the sets T r η slightly smaller we can assume that for any x T r η we have (3.18) e(x) η 1 r 2.

10 1 VITALI KAPOVITCH AND NAN LI Thus, for all x T r η we have (3.19) s h(ψ u (x) d (ψ u (x) du c(n, δ) s (η 1/2 r + r) <.1r if ε = ε(n, δ, η) is small enough. Therefore, by Lemma 2.5 and using (3.15) and (3.19) we get that (3.2) d(ψ s (x), φ s (y)).2r + d(x, y) < 1.1r. By the triangle inequality, (3.21) d(φ s (x), γ(t s)) r/ r 1.5r < 2r. By continuity the same holds for s and hence s S. Thus S is both open and closed in [, ε ] and therefore S = [, ε ]. Unwinding this further we see that this means that T r η(x) B t ε (r). Therefore, by (3.14) (3.22) 1 2 vol(bt ε (r)) vol(b r (γ(t))) when η was chosen small enough so that C(n, δ)η < 1/2. Hence ε S t. Therefore, S t is both open and closed and ε = ε. Remark 3.4. It follows from the proof above that the constant ɛ 1 can be chosen explicitly depending on the constant c in Theorem 2.4 and ɛ in Proposition 3.2. The proof of Lemma 3.3 shows that T r η(x) B t ε(r) for appropriately chosen ε depending on η. In view of (3.13) this means that the conclusion can be strengthened as follows (cf. [CN12]) Lemma 3.5. For every η η (n, δ) and r r (n, δ) that there exists ɛ ɛ(n, η, δ) such that the set B t ε(r) {z B r (γ(t)) : φ s (z) B 2r (γ(t s)) s ε} satisfies vol B t ε(r) 1 C(n, δ) η. vol(b r (γ(t))) When η is sufficiently small this means that most points in B r (γ(t)) remain close to the geodesic γ under the flow φ s. Also, because of the above lemma, (3.4) and (3.7) we have the following Lemma 3.6. Let F : M R be a nonnegative measurable function. Then F(φ s (x)) d vol x C(n, δ, η) F( x) d vol x B t ε(r) B 2r (γ(t s)) for any s ε as in Lemma 3.5. Remark 3.7. It is obvious that all results of this section concerning the flow of h r 2 also true for the flow of h +. r 2 are 4. Bilipschitz control The goal of this section is to prove the following somewhat stronger version of Theorem 1.1

11 ON DIMENSIONS OF TANGENT CONES IN LIMIT SPACES WITH LOWER RICCI CURVATURE BOUNDS 11 Theorem 4.1. Given H R, < δ < 1/3, < η < 1 there exist r (n, δ, H, η), ε(n, δ, H, η) > such that the following holds: Suppose (M n, g) is complete with Ric M (n 1)H and let γ: [, 1] M n be a unit speed minimizing geodesic. Then for any t 1, t 2 (δ, 1 δ) with t 1 t 2 < ε and any r < r there are subsets C r i B r (γ(t i )) (i = 1, 2) such that vol C r i vol B r (γ(t i )) 1 η and there exists a 4-Bilipschitz map f r : C r 1 Cr 2. As before, to simplify notation we will assume that H = 1 and Ric M n (n 1). Corollary 2.7 essentially means that all estimates involving integrals of Hess h from the previous section remain true for Mx ρ Hess h. In particular, for r as in Theorem 2.4 and any r r /1 we have 1 δ ( ) (4.1) Hess h16r 2 2 dt C δ B 4r (γ(t)) δ. By Corollary 2.7 this implies that 1 δ ( ) (4.2) (Mx r Hess h ) 2 dt C B 2r (γ(t)) δ. δ where h = h. It is clear that all results from the previous section work for this h as well 16r 2 as h. Therefore, everywhere in the previous section where we used (4.1) we could have r 2 used (4.2) instead. Indeed, we have for any 2δ < t < 1 2δ, ε ( ) (4.3) Mx r Hess h ds B 2r (γ(t s)) ε ( 2 ε Mx r Hess h ) ds B 2r (γ(t s)) ε ε (Mx r Hess h ) 2 ds C(n, δ) ε. B 2r (γ(t s)) In view of Lemma 3.6 this implies that (4.4) B t ε(r)) ε Mx r Hess h (φ s (x)) ds C(n, δ, η) ε. This means that for most points x B t ε(r) the integral ε Mx r Hess h (φ s (x)) ds is bounded. More precisely, given any < ν < 1 let (4.5) B t ε(r, ν) Then (4.6) { x B t ε(r) : ε Mx r Hess h (φ s (x)) ds C(n, δ, η) } ε. ν vol B t ε(r, ν) vol B t ε(r) > 1 ν. We will need the following slight modification of Lemma 3.7 from [KW11]

12 12 VITALI KAPOVITCH AND NAN LI Lemma 4.2. Given c >, there exists (explicit) C = C(n, λ) such that the following holds. Suppose (M n, g) has Ric M n (n 1) and X t is a vector field with compact support, which depends on time but is piecewise constant in time. Let c(t) be the integral curve of X t with c() = p M n and assume that div X t λ on B 1 (c(t)) for all t [, 1]. Let φ t be the flow of X t. Define the distortion function dt r (t)(p, q) of the flow on scale r by the formula dt r (t)(p, q) := min { (4.7) r, max d(p, q) d(φ τ (p), φ τ (q)) }. τ t Put ε = 1 Mx 1( X t )(c(t)) dt. Then for any r 1/1 we have (4.8) dt r (1)(p, q) d vol p d vol q Cr ε B r (p ) B r (p ) and there exists B r (p ) B r (p ) such that (4.9) and φ t (B r (p ) ) B 2r (c(t)) for all t [, 1]. This lemma immediately implies vol(b r (p ) ) vol(b r (p )) 1 Cε Corollary 4.3. Under the assumptions of Lemma 4.2 for every r 1/1 there exists B r (p ) B r (p ) such that vol(b r (p ) ) vol(b r (p )) 1 Cε and φ t (B r (p ) ) B 2r (c(t)) for all t [, 1] p, q B r (p ), dt r (1)(p, q) Cr ε. In [KW11] Lemma 4.2 is stated for divergence free vector fields. We want to apply it to X = h which is not divergence free. However, recall that it does satisfy div X λ and hence the Jacobian of its flow map satisfies (4.1) J φs e λs pointwise. The proof given in [KW11] goes through verbatim with a straightforward change in one place using (4.1) instead of the flow of X being volume-preserving. We include the proof here for reader s convenience. Proof of Lemma 4.2. We prove the statement for a constant in time vector field X t. The general case is completely analogous except for additional notational problems. Notice that all estimates are trivial if ε 2 C. Therefore it suffices to prove the statement with a universal constant C(n, λ) for all ε ε (n, λ). We put ε = 1 2C and determine C 2 in the process. We again proceed by induction on the size of r. Notice that the differential of φ s at c() is Bilipschitz with Bilipschitz constant (4.11) Thus the Lemma holds for very small r. s e X (c(t))dt 1 + 2ε.

13 ON DIMENSIONS OF TANGENT CONES IN LIMIT SPACES WITH LOWER RICCI CURVATURE BOUNDS 13 Suppose the result holds for some r/1 1/1. It suffices to prove that it then holds for r. By induction assumption we know that for any t there exists B r/1 (c(t)) B r/1 (c(t)) such that for any s [ t, 1 t] we have (4.12) vol(b r/1 (c(t)) ) (1 Cε) vol(b r/1 (c(t))) 1 2 vol(b r/1(c(t))) and (4.13) φ s (B r/1 (c(t)) ) B r/5 (c(t + s)), where we used ε 1 2C in the inequality. This easily implies that vol(b r/1(c(t))) are comparable for all t. More precisely, for any t 1, t 2 [, 1] we have that (4.14) 1 C vol B r/1 (c(t 1 )) vol B r/1 (c(t 2 )) C vol B r/1 (c(t 1 )) with a computable universal C = C (n). Put (4.15) h(s) = dt r (s)(p, q) d vol p d vol q, (4.16) B r/1 (c()) B r (c()) U s := {(p, q) B r/1 (c()) B r (c()) dt r (s)(p, q) < r}, φ s (U s ) := {(φ s (p), φ s (q)) (p, q) U s }, and dt r(s)(p, q) := lim sup h dt r (s+h)(p,q) dt r (s)(p,q) h. As dt r (t) r is monotonously increasing, we deduce that if dt r (s)(p, q) = r, then dt r(s)(p, q) =. Since dt r (s + h)(p, q) dt r (s)(p, q) + dt r (h)(φ s (p), φ s (q)) and φ s satisfies J φt e λt it follows (4.17) h (s) dt r(φ 1 (x), φ s (y)) e sλ dt r()(p, q) U s φ s (U s ) e sλ 4 vol B 3r(c(s)) 2 vol B r/1 (c()) 2 B 3r (c(s)) 2 dt r()(p, q), where we used that φ s (B r/1 (p ) ) 2 φ s (U s ) B 3r (c(s)) 2. We would like to point out that using J φs e λs instead of J φs = 1 (which is true for flows of harmonic maps) in the above inequality is the only place where the proof of Lemma 4.2 differs from the proof of [KW11, Lemma 3.7].) If p is not in the cut locus of q and γ pq : [, 1] M is a minimal geodesic between p and q, then by Lemma 2.5 (4.18) dt r()(p, q) d(p, q) + 1 X (γ pq (t)) dt. Combining the last two inequalities with the segment inequality we deduce (4.19) h (s) C 1 (n, λ)r X B 6r (c(s)) C 1 (n, λ)r Mx 1 X (c(s)). Note that the choice of the constant C 1 (n, λ) can be made explicit and independent of the induction assumption. We deduce h(1) C 1 (n, λ)rε and thus the subset { B r (p ) } (4.2) := p B r (p ) dt r (1)(p, q) d vol q r/2 B r/1 (p )

14 14 VITALI KAPOVITCH AND NAN LI satisfies (4.21) It is elementary to check that (4.22) vol(b r (p ) ) (1 2C 1 (n, λ)ε) vol(b r (p )). φ t (B r (p ) ) B 2r (c(t)) for all t [, 1]. Then arguing as before we estimate that (4.23) dt r (1)(p, q)d vol p d vol q C 2 (n, λ) r ε. B r (p ) B r (p ) Using dt r (1) r and the volume estimate (4.21) this gives (4.24) dt r (1)(p, q) d vol p d vol q C 2 r ε + 2rC 1 ε =: C 3 rε. B r (c(p )) 2 This completes the induction step with C(n, λ) = C 3 and ε = 1 2C 3. In order to remove the restriction ε ε one can just increase C(n, λ) by the factor 4, as indicated at the beginning. Remark 4.4. It s obvious from the proof that Lemma 4.2 and Corollary 4.3 remain valid for r ρ/1 if we change Mx 1 to Mx ρ in the assumptions. We can now finish the proof of Theorem 4.1 by establishing the following Lemma 4.5. Fix a small R < r /1 and let ν = η 1 and let ε ν 4 C 2 (n,δ,η) satisfies Lemma 3.5. Then for any s ε the map φ s B t ε (R,ν) is 4-Bilipschitz onto its image. ν Proof. Since ε 4 C 2 (n,δ,η) we have C(n,δ,η) ε ν ν and therefore by the definition of B t ε(r, ν) for all x B t ε(r, ν) it holds (4.25) ε Mx R Hess h (φ s (x)) ds ν. This means that we can apply Corollary 4.3 at all such points. Let x, y B t ε(r, ν). Set r =.6 xy. By Bishop-Gromov we can assume that B r (x) B r (y) provided ν = ν(n) is chosen small enough. Pick any z B r (x) B r (y). Then by Corollary 4.3 φ t (z) B 2r (φ t (x)) B 2r (φ t (x)) i.e d(φ t (x), φ t (z)) 2r and d(φ t (y), φ t (z)) 2r and therefore d(φ t (x), φ t (y)) 4r = 2.4d(x, y). This shows that φ t is 4-Lipschitz on B t ε(r, ν). To get that it s Bilipschitz let r 1 =.99d(x, y) and r 2 =.2d(x, y). As before, using Bishop-Gromov, we can find p B r1 (x) B r2 (y) and q B r1 (x) B r1 /1(x). This implies that d(φ t (p), φ t (y)) 2r 2 =.4d(x, y), d(φ t (q), φ t (x)) 2r 1 /1.2d(x, y) and d(p, q) d(φ t (p), φ t (q)).1r 1.1d(x, y). By the triangle inequality this yields (4.26) d(φ t (x), φ t (y)) d(p, q).4d(x, y).2d(x, y).1d(x, y) which finally proves Lemma d(x, y) Remark 4.6. It s easy to see form the proof that the constant 2 in Lemma 4.2 and in the definition of B t ε(r) (Definition 3.1) can be replaced by any constant > 1. With that change the proof of Theorem 4.1 can be easily adapted to show that the Bilipschitz constant 4 in in the statement of the theorem can be replaced by any constant β > 1. However, as to be expected, ε(λ) as β 1.

15 ON DIMENSIONS OF TANGENT CONES IN LIMIT SPACES WITH LOWER RICCI CURVATURE BOUNDS 15 References [AG9] U. Abresch and D. Gromoll. On Complete Manifolds With Nonnegative Ricci Curvature. Journal of the American Mathematical Society, 3(2): , April 199. [CC96] J. Cheeger and T. H. Colding. Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. of Math., 144(1): , [CC97] J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom., 46(3):46 48, [CCa] J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. II. J. Differential Geom., 54(1):13 35, 2. [CCb] J. Cheeger and T. H. Colding. On the structure of spaces with Ricci curvature bounded below. III. J. Differential Geom., 54(1):37 74, 2. [CN12] T. H. Colding and A. Naber. Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. of Math. (2), 176(2): , 212. [Hon13a] S. Honda. On low-dimensional Ricci limit spaces. Nagoya Math. J., 29:1 22, 213. [Hon13b] S. Honda. Ricci curvature and L p -convergence. to appear in Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal), 213. [KW11] V. Kapovitch and B. Wilking. Structure of fundamental groups of manifolds with Ricci curvature bounded below [Ste93] E. Stein. Harmonic Analysis. Princeton University Press, Vitali Kapovitch, Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4 address: vtk@math.toronto.edu Nan Li, Department of Mathematics, CUNY New York City College of Technology, 3 Jay Street, Brooklyn, NY 1121, USA address: NLi@citytech.cuny.edu