HOLDER CONTINUITY OF TANGENT CONES IN RCD(K,N) SPACES ANDAPPLICATIONS TO NON-BRANCHING

QIN DENG

AbstractIn this paper we prove that a metric measure space (X, d,m) satisfying the finite Riemannian curvature-

dimension condition RCD(K,N) is non-branching and that tangent cones from the same sequence of rescal-ings are Holder continuous along the interior of every geodesic in X. More precisely, we show that thegeometry of balls of small radius centred in the interior of any geodesic changes in at most a Holder con-tinuous way along the geodesic in pointed Gromov-Hausdorff distance. This improves a result in the Riccilimit setting by Colding-Naber where the existence of at least one geodesic with such properties betweenany two points is shown. As in the Ricci limit case, this implies that the regular set of an RCD(K,N) spacehas m-a.e. constant dimension, a result already established by Brue-Semola, and is m-a.e convex. It alsoimplies that the top dimension regular set is weakly convex and, therefore, connected. In proving the maintheorems, we develop in the RCD(K,N) setting the expected second order interpolation formula for thedistance function along the Regular Lagrangian flow of some vector field using its covariant derivative.

Contents

1. Introduction 21.1. Outline of Paper and Proof 31.2. Acknowledgements 6

2. Preliminaries 62.1. Curvature-dimension condition preliminaries 62.2. First order calculus on metric measure spaces 82.3. Tangent, cotangent, and tensor modules 92.4. RCD(K,N) and Bakry-Emery conditions 102.5. Heat flow and Bakry-Ledoux estimates 112.6. Second order calculus and improved Bochner inequality 122.7. Non-branching and essentially non-branching spaces 132.8. RCD(K,N) structure theory 142.9. Additional RCD(K,N) theory 152.10. Mean value and integral excess inequalities 16

3. Differentiation formulas for Regular Lagrangian flows 183.1. Regular Lagrangian flow 183.2. Continuity equation 193.3. Second order interpolation formula 23

4. Estimates on the heat flow approximations of distance and excess functions 295. Gromov-Hausdorff approximation 36

5.1. Proof of main lemma 375.2. Construction of limit geodesics 455.3. Proof of main theorem 56

6. Applications 60

University of Toronto, qin.deng@mail.utoronto.ca.

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6.1. Non-branching 606.2. Dimension and weak convexity of the regular set 62

References 62

1. Introduction

In this paper, we prove that RCD(K,N) spaces are non-branching and generalize to the RCD(K,N) settingan improved version of the main result from Colding-Naber [CN12]. We begin by stating two formulations ofthe latter, which is be the main technical result of this paper.

Theorem 1.1. (Holder continuity of geometry of small balls with same radius) Let (X, d,m) be an RCD(K,N)space for some K ∈ R and N ∈ (1,∞). Let p, q ∈ X and d(p, q) = `. Define K′ = ( K

−(N−1) ∨ 1)1/2 and `′ = `∧ 1.For any unit speed geodesic γ : [0, `]→ X between p and q, there exist constants C(N), α(N) and r0(N) > 0 sothat for any δ > 0 with 0 < r < r0

δ`′

K′ and δ` < s < t < ` − δ`,

dpGH((Br(γ(s)), γ(s)), (Br(γ(t)), γ(t))

)<

CK′

δ`′r|s − t|α. (1)

In order to pass the result to tangents, we use the following terminology: Let x1, x2 ∈ X, (Y, dY ,mY , y)∈ Tan(X, d,m, x1) and (Z, dZ ,mZ , z) ∈ Tan(X, d,m, x2). We say Y and Z come from the same sequence ofrescalings if there exists s j ↓ 0 so that

(X, s−1j d,mx1

s j, x1)

pmGH−−−−−→ (Y, dY ,mY , y) and (X, s−1

j d,mx2s j, x2)

pmGH−−−−−→ (Z, dZ ,mZ , z). (2)

The following estimate on tangents from the same sequence of rescaling follows from Theorem 1.1.

Theorem 1.2. (Holder continuity of tangent cones) In the notations of Theorem 1.1, for any unit speed ge-odesic γ : [0, `] → X between p and q, there exist constants C(N), α(N) > 0 so that if (Ys, dYs ,mYs , ys) ∈Tan(X, d,m, γ(s)) and (Yt, dYt ,mYt , yt) ∈ Tan(X, d,m, γ(t)) come from the same squence of rescalings, then

dpGH((Br(ys), ys), (Br(yt), yt)

)<

CK′

δ`′r|s − t|α (3)

for all r > 0.

To prove these we first construct at least one geodesic between any two points satisfying the conclusionof Theorem 1.1, which is the main result of [CN12]. We then use this construction to prove that RCD(K,N)spaces, and so, in particular, Ricci limit spaces, are non-braching in Subsection 6.1.

Theorem 1.3. Let (X, d,m) be an RCD(K,N) space for some K ∈ R and N ∈ (1,∞). (X, d,m) is non-branching.

This has been a natural open problem for Ricci limits since the seminal work of Cheeger-Colding in [CC96,CC97, CC00a, CC00b]. Potential branching which can come from some simple examples were ruled out in[CC00a, Section 5] and some partial results were obtained in [CN12] for noncollapsed limits of manifolds withuniform two-sided Ricci curvature bounds. To compensate for the lack of non-branching, the weaker notion ofessentially non-branching was introduced and shown to hold for RCD(K,∞) spaces in [RS14]. This has foundwide use in RCD theory and serves as a suitable replacement for the non-branching condition in most measuretheoretic arguments. We point out finally that non-branching would follow for Ricci limits directly from theresults of [CN12] and the argument we use if one were able to, for example, prove that all geodesics in anyRicci limit space are limit geodesics. Our intrinsic construction of a geodesic in Subsection 5.2 which satisfiesthe conclusion of Theorem 1.1 offers a little more freedom than the extrinsic construction of limit geodesicsand this was enough to prove Theorem 1.3. Theorem 1.3 is then used to pass from the existence of a geodesicbetween any two points which satisfies the conclusion of Theorem 1.1 to the theorem for all geodesics.

In the case of Ricci limits, the Holder continuity of tangent cones had several key applications.2

Theorem 1.4. ( [CN12, Theorems 1.18, 1.20 and 1.21]) Let (X, d) be the Ricci limit of (Mni , gi)i∈N and m be its

canonical limit measure. The following holds:1. There is a unique k ∈ N, 0 ≤ k ≤ n so that m(X\Rk) = 0, where Rk is the k-dimensional regular set;2. Rk from statement 1 is m-a.e. convex and weakly convex. In particular, Rk is connected;3. The isometry group of X is a Lie group.

Statements 1 and 3 have since been proved by other means in the case of RCD(K,N) spaces, see [BS20] for1 and [SOS18, SRG19] for 3. We prove statement 2 in Subsection 6.2 following [CN12]. Since the proofs ofstatements 1 and 2 are intricately related, we will prove statement 1 as well.

1.1. Outline of Paper and Proof. We begin this subsection by introducing the strategy of the proof in [CN12],which we will largely follow. We then discuss the issues that arise when extending this to the metric measuresetting and give an outline of their solutions.

The existence of at least one geodesic satisfying the conclusion of Theorem 1.1 was shown in [CN12]. Theproof there was extrinsic and obtained by proving the theorem for manifolds. As such, consider a Riemannianmanifold with (Mn, g) with RicM ≥ −(n − 1) and a unit speed minimizing geodesic γ : [0, 1] → M from somep to q. Fix some δ > 0 and δ < s0 < s1 < 1− δ. The desired Gromov-Hausdorff approximations in the proof ofTheorem 1.1 for γ are constructed on a large subset of the ball Br(γ(s1)) from the gradient flow Ψ of −dp (moreon the definition of this later). This is not altogether surprising since in the interior of γ, d is a smooth functionand the Laplacian of d has a two-sided bound. A simple argument applying the Bochner formula to d gives∫ 1−δ

δ|Hess dp|

2(γ(t)) dt ≤c(n)δ. (4)

The fact that Hess dp = 12L∇dpg shows that D(Ψs1−s0) : Tγ(s1)M → Tγ(s0)M satisfies the estimates of Theorem

1.2. Of course, Theorem 1.2 is completely trivial in the case of Riemannian manifolds but the point is that thismap comes from a construction on the manifolds itself. Smoothness then allows one to use Ψs1−s0 to constructGromov-Hausdorff aproximations from Br(γ(s1)) to Br(γ(s0)), where r needs to be sufficiently small dependingon γ and δ instead of just n and δ. This property does not pass through Ricci limits and so the challenge, then,is to remove the dependence on γ.

One might consider using Hess dp to control the geometry of balls of uniform (i.e. independent of γ) radiusunder Ψ. However, we do not have estimates on Hess dp for such balls along γ; we do not even have smoothnessof dp. We mention that due to this lack of smoothness, Ψ is not globally defined. The integral curves of Ψ

starting at any x ∈ M should be thought of simply as a (choice of) unit speed geodesic from x to p. In this wayΨ is defined locally away from p for a definite amount of time.

Nevertheless, it turns out that one can still control the geometry under Ψ by utilizing the next formula, whichfollows from the standard first variation formula and second order interpolation formula using the Hessian.∣∣∣∣ d

dtd(σ1(t), σ2(t))

∣∣∣∣ ≤ |∇h − σ′1|(σ1(t)) + |∇h − σ′2|(σ2(t)) + inf∫γσ1(t),σ2(t)

|Hess h| (5)

for a.e. t, where σ1, σ2 are unit speed geodesics in M, h : M → R is a smooth function, and inf is taken acrossall minimizing geodesics connecting σ1(t), σ2(t). This means as long as we can closely approximate dp by asmooth function h where we have reasonable control on |∇h−∇dp| along two geodesics and on Hess h, then wecan control the distance between those two geodesics.

We mention a similar strategy was used to prove the almost splitting theorem in [CC96]. In that setting,a single ball Br(γ(s)) of small radius for a fixed s ∈ (δ, 1 − δ) is considered. It turns out that the correctapproximation to take for dp is the harmonic replacement b of dp on B2r(γ(s)). One is able to obtain the betterthan scale invariant estimate ?

Br(γ(s))|Hess b|2 dV ≤ c(n, δ)r−2+α(n) (6)

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using the Bochner formula, which is enough to prove almost splitting. The almost splitting theorem has sincebeen proved through other means for RCD spaces in [GIG13], see also [MN19].

For our purposes, (6) and the resulting almost splitting theorem is not good enough because it only allowsone to compare two balls of radius r that are distance r away from each other. As discussed in detail in[CN12, Section 2], this estimate blows up as r → 0 if one iterates along γ in r-length intervals. The crucialidea in [CN12] was then to use the heat flow approximation h to (some cutoff of) dp instead. For such anapproximation, they were able to obtain the estimate∫ 1−δ

δ

?Br(γ(t))

|Hess h|2 dV dt ≤ c(n, δ), (7)

where h is the heat flow taken to some time on the scale of r2 (see Theorem 4.12, statement 4). Moreover,∫ 1−δδ|∇dp−∇h| can also be bounded to the correct order (see Theorem 4.13) for most geodesics. These estimates

can then be used along with the segment inequality of Cheeger-Colding [CC96, Theorem 2.11] and (5) tocontrol the total integral change in distances between elements of two sets of large measure in Br(γ(s1)) underthe flow Ψ. This is ultimately good enough to construct a Gromov-Hausdorff approximation using Ψs1−s0 . Wemention that since we are using segment inequality and integral bounds, the smaller the relative measure of thesets compared to the region where we have Hessian estimates, the worse the control we have on total distancechange for those sets under Ψ.

A crucial detail in this is that in order to make use of estimate (7), it is important that most of Br(γ(t)) staysclose, on the scale of r, to γ under the gradient flow for an amount of time independent of r and γ. Sincethe control one has over distance is for sets of large relative measure and γ is trivial in measure, one cannotguarantee using the argument outlined in the previous paragraph that most of Br(γ(t)) does not simply driftaway from γ quickly. In [CN12], this was overcome by using (4). As mentioned previously, by smoothness,(4) implies balls of sufficiently small radius depending on γ stays close to γ under Ψ for some fixed amount oftime depending only on n and δ. Induction with geometrically increasing radii along with the argument fromthe previous paragraph can then be used to guarantee large proportions of balls up to some radius independentof γ also stay close to γ under the flow Ψ.

We now outline the issues with extending this argument to the metric measure setting and the ideas we willuse to resolve them:• (5) is essential in utilizing the Hessian and gradient estimates of the approximating function to control the

geometry under the flow Ψ. In the smooth setting, it stems from the first variation formula along σ1 and σ2and the following interpolation formula along a unit speed geodesic α, which should be thought of as goingbetween σ1(t) and σ2(t) for some t in this application.

〈α′(τ1),V〉 − 〈α′(τ0),V〉 =

∫ τ1

τ0

〈∇α′(τ)V, α′(τ)〉 dτ, (8)

where V is a vector field along α and ∇V is its covariant derivative. We will apply (5) to control the integraldistance change between all elements of two sets under the flow Ψ and so an integral version of the formulasuffices. The first variation formula for “almost every” pair of σ1 and σ2 we are interested in follows easilyfrom the first order differentiation formula for Wassersein geodesics (see [GIG13]).

In the direction of (8), the same formula (with obvious changes) was proved along Wasserstein geodesicswith bounded density in [GT18, Theorem 5.13]. While this does most of the work, a suitable interpretationis required to obtain the integral interpolation formula between two sets S 1 and S 2. To see the difficulty, onemight try to decompose the set of all geodesics between S 1 and S 2 by grouping together all geodesics thatstart at the same x ∈ S 1. In this way, one obtains a family of Wasserstein geodesics parameterized by x ∈ S 1.However, these end at a δ measure and therefore the interpolation formula of [GT18, Theorem 5.13] doesnot apply. This is, of course, expected because 〈∇dx,V〉 is not well-defined at x. The correct decompositionthen is to break all the geodesics between S 1 and S 2 down the middle, parameterize the half that start in S 2

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by the elements of S 1 they each go toward and vice versa. We point out that the same decomposition is usedin the proof of the segment inequality. Some work then needs to be done to check the boundary terms thatarise in interpolating between each of the halves match correctly. These are the contents of Section 3.• The Hessian estimate (7) and several other estimates on the heat flow approximation of the distance function

need to be shown in the RCD(K,N) setting. This simply comes down to verifying the proofs of [CN12] alltranslate to the metric measure setting with minor adjustments. These are the contents of Section 4.• Lastly, the argument in [CN12] relies on (4) to obtain estimates for small balls centred along in the interior γ

under Ψ in order to start an induction process. Such an inequality is not available in the RCD setting since theHessian is a measure-theoretic object, although progress has been made in this direction, see [BC13,CAV14,CM17b, CM18]. Even if it were well-defined, one does not have Jacobi fields or smoothness arguments totranslate such an inequality to a statement about tangent cones or small balls along γ. This is, in many ways,the main obstruction to extending the arguments of [CN12]. We will not attempt to develop all this theory.The key observation is that in fact we can do without the start of induction in radius.

Recall that the need for this start of induction argument stems from the failure of (7) to control distance be-tween a set of small measure and another set under Ψ. As such, it is possible for most of Ψt(Br(γ(s1))) to dis-tance from γ(s1− t) quickly, after which we can no longer apply (7) to control the geometry of Ψt(Br(γ(s1))).To deal with this, consider for each x ∈ Br(γ(s1)) a piecewise geodesic which goes from p to x and thenx to q. It was shown in [CN12] that for a significant amount of x (those with relatively low excess) andtheir corresponding piecewise geodesics, the estimate (7) still holds on the scale of r. The same is true forRCD(−(N−1),N) spaces, see Theorem 4.12. Using this, one can make an induction argument in time instead.Suppose for some small time t most of Ψt(Br(γ(s1))) stays close to γ(s1 − t), after which it leaves. Due tothe control we had on the geometry of Ψt(Br(γ(s1))) in that time, we can guarantee that Ψt(Br(γ(s1))) is stillvery close to one of (in fact, much of) these other piecewise geodesics with a good estimate (7). Therefore,we can use the estimate for that piecewise geodesic for a little longer. The start of induction is trivial sincethe integral curves of Ψ are 1-Lipschitz. In this way, we arrive at an x ∈ Br(γ(s1)) whose trajectory under Ψ

well represents the behaviour of Br(γ(s1)) under Ψ, in the sense that most of Br(γ(s1)) stays close to x on thescale of r under Ψ for a definite amount of time. Multiple limiting and gluing arguments then allow for theselection of a geodesic from p to q, perhaps different from γ, which well represents the behaviours of smallballs centred in its interior under Ψ. The original argument of [CN12] gives the required Gromov-Hausdorffapproximations in the interior of such a geodesic. Notice that, analogous to [CN12], we have at this pointonly shown the existence of a geodesic between p and q which satisfies the main theorem. These are thecontents of Section 5.The ideas outlined above overcome the difficulties of generalizing the arguments of [CN12] to the RCD

setting. To finish, we will first show that RCD spaces are non-branching before proving Theorem 1.1. In orderto prove non-branching, first notice that any two geodesics having the property above cannot branch. To seethis, let γ1 and γ2 be two branching geodesics starting at some p ∈ X which can be constructed by the methodsof Section 5. In the interior, most of an arbitrarily small ball centred around γ1 (resp. γ2) must stay close toγ1 (resp. γ2) for some definite amount of time under the flow of Ψ, where the closeness is Holder dependenton time. Moreover, it is possible to control how the volumes of balls changes along each geodesic. Combiningthese obesrvations with the essentially non-branching property of RCD(K,N) spaces show that there cannotbe any splitting because there is simply not enough room to flow disjoint small balls around γ1 and γ2 into asmall ball around a branching point. While we do not initially claim all geodesics can be constructed with themethods of Section 5, our construction does give a certain amount of freedom. For any δ > 0, it allows us toconstruct a geodesic γδ with nice properties on [δ, 1 − δ] which agrees with the initial geodesic γ at δ. As itturns out, combining this with the previous observation is enough to show that in fact no pair of geodesics canbranch. Theorem 1.1 follows easily from the results of Section 5 and non-branching. These are the contentsof Subsection 6.1. In Subsection 6.2, we generalize to the RCD setting the applications of the main result forRicci limits outlined in [CN12] using verbatim arguments.

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1.2. Acknowledgements. I thank Vitali Kapovitch for introducing me to this problem and sharing his manyinsights. The many hours of his time spent in discussing ideas, answering my questions, and editing my workwere indispensible to the writing of this paper. I thank Christian Ketterer for helpful discussions and reviewingan early draft of this paper. I thank Nicola Gigli for answering several technical questions in the early stages ofthe paper and, along with Andrea Mondino, for many helpful comments.

2. Preliminaries

2.1. Curvature-dimension condition preliminaries. A metric measure space (m.m.s.) is a triple (X, d,m)where (X, d) is a complete, separable metric space and m is a nonnegative, locally finite Borel measure.As a matter of convention, m-measurable in this paper means measurable with respect to the completion of(X,B(X),m). We take the same convention for all other Borel measures as well.

Given a complete and separable metric space (X, d), we denote by P(X) the set of Borel probability measuresand by P2(X) the set of Borel probability measure with finite second moment, that is, the set of µ ∈ P(X) where∫

X d(x, x0)dµ(x) < ∞ for some x0 ∈ X. Given µ1, µ2 ∈ P2(X), the L2-Wasserstein distance W2 between them isdefined as

W22 (µ1, µ2) := inf

γ

∫X×X

d2(x, y)dγ(x, y),

where the infimum is taken over all γ ∈ P(X × X) with (π1)∗(γ) = µ1 and (π2)∗(γ) = µ2. Such measures γare called admissible plans for the pair (µ1, µ2). (P2(X),W2) is called the L2-Wasserstein space of (X, d) andhas been well-studied in the theory of optimal transportation. A W2-geodesic between µ0, µ1 ∈ P2(X) is anypath (µt)t∈[0,1] in P2(X) satisfying W2(µs, µt) = |s − t|W2(µ0, µ1) for any s, t ∈ [0, 1]. If (X, d) is a geodesicspace then (P2(X),W2) is as well. A c-concave solution ϕ to the coresponding dual problem of maximizing∫ϕdµ0 +

∫ϕcdµ1 is called a Kantorovich potential. We refer to [AG13] and [VIL09] for definitions and details.

The various notions of the classical curvature-dimension condition were first proposed independently in[LV09] and [STU06a, STU06b] and are defined as certain convexity conditions on the L2-Wasserstein space ofa metric measure space. We follow closely the formulations of [BS10].

Given a m.m.s. (X, d,m), for any µ ∈ P(X), the Shannon-Boltzmann entropy is defined as

Entm(·) : P(X)→ (−∞,∞], Entm(µ) :=∫

log ρ dµ , if µ = ρm and (ρ log ρ)− is m-integrable

and∞ otherwise.

Definition 2.1. (CD(K,∞) condition) Let K ∈ R. A m.m.s. (X, d,m) is a CD(K,∞) space iff for any twoabsolutely continuous measures µ0, µ1 ∈ P(X) with bounded support, there exists a W2-geodsic µtt∈[0,1] suchthat for any t ∈ [0, 1],

Entm(µt) ≤ (1 − t) Entm(µ0) + t Entm(µ1) −K2

t(1 − t)W22 (µ0, µ1).

The N-Renyi entropy is defined as

S N(·|m) : P(X)→ (−∞, 0], S N(µ|m) := −∫

ρ1− 1N dm,

if ρ1− 1N ∈ L1(m), where µ = ρm, and 0 otherwise.

6

Let K ∈ R and N ∈ [1,∞), the distortion coefficents σ(t)K,N and τ(t)

K,N are defined as follows:

(t, θ) ∈ [0, 1] × R+ → σ(t)K,N(θ) :=

∞ if Kθ2 ≥ Nπ2

sin(tθ√

K/N)

sin(θ√

K/N)if 0 < Kθ2 < Nπ2,

t if Kθ2 = 0,sinh(tθ

√K/N)

sinh(θ√

K/N)if 0 < Kθ2 < 0,

andτ(t)

K,N(θ) := t1Nσ(t)

K,N(θ)1− 1N .

The standard finite dimensional curvature-dimension condition was introduced in [STU06b, LV09].

Definition 2.2. (CD(K,N) condition) Let K ∈ R and N ∈ [1,∞). We say that a m.m.s. (X, d,m) is a CD(K,N)space if for any two absolutely continuous measures µ0 = ρ0m, µ1 = ρ1m ∈ P(X) with bounded supportthere exists a W2-geodesic µtt∈[0,1] and an associated optimal coupling π between µ0 and µ1 such that for anyt ∈ [0, 1] and N′ ≥ N,

S N(µt|m) ≤ −∫ (

τ(1−t)K,N′ (d(x, y))ρ0(x)−

1N′ + τ(t)

K,N′(d(x, y))ρ1(y)−1

N′)

dπ(x, y).

The reduced curvature-dimension condition CD∗(K,N) was introduced in [BS10] for its seemingly bettertensorization and globalization properties. It is defined by replacing τ with σ in Definition 2.2. The CD(K,N)and CD∗(K,N) conditions generalize to the metric measure setting the notion of Ricci curvature bounded belowby K and dimension bounded above by N. Examples include (possibily weighted) Riemannian manifolds[STU06b], Finsler manifolds [OHT09] and Alexandrov spaces [PET11].

Remark 2.3. CD(K,N) implies CD(K′,N′) and CD∗(K′,N′) for all K′ ≤ K and N′ ≥ N as well as CD(K′,∞).A host of results that we cite were shown in the RCD∗(K,N) and RCD(K,∞) setting, see 2.12 for definitions,and therefore apply in the RCD(K,N) setting. Going in the other direction, it was shown in [CM16] thatRCD∗(K,N) is equivalent to RCD(K,N) when m(X) < ∞. It is believed that this argument can be takento the noncompact case. We mention that the proofs of this paper carry forward without modication to theRCD∗(K,N) setting. However, since several papers we cite use the stronger RCD assumption (though it can bechecked this is not needed for the particular results we cite from them), we will do so as well to ease the burdenof exposition.

It is known that if (X, d,m) is CD(K,N) then supp(m) is a geodesic space which also satisfies the CD(K,N)condition. Due to this, we will always assume X = supp(m). One can check that for any λ, c > 0, if (X, d,m) isCD(K,N), then (X, λd, cm) is CD( K

λ2 ,N). CD(K,N) spaces, like their smooth counterparts, satisfy the standardBishop-Gromov volume comparison.

Theorem 2.4. (Bishop-Gromov volume comparison [STU06b, Theorem 2.3]) Let (X, d,m) be a CD(K, N) spacefor some K ∈ R and N ∈ (1,∞). Then for all x0 ∈ X and all 0 < r < R ≤ π

√N − 1/(K ∨ 0) it holds:

m(Br(x0))m(BR(x0))

≥ VK,N(r,R) :=

∫ r0

(sin(t

√K/(N − 1))

)N−1

dt

∫ R0

(sin(t

√K/(N − 1))

)N−1

dtif K > 0,

(rR

)Nif K = 0,

∫ r0

(sinh(t

√K/(N − 1))

)N−1

dt

∫ R0

(sinh(t

√K/(N − 1))

)N−1

dtif K < 0.

(9)

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In contexts where K and N are clear, we will simply write V(r,R) for VK,N(r,R).For N < ∞, CD(K,N) spaces are locally doubling, by Theorem 2.4, and are therefore proper. They satisfy a

1-1 Poincare inequality by [RAJ12, Theorem 1.1].

2.2. First order calculus on metric measure spaces. We follow the framework for calculus on metric mea-sure space developed by Ambrosio, Gigli and Savare in [AGS13, AGS14a, AGS14b, GIG15, GIG18]. Let(X, d,m) be a metric measure space. Let lip(X), liploc(X), lipb(X) be its class of Lipschitz, locally Lipschitz,and bounded Lipschitz functions respectively. Given f ∈ liploc(X), the local Lipschitz constant (or local slope)lip( f ) : X → R is defined by

lip( f )(x) := lim supy→x

| f (y) − f (x)|d(y, x)

. (10)

By convention, lip( f )(x) := 0 at any isolated point x. Given f ∈ L2(m), a function g ∈ L2(m) is called a relaxedgradient if there exists a sequence fn ∈ lip(X) and g ∈ L2(m) so that

1. fn → f in L2(m) and lip( fn) converges weakly to g in L2(m)2. g ≥ g m-a.e..

A minimal relaxed gradient is a relaxed gradient that is minimal in L2-norm in the family of relaxed gradientsof f . If this family is non-empty, one can check that such a function exists and is unique m-a.e.. The minimalrelaxed gradient is denoted by |D f |. The domain of the Cheeger energy D(Ch) ⊆ L2(m) is the subset of L2

functions with a minimal relaxed gradient. For f ∈ L2(m), the Cheeger energy is defined as

Ch( f ) =

12

∫|D f |2 dm if f ∈ D(Ch),

∞ otherwise.

Ch is a convex and lower semicontinuous functional on L2(m). The Cheeger energy first introduced in [CHE99]was defined using a slightly different relaxation procedure. It is also possible to define a similar functional usingthe idea of minimal weak upper gradients, see [AGS14a, Section 5.1]. It is shown in [AGS14a, Section 6] thatunder mild assumptions on the metric measure space, satisfied, for example, by the various curvature-dimensionconditions, all these notions are equivalent.

Remark 2.5. Let (X, d,m) be an CD(K,N) space with N < ∞. For any Lipschitz function f on X, lip( f ) =

|D f | m-a.e. This follows from [CHE99], where it is shown that a metric measure space satisfying a Poincareinequality and a doubling inequality has lip( f ) = |D f | m-a.e..

W1,2(X) := D(Ch) is a Banach space endowed with the norm ‖ f ‖2W1,2(X):= ‖ f ‖2

L2(m)+‖ |D f | ‖2L2(m). We

define W1,2Loc(X) as the space of all function f ∈ L2(X,m) so that g f ∈ W1,2(X) for every compactly supported,

Lipschitz g. By the strong locality property of the minimal relaxed gradient (ie. |Dg|= |Dh| m-a.e. in g = hfor any g, h ∈ W1,2(X)), any f ∈ W1,2

Loc(X) has an associated differential |D f |∈ L2loc(m).

(X, d,m) is said to be infinitesimally Hilbertian if W1,2(X) is a Hilbert space. In this case, for f , g ∈ W1,2(X),one may define 〈D f ,Dg〉 using polarization: 〈D f ,Dg〉 := 1

2 (|D( f + g)|2−|D f |2−|Dg|2) ∈ L1(m).From here one can define the Laplacian. f ∈ W1,2(X) is said to be in the domain of the Laplacian ( f ∈ D(∆))

if there exists ∆ f ∈ L2(m) so that∫g∆ f dm +

∫〈Dg,D f 〉 dm = 0 for any g ∈ W1,2(X).

Given a subspace V ∈ L2(m), we denote DV (∆) := f ∈ D(∆) : ∆( f ) ∈ V. More generally, one may define themeasure valued Laplacian.

8

Definition 2.6. (Measure valued Laplacian [GIG18, Definition 3.1.2]) The space D(∆) ⊂ W1,2(X) is the spaceof f ∈ W1,2(X) such that there is a signed Radon measure µ satisfying∫

gdµ = −

∫〈Dg,D f 〉dm ∀g : X → R Lipschitz with bounded support.

In this case the measure µ is unique and is denoted by ∆ f .

2.3. Tangent, cotangent, and tensor modules. A technical framework for describing first order calculus onmetric measure spaces and second order calculus on RCD(K,N) spaces was developed by Gigli in [GIG18].While aspects of second order calculus can be effectively developed without this framework (see for exam-ple [SAV14, AGS15]), [GIG18] crucially gives constructions which generalize the notion of tensor fields. Inthe next few subsections, we will quickly introduce, sometimes informally, the necessary definitions givenin [GIG18] and refer to the original article for details and insights.

Let (X, d,m) be a metric measure space. The various collections of tensor fields of interest will be objects inthe category of Lp(m)-normed L∞(m)-modules.

Definition 2.7. (Lp-normed L∞-premodules [GIG18, Definition 1.2.1 1.2.10]) Let p ∈ [0,∞]. Let (M, ‖ · ‖M)be a Banach space endowed with a bilinear map L∞(m)×M 3 ( f , v) 7→ f ·v ∈ M and a function |·|:M→ Lp

+(m).We say (M, ‖ · ‖M, ·, |·|) is an Lp-normed L∞-premodule iff the following holds

1. ( f g) · v = f · (g · v) for all f , g ∈ L∞(m) and v ∈ M2. 1 · v = v for all v ∈ M where 1 is the constant function equal to 13. ‖ |v| ‖Lp(m)= ‖ v ‖M for all v ∈ M4. | f · v|= | f ||v|m-a.e. for all f ∈ L∞(m) and v ∈ M.

We will often simply write f v for f · v and call |·| the pointwise norm. If an Lp(m)-normed L∞(m)-premodule satisfies additional locality and gluing properties ( [GIG18, Definition 1.2.1]), we say it is anLp(m)-normed L∞(m)-module. One may localize such an object to some A ∈ B(X) by defining M|A :=v ∈ M : |v|= 0 m-a.e. on Ac, which is again canonically an Lp(m)-normed L∞(m)-module.

An L2-normed L∞-module which is a Hilbert space under ‖ · ‖M is called a Hilbert module. In this caseone can define a pointwise inner product by polarizing the pointwise norm |·|. The prototypical example of aHilbert module one has in mind is the collection of L2 vector fields on a Riemannian manifold, where |·| is theRiemannian pointwise norm.

Given L∞-modules M and N , we say a map T : M → N is a module morphism if it is a bounded linearmap betweenM and N as Banach spaces satisfying in addition T ( f v) = f T (v) for all f ∈ L∞(m) and v ∈ M.The dual module M∗ is the space of all module morphisms between M and L1(m) and is an Lp∗(m)-normedL∞(m)-module, where 1

p + 1p∗

= 1. A Hilbert moduleH is canonically isomoprhic to its dual.Given two Hilbert modulesH1 andH2, one can construct the Hilbert modules: the tensor productH1 ⊗H2

( [GIG18, Definition 1.5.1]) and the exterior productH1ΛH2 ( [GIG18, Definition 1.5.4]).

Definition 2.8. ( [GIG18, Definition 2.19]) LetM be an L2(m)-normed and V ⊆ M. Span(V) is defined as thecollection of v ∈ M for which there is a Borel decomposition (Xn)n∈N of X and, for each n ∈ N, collectionsv1,n, ..., vkn,n ∈ V and f1,n, ..., fkn,n ∈ L∞(m) so that

1Xnv =

kn∑i=1

fi,nvi,n , for each n ∈ N,

where 1Xn is the characteristic function of Xn. We say that V genereatesM iff Span V = X.

From this point we will assume (X, d,m) is a infinitesimally Hilbertian metric measure space. We now definethe tangent and cotangent modules of (X, d,m).

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Theorem 2.9. ( [GIG18, Proposition 2.2.5]) There exists a unique, up to isomorphism, Hilbert module Hendowed with a linear map d : W1,2(X)→ H satisfying:

1. |d f |= |D f | m-a.e. for all f ∈ W1,2(X)2. d(W1,2(X)) generatesH .

Such anH is called the cotagent module of (X, d,m) and denoted by L2(T ∗X).The dual of L2(T ∗X) is called the tangent module of (X, d,m) and denoted by L2(T X). Elements of L2(T X) arecalled vector fields.We denote by ∇ f the dual of d f (i.e. the unique element ∇ f ∈ L2(T X) so that v(∇ f ) = 〈v, d f 〉 for all v ∈L2(T ∗X)).

Notice 〈d f , dg〉 = d f (∇g) = 〈∇ f ,∇g〉 = 〈D f ,Dg〉 m-a.e. and we will use these interchangeably in the rest ofthe paper depending on the convention of the theorems we are quoting. For a discussion of the philosophicaldifferences of the objects involved, see [AGS14a, Section 2.2].

Definition 2.10. [GIG18, Definition 2.3.11] D(div) ⊆ L2(T X) is the space of all vector fields v ∈ L2(T X) forwhich there exists f ∈ L2(m) so that for any g ∈ W1,2(X) the equality∫

f g dm = −

∫dg(v) dm

holds. In this case f is called the divergence of v and denoted by div(v). In particular, if f ∈ D(∆), then∇ f ∈ D(div) and div(∇ f ) = ∆ f .

Definition 2.11. L2((T ∗)⊗2(X)) denotes the tensor product of L2(T ∗X) with itself (see [GIG18, Definition1.5.1])). Similarly, L2(T⊗2(X)) denotes the tensor product of L2(T X) with itself. We will use |·|HS and · : · todenote the pointwise norm (Hilbert-Schmidt norm) and the pointwise inner product of Hilbert modules whicharise from tensors.

L2((T ∗)⊗2(X)) and L2(T⊗2(X)) are Hilbert module duals of each other. We mention that for any elementA ∈ L2((T ∗)⊗2(X)), we will often write A(V,W) = A(V ⊗W). We will sometimes write this even when V and Ware so that V ⊗W is not in L2(T⊗2(X)). In all these cases, V ⊗W when multipled by the characteristic functionof a compact set will be in L2(T⊗2(X)) and so A(V,W) is well-defined as a measurable function by locality andsatisfies

A(V,W) ≤ |A|HS|V ||W |. (11)We will usually have additional assumptions on |V | and |W | so that A(V,W) ∈ L1

loc(m).

2.4. RCD(K,N) and Bakry-Emery conditions. We now introduce the notion of RCD spaces, which are themain objects of interest for this paper. These were proposed and carefully analyzed in a series of papers includ-ing [GIG15,EKS15,AMS19] in the finite dimensional case and [AGS14b,AGMR15] in the infinite dimensionalcase.

Definition 2.12. ( [AGS14b, GIG15]) Let (X, d,m) be a metric measure space, K ∈ R, and N ∈ [1,∞). We say(X, d,m) satisfies the Riemmanian curvature-dimension condition RCD(K,N) iff (X, d,m) satisfies the CD(K,N)condition and is infinitesimally Hilbertian. Similarly one defines the Riemannian curvature-dimension condi-tions RCD∗(K,N) and RCD(K,∞) using CD∗(K,N) and CD(K,∞) respectively.

The RCD condition is stable under measured Gromov-Hausdorff convergence and tensorization. Examplesof RCD spaces include Ricci limits and Alexandrov spaces but non-Riemannian Finsler geometries are ruledout. We now state some equivalent formulations of the RCD(K,N) property. We will in general assume (X, d,m)is infinitesimally Hilbertian in this subsection.

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As in [AGS15], we define the Carre du champ operator for f ∈ DW1,2(X)(∆) and ϕ ∈ DL∞(m)∩L2(m)(∆)∩L∞(m)by

Γ2( f ;ϕ) :=∫

12|∇ f |2∆ϕdm −

∫〈∇ f ,∇∆ f 〉ϕdm.

This enables us to state the non-smooth Bakry-Emery condition BE(K,N).

Definition 2.13. (Bakry-Emery condition [AGS15, EKS15]) Let K ∈ R and N ∈ [1,∞]. We say (X, d,m)satisfies the BE(K,N) condition iff

Γ2( f ;ϕ) ≥1N

∫(∆ f )2ϕdm + K

∫|∇ f |2ϕdm.

BE(K,N) is closely related to CD(K,N). We say (X, d,m) satisfies the Sobolev-to-Lipschitz property, [GH18,Definition 3.15], if any function f ∈ W1,2(X) with |∇ f |∈ L∞(m) has a Lipschitz representatitive f = f m-a.e.with Lipschitz constant equal to ess sup(|∇ f |).

Theorem 2.14. ( [AGS15, EKS15, AMS19]) Let (X, d,m) be a metric measure space satisfying an exponentialgrowth condition (see [AGS15, secion 3]), K ∈ R, and N ∈ (1,∞). (X, d,m) is RCD(K,N) iff (X, d,m) isinfinitesimally Hilbertian, satisfies the Sobolev-to-Lipschitz property and the BE(K,N) condition.

2.5. Heat flow and Bakry-Ledoux estimates. By applying the theory of the gradient flow of convex func-tionals on Hilbert spaces to Ch as in [AGS14a], one obtains for each f ∈ L2(m) a unique continuous curve(Ht( f ))t∈[0,∞) in L2(m) which is locally absolutely continuous in (0,∞) with H0( f ) = f so that

ddt

Ht( f ) = ∆′Ht( f ) for a.e. t ∈ (0,∞), (12)

where ∆′g is defined as the minimizer in L2 energy in ∂−(Ch) at g provided it is non-empty (see [AGS14a,Section 4.2]).

If (X, d,m) is infinitesimally Hilbertian, then for any t > 0, Ht( f ) ∈ D(∆) and one has the a priori estimates

‖Ht( f ) ‖L2≤ ‖ f ‖L2 , ‖ |DHt( f )| ‖2L2≤‖ f ‖2

L2

2t2 , ‖∆Ht( f ) ‖L2≤‖ f ‖L2

t.

Ht( f ) is linear and satisfies ∆(Ht( f )) = Ht(∆( f )) for any t > 0. In particular, (12) is true for all t ∈ (0,∞) and

Ht( f ) = f +

∫ t

0∆(Hs( f )) ds.

If (X, d,m) is RCD(K,∞), Ht can be identified with Ht, the gradient flow of Entm on P2(X). Due to contrac-tion properties coming from the RCD condition, Ht can be extended from D(Entm) to all of P2(X).

Definition 2.15. For t > 0 and x ∈ X, Ht(δx) is absolutely continuous with respect to m. Then Ht(δx) =

Ht(x, ·)m, where Ht(·, ·) is the heat kernel.

Ht(x, y) is symmetric and continuous in both variables. For each f ∈ L2(m), one has the representationformula, [AGS14b, Theorem 6.1],

Ht( f )(x) =

∫f (y)Ht(x, y) dm. (13)

The RCD(K,N) condition implies the Bakry-Ledoux estimate, which is a finite dimensional analogue of theBakry-Emery contraction estimate ( [AGS14b, Theorem 6.2]). Moreover, it was shown in [EKS15] that onehas equivalence in Theorem 2.14 with the BE(K,N) condition replaced by the (K,N) Bakry-Ledoux estimate.

Theorem 2.16. (Dimensional Bakry-Ledoux L2 gradient-Laplacian estimate [EKS15, Theorem 4.3]) Let (X, d,m)be an RCD(K,N) space for some K ∈ R and N ∈ [1,∞). For any f ∈ W1,2(X) and t > 0,

|∇(Ht( f ))|2+4Kt2

N(e2Kt − 1)|∆Ht( f )|2≤ e−2KtHt(|∇( f )|2) m-a.e..

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Remark 2.17. If |∇ f |∈ L∞, one can take continuous representatitves of ∆Ht( f ) and Ht(|∇( f )|2) and identify|∇(Ht( f ))| canonically with the local Lipschitz constant of Ht( f ) to obtain a pointwise Bakry-Ledoux bound,see [EKS15, Proposition 4.4]).

This implies the Sobolev-to-Lipschitz property by [AGS14b, Theorem 6.2]. Ht also has a L∞-to-Lipschitzproperty by [AGS14b, Theorem 6.8], where it was shown for t > 0 and f ∈ L2(m) that

2I2K(t)|∇Ht( f )|2≤ Ht( f 2) , m-a.e., (14)

where I2K(t) :=∫ t

0 e2Ks ds.

2.6. Second order calculus and improved Bochner inequality. The class of test functions was introducedin [SAV14] as

TestF(X) :=f ∈ D(∆) ∩ L∞ : |D f |∈ L∞ and ∆ f ∈ W1,2(X)

.

It is known that TestF(X) is an algebra and, on RCD(K,∞) spaces, it was shown by the results of [AGS14b]mentioned in the previous subsection that the heat flow approximations of an L∞∩L2 function are test functionsand so TestF(X) is dense in W1,2(X).

In [SAV14,HAN18], it was shown that under the BE(K,N) condition, |∇ f |2∈ D(∆) for any f ∈ TestF(X) andso one may define Γ2( f ) := 1

2∆|∇ f |2−〈∇ f ,∇∆ f 〉. One can then define a Hessian for f ∈ TestF(X) and showthe improved Bochner inequality (see 2.19). In section 3 of [GIG18], the same calculations were carried out inthe framework proposed therein. We outline the main definitions and results from there. In this subsection, weassume (X, d,m) is an RCD(K,N) space.

Definition 2.18. ( [GIG18, Definition 3.3.1]) W2,2(X) ⊆ W1,2(X) is the space of all functions f ∈ W1,2(X) forwhich there exists A ∈ L2((T ∗)⊗2(X)) so that for any g1, g2, h ∈ TestF(X) the equality

2∫

hA(∇g1,∇g2) dm

=

∫−〈∇ f ,∇g1〉 div(h∇g2) − 〈∇ f ,∇g2〉 div(h∇g1) − h〈∇ f ,∇〈∇g1,∇g2〉〉 dm

holds. In this case A is called the Hessian of f and denoted by Hess f . W2,2(X) is a Hilbert space under thenorm

‖ f ‖2W2,2(X):= ‖ f ‖2L2(m)+‖ d f ‖2L2(T ∗X)+‖Hess f ‖L2((T ∗)⊗2(X)).

It turns out that the test functions are contained in W2,2(X) and one has the improved Bochner inequality asin [SAV14, HAN18].

Theorem 2.19. (Improved Bochner inequality [GIG18, Theorem 3.3.8]) Let (X, d,m) be an RCD(K,N) spacefor K ∈ R and N ∈ [1,∞) and f ∈ TestF(X). Then f ∈ W2,2(X) and

Γ2( f ) ≥ [K|∇ f |2+|Hess( f )|2HS]m.

H2,2(X) is then defined as the closure of TestF(X) in W2,2(X). An approximation argument gives the follow-ing.

Corollary 2.20. ( [GIG18, Corollary 3.3.9]) D(∆) ⊆ W2,2(X) and for f ∈ D(∆),∫|Hess f |2HS dm ≤

∫ [(∆ f )2 − K|∇ f |2

]dm.

Finally, we introduce the analogue of vector fields which have a first order (covariant) derivative.

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Definition 2.21. ( [GIG18, Definition 3.4.1]) W1,2C (T X) ⊆ L2(T X) is the space of all v ∈ L2(T X) for which

there exists T ∈ L2(T⊗2(X)) so that for any g1.g2, h ∈ TestF(X) the equality∫hT : (∇g1 ⊗ ∇g2) =

∫−〈v,∇g2〉 div (h∇g1) − h Hess(g2)(v,∇g2) dm

holds. In this case T is called the covariant derivative of v and denoted by ∇v. W1,2C (T X) is a Hilbert space

under the norm‖ v ‖2

W1,2C (T X)

:= ‖ v ‖2L2(T X)+‖ ∇v ‖2L2(T⊗2(X)).

The class of test vector fields is defined as

TestV(X) :=

n∑i=1

gi∇ fi : n ∈ N, fi, gi ∈ TestF(X)

.By [GIG18, Theorem 3.4.2], TestV(X) ⊆ W1,2

C (T X) and so H1,2C (T X) is defined to be the closure of TestV(X) in

W1,2C (T X). For any f ∈ W2,2(X), Hess f and ∇(∇ f ) are dual under the duality of L2((T ∗)⊗2(X)) and L2(T⊗2(X)).

2.7. Non-branching and essentially non-branching spaces. Given a geodesic metric space (X, d), we definethe space of constant speed geodesics

Geo(X) := γ ∈ C([0, 1], X) : d(γ(s), γ(t)) = |s − t|d(γ(0), γ(1)) ∀s, t ∈ [0, 1] .

For each t ∈ [0, 1], et : Geo(X) → X defined by et(γ) := γ(t) denotes the evaluation map at time t. On acomplete and separable metric space (X, d), any W2-geodesic has a lifting to a measure on the space of geodesicsin the following sense.

Theorem 2.22. [LIS07, Theorem 3.2] Let (µt)t∈[0,1] be a W2-geodesic. Then there exists π ∈ P(Geo(X)) sothat

(et)∗(π) = µt ∀t ∈ [0, 1],

|µt|2=

∫|γt|

2 dπ(γ), for a.e. t ∈ [0.1],

where et(γ) := γ(t) is the evaluation map at time t.

These are called optimal dynamical plans. This motivates the following definition: for any µ0, µ1 ∈ P2(X),we denote by OptGeo(µ0, µ1) the space of all optimal dynamical plans from µ0 to µ1.

Definition 2.23. Given two geodesics γ1 6= γ2 on a geodesic metric space (X, d). Assume γ1, γ2 are constantspeed and parameterized on the unit interval. we say γ1 and γ2 branch if there exists 0 < t < 1 such thatγ1

s = γ2s for all s ∈ [0, t]. A subset S ⊆ Geo(X) is called a set of non-branching geodesics if there are no

branching pairs in S . A geodesic metric space for which Geo(X) is itself a set of non-branching geodesics iscalled non-branching.

Many results were shown for various types of CD spaces under the additional non-branching assumption.These include the local-to-global property, tensorization property and local Poincare inequality, see [STU06a,BS10, LV17]. A weaker assumption was introduced in [RS14] for which these results generalize.

Definition 2.24. A metric measure space (X, d,m) is called essentially non-branching if for any µ0, µ1 ∈ P2(X)absolutely continuous with respect to m, any element of OptGeo(µ0, µ1) is concentrated on a set of non-branching geodesics.

RCD(K,∞) spaces are shown to be essentially non-branching by the results of [DS08,AGS14b] and [RS14].We will frequently refer to the following theorem from [GRS16] shown for finite dimensional RCD(K,N)spaces, see also [RAJ12, RS14] for related results and [CM17a] for the same result in the case of essentiallynon-branching MCP(K,N) spaces.

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Theorem 2.25. ( [GRS16], [CM17a, Theorem 1.1]) Let (X, d,m) be an RCD(K,N) space for some K ∈ R andN ∈ [1,∞). If µ0, µ1 ∈ P2(X) with µ0 = ρ0m m, then there exists a unique ν ∈ OptGeo(µ0, µ1). (et)∗(ν) mfor any t ∈ [0, 1) and such ν is given by a unique map S : supp(µ0) → Geo(X) in the sense that ν = S ∗(µ).Moreover, if µ0, µ1 have bounded support and ‖ ρ0 ‖L∞(m)< ∞, then

‖ ρt ‖L∞(m)≤1

(1 − t)N eDt√

(N−1)K−‖ ρ0 ‖L∞(m), ∀t ∈ [0, 1), (15)

where D := diam(supp(µ0) ∪ supp(µ1)) and K− := max −K, 0

Remark 2.26. In particular, this implies for any p ∈ X, there is a unique geodesic between p and x for m-a.e.x ∈ X. Using the Kuratowski and Ryll-Nardzewski measurable selection theorem, one may select constantspeed geodesics γx,p from all x ∈ X to p so that the map X × [0, 1] 3 (x, t) 7→ γx,p(t) is Borel. This thenguarantees such a choice is unique up to a set of measure 0. Similarly, one may select constant speed geodesicsγx,y for all x, y ∈ X so that the map X×X×[0, 1] 3 (x, y, t) 7→ γx,y(t) is Borel. Using, for example, the argumentsin [CAV14, Section 4], the set of points (x, y) ∈ X × X connected by non-unique geodesics is analytic. (m×m)-almost everywhere uniquness of the Borel selection then follows using Fubini’s theorem. In cases where we fixgeodesics in this manner, we say we take a Borel selection of geodesics γx,p from all x ∈ X to p (or γx,y fromall x ∈ X to all y ∈ X).

2.8. RCD(K,N) structure theory. We review of the structure theory of RCD spaces in this subsection. Wewill assume basic familiarity with pointed Gromov-Hausdorff (pGH) and pointed measure Gromov-Hausdorff(pmGH) convergence and refer to [BBI01, VIL09, GMS15] for details.

A notion of considerable interest for RCD(K,N) spaces is that of measured tangents. Similar objects havebeen well-studied in the setting of Alexandrov spaces and Ricci limits (see, for example, [BGP92] and [CN13]for an overview). Given a m.m.s. (X, d,m), x ∈ X and r ∈ (0, 1), consider the normalized rescaled pointedm.m.s. (p.m.m.s.) (X, r−1d,mx

r , x) where

mxr :=

( ∫Br(x)

1 −d(x, y)

rdm(y)

)−1m.

In what follows let (X, d,m) be an RCD(K,N) space for some K ∈ R and N ∈ (1,∞). We define

Definition 2.27. (The collection of tangent spaces Tan(X, d,m, x)) Let x ∈ X. A p.m.m.s. (Y, dY ,mY , y) is calleda tangent (cone) of (X, d,m) at x if there exists a sequence of radii ri ↓ 0 so that (X, r−1

i d,mxri, x)→ (Y, dY ,mY , y)

as i→ ∞ in the pmGH topology. The collection of all tangents of (X, d,m) at x is denoted Tan(X, d,m, x).

A standard compactness argument by Gromov shows that Tan(X, d,m, x) is non-empty for any x ∈ X. Therescaling and stability properties of the RCD(K,N) condition under pmGH convergence (see [GMS15, Theorem7.2] and [AGS14b, Theorem 6.11]) show that every element of Tan(X, d,m, x) is an RCD(0,N) space.

Let ck =∫

B1(0) 1−|x| dL (x) be the normalization constant of the k-dimensional Lebesgue measure and definethe k-dimensional regular set Rk by

Rk := x ∈ X : Tan(X, d,m, x) = (Rk, dE , ckHk, 0k).

Define Rreg :=bNc⋃k=1Rk the regular set of X and S := X\Rreg the singular set. In [GMR15], it was shown that,

for m-a.e. x ∈ X, there exists some k ∈ N, 1 ≤ k ≤ N so that (Rk, dE , ckH k, 0k) ∈ Tan(X, d,m, x). This wasimproved in [MN19], where it was shown that each Rk is m-measurable and m(S) = 0. A final improvementwas made in [BS20] with the following theorem.

Theorem 2.28. (Constancy of the dimension [BS20, Theorem 3.8]) Let (X, d,m) be an RCD(K,N) m.m.s. forsome K ∈ R and 1 ≤ N < ∞. Assume X is not a point. There exists a unique n ∈ N, 1 ≤ n ≤ N so thatm(X\Rn) = 0.

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The same theorem was proved in the case of Ricci limits in [CN12] from the main result there and as suchalso follows from Theorem 1.2, see Theorem 6.2. By [KIT18, Theorem 1.2], see also [KL18, Theorem 1.9] inthe case of Ricci limits, it is known that the unique n in Theorem 2.28 is also the largest integer n for which Rnis non-empty.

2.9. Additional RCD(K,N) theory. In this subsection we record several theorems for RCD(K,N) spaceswhich will be of use later.

Define the coefficients σK,N(·) : [0,∞)→ R by

σK,N(θ) :=

θ

√KN

cotan

θ√KN

, if K > 0,

1 if K = 0,

θ

√−

KN

cotanh

θ√−KN

, if K < 0,

then one has the following sharp bound on the measure valued Laplacian of distance functions.

Theorem 2.29. (Laplacian comparison for the distance function [GIG15, Corollary 5.15]) Let (X, d, m) be acompact RCD(K,N) space for some K ∈ R and N ∈ (1,∞). For x0 ∈ X denote by dx0 : X → [0,∞) the functionx 7→ d(x, x0). Then

d2x0

2∈ D(∆) with ∆

d2x0

2≤ NσK,N(dx0)m ∀x0 ∈ X

and

dx0 ∈ D(∆, X\x0) with ∆dx0 |X\x0 ≤NσK,N(dx0) − 1

dx0

m ∀x0 ∈ X.

∆dx0 |X\x0 is defined similar to Definition 2.6, the difference being that the test functions g must be com-pactly supported in X\x0.

Remark 2.30. We mention, and will use the fact, that in the case where X is noncompact one can make es-sentially the same statement. Some small adjustments are needed since the Laplacians of dx0 on X\x0 andd2

x0on X are not naturally guaranteed to be signed Radon measures. To accomodate this, a weakening of the

definition of the measure valued Laplacian was given in [CM18, Definition 2.11, 2.12]. This definition allowsthe Laplacian to be, more generally, a Radon functional (i.e. in (CC(X))′). The difference is that a Radonfunctional has a representation as the difference of two possibly infinite positive Radon measures by the Riesz-Markov-Kakutani representation theorem, whereas in the case of a signed Radon measure, at least one of thesemust be finite. It was shown in [CM18, Corollary 4.17, 4.19] that the Laplacian comparison for the distancefunction holds as stated with this weaker definition. As such, we will, by a slight abuse of notation, treat theLaplacian of the distance function dx0 on X\x0 as a signed Radon measure in the few instances where we useTheorem 2.29 in integration against compactly supported functions. Note that due to the comparison theorem,this Laplacian is locally a signed Radon measure, having at most an infinite negative part.

We will also need the Li-Yau Harnack inequality [LY86] and the Li-Yau gradient inequality [BG11, LY86].These were proved for the RCD setting, in the finite measure case in [GM14], and in general in [JIA15].

Theorem 2.31. (Li-Yau Harnack inequality [GM14, JIA15]) Let (X, d,m) be an RCD(K,N) space for someK ∈ R and N ∈ [1,∞). Let f ∈ Lp(m) for some p ∈ [1,∞) be non-negative. If K ≥ 0, then for every x, y ∈ Xand 0 < s < t it holds that

(Ht f )(y) ≥ (Hs f )(x)ed2(x,y)

4(t−s)e2Ks

3(1 − e

2K3 s

1 − e2K3 t

) N2.

15

If instead K < 0, then

(Ht f )(y) ≥ (Hs f )(x)ed2(x,y)

4(t−s)e2Kt

3(1 − e

2K3 s

1 − e2K3 t

) N2.

Theorem 2.32. (Li-Yau gradient inequality [GM14, JIA15]) Let (X, d,m) be an RCD(K,N) space for someK ∈ R and N ∈ [1,∞). Let f ∈ Lp(m) for some p ∈ [1,∞) be non-negative. Then for every t > 0 it holds that

|∇Ht f |2≤ e−2Kt

3 (∆Ht f )Ht f +NK3

e−4Kt

3

1 − e−2Kt

3

(Ht f )2 m-a.e..

RCD(K,N) spaces also satisfy the parabolic maximum principle, see [LI18, Section 3] and [GH08, Section4.1] for full details.

Definition 2.33. ( [LI18, Definition 3.1]) Let (X, d,m) be an RCD(K,N) space with K ∈ R and N ∈ (1,∞). LetI be an open interval in R, Ω be an open seubset of X, and g ∈ L2(Ω). We say that a function u : I → W1,2(Ω)satisfies the parabolic equation

∂

∂tu − ∆u ≤ g, weakly in I ×Ω,

if for every t ∈ I, the Frechet derivative of u, denoted by ∂∂t u, exists in L2(Ω) and for any nonnegative function

ψ ∈ W1,2(Ω), it holds ∫Ω

∂

∂tu(t, ·)ψ dm + E(u(t, ·), ψ) ≤

∫Ω

gψ dm.

Theorem 2.34. (Parabolic maximum principle [LI18, Lemma 3.2]) Let (X, d,m) be an RCD(K,N) space withK ∈ R and N ∈ (1,∞). Fix T ∈ (0,∞] and open subset Ω ⊆ X. Assume that a function u : (0,T ) → W1,2(Ω),with u+(t, ·) = maxu(t, ·), 0 ∈ W1,2(Ω) for any t ∈ (0,T ), satisfies the following equation with initial valuecondition: ∂

∂t u − ∆u ≤ 0, weakly in (0,T ) ×Ω,

u+(t, ·)→ 0, in L2(Ω) as t → 0.Then u(t, x) ≤ 0 for any t in (0,T ) and m-a.e. x in Ω.

2.10. Mean value and integral excess inequalities. We refer to [CN12] in the smooth case, and [MN19] inthe RCD case, for the proofs of the statements in this subsection. We start with the existence of good cut off

functions.

Lemma 2.35. (Existence of good cut off functions [MN19, Lemma 3.1]) Let (X, d,m) be an RCD(K,N) spacefor some K ∈ R and N ∈ [1,∞). Then for every x ⊂ X, for every R > 0 and 0 < r < R there exists a Lipschitzfunction ψr : X → R satisfiying:

1. 0 ≤ ψr ≤ 1 on X, ψr ≡ 1 on Br(x) and supp(ψ) ⊂ B2r(x);2. r2|∆ψr |+r|∇ψr |≤ C(K,N,R) m-a.e..

For any subset C in a metric space, we denote by Tr(C) the r-tubular neighbourhood of C and for r1 > r0 > 0,Ar0,r1(C) := Tr1(C)\Tr0(C) the (r0, r1)-annular neighbourhood of C.

Lemma 2.36. (Existence of good cut off functions on annular neighbourhoods [MN19, Lemma 3.2]) Let(X, d,m) be an RCD(K,N) space for some K ∈ R and N ∈ [1,∞). Then for every closed subset C ⊂ X,for every R > 0 and 0 < 10r0 < r1 ≤ R there exists a Lipschitz function ψ : X → R satisfiying:

1. 0 ≤ ψ ≤ 1 on X, ψ ≡ 1 on A3r0,r13

(C) and supp(ψ) ⊂ A2r0,r12

(C);

2. r20 |∆ψ|+r0|∇ψ|≤ C(K,N,R) m-a.e. on A2r0,3r0(C);

3. r21 |∆ψ|+r1|∇ψ|≤ C(K,N,R) m-a.e. on A r1

3 ,r12

(C).

16

Remark 2.37. Note that the gradient bounds in 2.35 and 2.36 are naturally m-a.e.. However, since the proofinvolves only 2.16, by Remark 2.17, choosing continuous representatitves and using the local Lipschitz constantof ψ for |∇ψ|, these statements can be made pointwise.

As demonstrated in [CN12], and later for the RCD setting in [MN19], several key estimates, including heatkernel bounds, the mean value, L1-Harnack, and integral Abresch-Gromoll inequalities can be proved startingfrom the existence of good cut off functions and the Li-Yau Harnack inequality 2.31.

Lemma 2.38. (Heat kernel bounds [MN19, lemma 3.3]) Let (X, d,m) be an RCD(K,N) space for some K ∈ R,N ∈ (1,∞) and let Ht(x, y) be the heat kernel for some x ∈ X. Then for every R > 0, for all 0 < r < R andt ≤ R2,

1. if y ∈ B10√

t(x), then C−1(K,N,R)m(B

10√

t(x)) ≤ Ht(x, y) ≤ C(K,N,R)m(B

10√

t(x))

2.∫

X\Br(x)Ht(x, y)dm(y) ≤ C(K,N,R)r−2t.

Lemma 2.39. (Mean value and L1-Harnack inequality [MN19, Lemma 3.4]) Let (X, d,m) be an RCD(K,N)space for some K ∈ R, N ∈ (1,∞) and let 0 < r < R. If u : X × [0, r2] → R, u(x, t) = ut(x), is a nonnegativeBorel function with compact support at each time t and satisfiying (∂t − ∆)u ≥ −c0 in the weak sense, then,?

Br(x)

u0 ≤ C(K,N,R)[ur2(x) + c0r2] for m-a.e. x.

More generally the following L1-Harnack inequality holds?Br(x)

u0 ≤ C(K,N,R)[ess infy∈Br(x)

ur2(y) + c0r2] ∀x ∈ X.

Remark 2.40. Lemma 3.4 of [MN19], whose proof follows [CN12, Lemma 2.1], treats the continuous case ofu. However, in what follows we will want to use this inequality for ut = |∇ht( f )|, which is not known to havea continuous representatitive. The proof of this statement for Borel u follows exactly as in the continuous casewith the obvious measure-theoretic adjustments.

Applying 2.39 to a function which is constant in time gives the following classical mean value inequality.

Corollary 2.41. (Classical mean value inequality [MN19, Corollary 3.5]) Let (X, d,m) be an RCD(K, N) spacefor some K ∈ R, N ∈ (1,∞) and let 0 < r < R. If u : X → R is a nonnegative Borel function with compactsupport with u ∈ D(∆) and satisfies ∆u ≤ c0m in the sense of measures, then for 0 < r ≤ R,?

Br(x)

u ≤ C(K,N,R)[u(x) + c0r2] for m-a.e. x.

This, in combination with the existence of good cut off functions and Laplacian estimates on distance func-tions, allows one to prove an integral Abresch-Gromoll inequality. For points p and q in a metric space, wedefine the excess function ep,q(x) := d(p, x) + d(x, q) − d(p, q).

Theorem 2.42. (Integral Abresch-Gromoll inequality [MN19, Theorem 3.6]) Let (X, d,m) be an RCD(K,N)space for some K ∈ R, N ∈ (1,∞); let p, q ∈ X with dp,q := d(p, q) ≤ 1 and fix 0 < ε < 1.If x ∈ Aεdp,q,2dp,q(p, q) satisfies ep,q(x) ≤ r2dp,q ≤ r(K,N, ε)2dp,q, then?

Brdp,q (x)

ep,q(y)dm(y) ≤ C(K,N, ε)r2dp,q.

17

Combined with Bishop-Gromov volume comparison, this immediately implies the classical Abresch-Gromollinequality [AG90].

Corollary 2.43. (Classical Abresch-Gromoll inequaltiy [MN19, Corollary 3.7]) Let (X, d,m) be an RCD(K,N)space for some K ∈ R, N ∈ (1,∞); let p, q ∈ X with dp,q := d(p, q) ≤ 1 and fix 0 < ε < 1.If x ∈ Aεdp,q,2dp,q(p, q) satisfies ep,q(x) ≤ r2dp,q ≤ r(K,N, ε)2dp,q, then there exists α(N) ∈ (0, 1) such that

ep,q(y) ≤ C(K,N, ε)r1+α(N)dp,q , ∀y ∈ Brdp,q(x).

3. Differentiation formulas for Regular Lagrangian flows

3.1. Regular Lagrangian flow. In what follows, we will always be on some RCD(K,N) space (X, d,m) forK ∈ R and N ∈ [1,∞). In [CN12], the crucial idea is to understand the geometric properties of the gradientflow with respect to heat flow approximations of the distance function. We will do the same with the RegularLagrangian flow which was first introduced by Ambrosio in [AMB04] on Rd and generalized to the metricmeasure setting by Ambrosio-Trevisan in [AT14]. The setup is quite general and we refer to [AT14] for fulldetails. We will be interested in applying this theory specificially for vector fields in the L2 tangent module ofan RCD space.

Definition 3.1. (Time-dependent L2 vector fields) Let T > 0. V : [0,T ] → L2(T X) is a time-dependent L2

vector field iff the map [0,T ] 3 t 7→ Vt ∈ L2(T X) is Borel.V is bounded iff

‖V ‖L∞ := ‖ |V | ‖L∞([0,T ]×X)< ∞.

V ∈ L1([0,T ], L2(T X)) iffT∫

0

‖Vt ‖L2(T X)dt < ∞.

Definition 3.2. (Regular Lagrangian flow) Given a time-dependent L2 vector field (Vt). A Borel map F :[0,T ] × X → X is a Regular Lagrangian flow (RLF) to Vt iff the following holds:

1. F0(x) = x and [0,T ] 3 t 7→ Ft(x) is continuous for every x ∈ X;2. For every f ∈ TestF(X) and m-a.e. x ∈ X, t 7→ f (Ft(x)) is in W1,1([0,T ]) and

ddt

f (Ft(x)) = d f (Vt)(Ft(x)) for a.e. t ∈ [0,T ]; (16)

3. There exists a constant C := C(V) so that (Ft)∗m ≤ Cm for all t in [0,T ].

Remark 3.3. In the case where V ∈ L1([0,T ], L2(T X)) and F is an RLF of V , using a standard Fubini’s theoremargument and that TestF(X) is dense in W1,2(X), we have for every f ∈ W1,2

loc (X), t 7→ f (Ft(x)) is in W1,1([0,T ])and

ddt

f (Ft(x)) = d f (Vt)(Ft(x)) for a.e. t ∈ [0,T ]. (17)

[AT14] gives the existence and uniqueness of RLFs to (Vt) in a certain class of vector fields. We use thefollowing weaker formulation of their result and note that only a bound on the symmetric part of ∇Vt is needed.

Theorem 3.4. (Existence and uniqueness of Regular Lagrangian flow [AT14]) Let (Vt) ∈ L1([0,T ], L2(T X))satisfy Vt ∈ D(div) for a.e. t ∈ [0,T ] with

div(Vt) ∈ L1([0,T ], L2(m)) (div(Vt))− ∈ L1([0,T ], L∞(m)) ∇Vt ∈ L1([0,T ], L2(T⊗2X)).

18

There exists a unique, up to m-a.e. equality, RLF (Ft)t∈[0,T ] for (Vt). The bound

(Ft)∗(m) ≤ exp(

t∫0

‖ div(Vs)− ‖L∞(m) ds)m (18)

holds for every t ∈ [0,T ].

Remark 3.5. It was pointed out to the author by Nicola Gigli that the estimate (18) can be localized for anyS ∈ B(X),

(Ft)∗(m|S ) ≤ exp(

t∫0

‖ div(Vs)− ‖L∞((Fs)∗(m|S )) ds)m.

This follows from [AT14, (4-22)] choosing β(z) := zp for p→ ∞.

For (Ft) an RLF to some (Vt), we will be interested in expressing ddt d(Ft(x), Ft(y)) in two ways: using Vt in

a first order variation formula and ∇Vt in a second order formula, see (8), which we show in Subection 3.3.

Proposition 3.6. (First order differentiation formula along RLFs) Let T > 0 and U,V ∈ L1([0,T ], L2(T X)). If(Ft), (Gt) are the Regular Lagrangian flows of (Ut), (Vt) respectively, then for m-a.e. x, y ∈ X, d(Ft(x),Gt(y)) ∈W1,1([0,T ]) and

ddt

d(Ft(x),Gt(y)) = 〈∇dGt(y),Ut〉(Ft(x)) + 〈∇dFt(x),Vt〉(Gt(y)) for a.e. t ∈ [0,T ].

Proof. It is known that RCD(K,∞) spaces have the tensorization of Cheeger energy property from [AGS14b,Theorem 6.17] and the density of the product algebra property from [BS20, Proposition A.1], see also [GR18,Defintion 3.8, 3.9] for definitions. Consider the vector field (Wt) defined by requiring, for all f ∈ W1,2(X × X),

〈Wt,∇ f 〉(x, y) = 〈Ut,∇ fy〉(x) + 〈Vt,∇ fx〉(y),

for (m×m)-a.e. (x, y) ∈ X × X. The tensorization of Cheeger energy is used implicitly in this definition and thevector field is naturally in L1([0,T ], L2

loc(T (X × X))). We refer to [GR18] for a rigorous treatment of locally L2

vector fields and the corresponding theory of RLFs. We mention a slightly more careful, alternative definitionof (Wt) was also given in [GR18, Proposition 3.7, Theorem 3.13], where the expected decomposition of themodule L0(T ∗(X × X)) was shown for spaces with tensorization of Cheeger energy and density of the productalgebra properties. By [BS20, Proposition A.2], (Ft,Gt) is an RLF of (Wt), from which the proposition followsby definition of (Wt).

3.2. Continuity equation. We give a brief summary of the theory of continuity equations in this section. Theseare intimately related to Regular Lagrangian flows but provide a more convenient language for the discussionof local flows in cases where RLFs, which as defined are of a global nature, may not exist.

Definition 3.7. Curves of bounded compression [GIG18, Definition 2.3.21] We say a curve (µt)t∈[0,T ] ⊆ P2(X)is a curve of bounded compression iff

1. It is W2-continuous;2. For some C > 0, µt ≤ Cm for every t ∈ [0,T ].

Definition 3.8. (Solutions of continuity equation [GH15], [GIG18, Definition 2.3.22]) Let (µt)t∈[0,T ] ⊆ P2(X)be a curve of bounded compression and V ∈ L1([0,T ],L2(T X)). We say that (µt,Vt) solves the continuityequation

ddtµt + div(Vtµt) = 0

19

iff, for every f ∈ W1,2(X), the map t 7→∫

f dµt is absolutely continuous and satisfies

ddt

∫f dµt =

∫d f (Vt) dµt for a.e. t ∈ [0,T ]. (19)

Remark 3.9. By abuse of notation we will sometimes say (µt) solves the continuity equation ddtµt +div(Vtµt) = 0

for some vector field (Vt) which is only locally L2, for example, Vt = −∇dp for some p ∈ X. In this case, (µt)is always compactly supported for every t and it is understood that we cut off the vector field Vt outside of thissupport.

As shown in [AT14], RLFs are very closely related to the solutions of continuity equations; they can bethought of as realizations of these solutions as maps on the space itself.

Theorem 3.10. ( [AT14]) Let (Vt) satisfy the conditions of 3.4 and (Ft) be the corresponding unique RegularLagrangian flow. If µ0 ∈ P2(X) with bounded density, then µt := (Ft)∗(µ0) is a distributional solution of thecontinuity equation d

dtµt + div(Vtµt) = 0.

Remark 3.11. The existence and uniqueness of solutions to ddtµt + div(Vtµt) = 0 starting at some µ0 of bounded

density is proved in [AT14] for (Vt) satisfying the conditions of Theorem 3.4. In fact, the existence and unique-ness of RLFs in 3.4 is shown in part by using the existence and uniqueness on the level of continuity equationscombined with a superposition principle.

RLFs from vector fields with a two-sided divergence bound are m-a.e. invertible. To be precise,

Proposition 3.12. Let (Vt) ∈ L1([0,T ], L2(T X)) satisfy Vt ∈ D(div) for a.e. t ∈ [0,T ] with

div(Vt) ∈ L1([0,T ], L2(m)) div(Vt) ∈ L1([0,T ], L∞(m)) ∇Vt ∈ L1([0,T ], L2(T⊗2X)).

Let (Ft) be the unique RLF of (Vt)t∈[0,T ] and (Gt) be the unique RLF of (−VT−t)t∈[0,T ]. For m-a.e. x ∈ X and any0 ≤ t ≤ T,

Gt(FT (x)) = FT−t(x).

Proof. We first show GT (FT (x)) = x for m-a.e. x ∈ X. Define the time-dependent L2 vector field (Wt)t∈[0,2T ] by

Wt :=

Vt if 0 ≤ t ≤ T−V2T−t if T < t ≤ 2T.

For any µ with compact support and bounded density, (µt)t∈[0,2T ] defined by

µt :=

(Ft)∗(µ) if 0 ≤ t ≤ T(Gt−T )∗((FT )∗(µ)) if T < t ≤ 2T

solves the continuity equation ddtµt + div(Wtµt) = 0 by Theorem 3.10. This in particular means (µt)t∈[0,T ] solves

the continuity equation ddtµt + div(Vtµt) = 0 on [0,T ]. It is then easy to check by definition that

µ′t :=

µt if 0 ≤ t ≤ Tµ2T−t if T < t ≤ 2T

solves the continuity equation ddtµ′t +div(Wtµ

′t) = 0 as well. By uniqueness, see Remark 3.11, (GT )∗((FT )∗(µ)) =

µ. Since this is true for any µwith compact support and bounded density, we conclude GT (FT (x)) = x for m-a.e.x ∈ X.

By the same argument for each t in the countable set Q ∩ [0,T ], we have for m-a.e. x ∈ X and any t ∈Q ∩ [0,T ],

Gt(FT (x)) = FT−t(x).The proposition follows by continuity of Gt(x) and Ft(x) in t for all x ∈ X.

20

We recall the following result fom [GIG13] which in particular implies W2-geodesics with uniformly boundeddensities are solutions of continuity equations.

Theorem 3.13. ( [GT18, Theorem 1.1]) Let µt be a W2-geodesic with compact support and µt ≤ Cm for everyt ∈ [0, 1] and some C > 0. If f ∈ W1,2(X) then the map [0, 1] 3 t 7→

∫f dµt is C1([0, 1]) and

ddt

∫f dµt =

∫〈∇ f ,∇φt〉 dµt,

where φt is any function such that for some s 6= t, s ∈ [0, 1], the function −(s − t)φt is a Kantorovich potentialfrom µt to µs.

The corollary below then follows by making the same type of arguments as in [AT14, Section 7].

Corollary 3.14. Let p ∈ X and f ∈ W1,2(X) fixing a representatitive. For m-a.e. x ∈ X, the map t 7→ f (γx,p(t))is in W1,1

loc ([0, dx,p)) and

ddt

f (γx,p(t)) = −d f (∇dp)(γx,p(t)) for a.e. t ∈ [0, dx,p),

where γx,p is a unit speed geodesic from x to p.

Proof. By Remark 2.26, we take a Borel selection of γx,p which is unique for m-a.e. x. For each x ∈ X, letγx,p : [0, 1]→ X be the constant speed reparameterization of γ.

First consider a Lipschitz representative of f ∈ TestF(X). Clearly f (γx,p(t)) is continuous on t ∈ [0, 1] foreach x. We show

1. for m-a.e. x, −d(γx,p(t),p)1−t 〈∇ f ,∇dp〉(γx,p(t)) ∈ L1

loc([0, 1));

2. for m-a.e. x, f (γx,p(b)) − f (γx,p(a)) =∫ b

a−d(γx,p(t),p)

1−t 〈∇ f ,∇dp〉(γx,p(t)) dt for any 0 ≤ a < b < 1.For any µwith compact support and bounded density with respect to m, define µt := (γ·,p(t))∗(µ) for t ∈ [0, 1].

(µt)t∈[0,1] is a W2-geodesic. By 2.25, for any δ > 0, (µt)t∈[0,1−δ] is of uniformly bounded density. By Theorem3.13, the map [0, 1 − δ] 3 t 7→

∫f dµt is in C1([0, 1 − δ]) and

ddt

∫f dµt =

∫−d(x, p)

1 − t〈∇ f ,∇dp〉(x)dµt(x). (20)

Fix a representative of 〈∇ f ,∇dp〉 ∈ L2(m).Proof of 1: The map t 7→

∫−d(x,p)

1−t 〈∇ f ,∇dp〉(x) dµt(x) is in L1loc([0, 1)) since 〈∇ f ,∇dp〉(x) ∈ L1

loc(m) andµt has uniformly bounded support and locally unifomly bounded density on [0, 1). By Fubini’s theorem, thisimplies for µ-a.e. x, −d(γx,p(t),p)

1−t 〈∇ f ,∇dp〉(γx,p(t)) ∈ L1loc([0, 1)). Since this is true starting at any measure µ with

compact support and bounded density with respect to m, statement 1 follows.Proof of 2: By continuity of f (γx,p(t)) in t, it is enough to show that, for m-a.e. x,

f (γx,p(kn

)) − f (γx,p(k − 1

n)) =

∫ kn

k−1n

−d(γx,p(t), p)1 − t

〈∇ f ,∇dp〉(γx,p(t)) dt

for any n ∈ N and 1 ≤ k ≤ n − 1. Assume this is not the case, then there exists some n, k and a boundset S with 0 < m(S ) < ∞ so that for each x ∈ S , without loss of generality, f (γx,p( k

n )) − f (γx,p( k−1n )) >∫ k

nk−1

n

−d(γx,p(t),p)1−t 〈∇ f ,∇dp〉(γx,p(t)) dt. Applying (20) to a part of the Wasserstein geodesic from the normalization

of m|S to δp gives a contradiction.The general case of f ∈ W1,2(X) then follows by an approximation argument. Choose a sequence fi ∈

TestF(X) converging to f in W1,2(X). Using a diagonalization argument with Borel-Cantelli lemma and Fubini’stheorem, there exists some subsequence fi so that for m-a.e. x ∈ X, fi(γx,p(t)) → f (γx,p(t)) in L1

loc([0, 1)) asfunctions of t.

21

For any µ with compact support and bounded density, and µt defined as before, we also have∫−d(x, p)

1 − t〈∇ fi,∇dp〉(x) dµt(x)→

∫−d(x, p)

1 − t〈∇ f ,∇dp〉(x) dµt(x) in L1

loc([0, 1)).

Another diagonalization argument with Borel-Cantelli lemma and Fubini’s theorem gives a further subsequencefi so that for m-a.e. x ∈ X,

−d(γx,p(t), p)1 − t

〈∇ fi,∇dp〉(γx,p(t))→−d(γx,p(t), p)

1 − t〈∇ f ,∇dp〉(γx,p(t)) in L1

loc([0, 1)).

Combining these with statements 1 and 2, we have for any f ∈ W1,2(X), for m-a.e. x ∈ X, the map t 7→ f (γx,p(t))is in W1,1

loc ([0, 1)) and

ddt

f (γx,p(t)) =−d(γx,p(t), p)

1 − td f (∇dp)(γx,p(t)) for a.e. t ∈ [0, 1).

The corollary then follows by a reparameterization of γx,p.

We will be particularly interested in the following type of object: Let p ∈ X and µ ∈ P2(X) be of boundeddensity with respect to m. Take a Borel selection (2.26) of unit speed geodesics γx,p from all x ∈ X to pand define T := d(supp(µ), p). For 0 ≤ t ≤ T , define µt := (γ·,p(t))∗(µ). (µt) defined this way are morenaturally considered L1-Wasserstein geodesics and are well-studied in the theory of needle decomposition ofRCD spaces, see [BC13, CAV14, CM17b]. We record some properties of these objects which will be neededlater.

Theorem 3.15. Let 0 < δ < T and µ ≤ Am for some A > 0. Let (µt)t∈[0,T−δ] be as defined in the previousparagraph and D := supx∈supp(µ) d(x, p) ≤ D. Then

1. (µt)t∈[0,T−δ] is a W2-geodesic;2. There exists C(K,N, D, δ) so that µt ≤ A(1 + Ct)Nm for all t ∈ [0,T − δ]. In particular, the densities of

(µt)t∈[0,T−δ] are uniformly bounded with respect to m;3. (µt)t∈[0,T−δ] solves the continuity equation

ddtµt + div(−∇dpµt) = 0.

Proof. Statement 1 was proved in [CAV14, Lemma 4.4], where d-monotonicity of(x, γx,p(T − δ)) : x ∈ supp µ

is used to show its d2-montonicity. Statement 2 was proved in [BC13, Section 9], see also [CM18, Section 3.2]for a discussion. Statements 1 and 2 give that (µt) is of bounded compression. Statement 3 then follows fromCorollary 3.14.

In order to have terminology which includes the globally defined RLFs as well as the type of locally definedflows such as the example above, we will use the following definition.

Definition 3.16. Let S ∈ B(X) with m(S ) > 0 and (Vt)t∈[0,T ] ∈ L1([0,T ], L2(T X)). A Borel map F : [0,T ] ×S → X is a local flow of Vt from S if the following holds:

1. F0(x) = x and [0,T ] 3 t 7→ Ft(x) is continuous for every x ∈ S ;2. For every f ∈ TestF(X) and m-a.e. x ∈ S , t 7→ f (Ft(x)) is in W1,1([0,T ]) and

ddt

f (Ft(x)) = d f (Vt)(Ft(x)) for a.e. t ∈ [0,T ]; (21)

3. There exists a constant C := C(V, S ) so that (Ft)∗(m|S ) ≤ Cm for all t in [0,T ].

As before, by abuse of notation we will often say (Ft) is the local flow of (Vt) from S for some vector fieldV which is only locally L2. In this case, it is understood that Ft(S ) is essentially bounded for each t and we cutoff Vt outside of this region.

22

Remark 3.17. We will be primarily intersted in the following examples:1. For any p ∈ X and bounded S ∈ B(X), Ft(x) := γx,p(t) defined on (t, x) ∈ [0,T − δ] × S , where

T := ess infx∈S d(x, p), δ > 0, and γx,p is a unit speed geodesic from x to p is Borel selected (see 2.26),is a local flow of −∇dp from S by Corollary 3.14 and Theorem 3.15.

2. The restriction of any RLF onto some S ∈ B(X) is a local flow of the corresponding (Vt) from S bydefinition.

The following differentiation formula follows by the same argument for RLFs in Proposition 3.6.

Proposition 3.18. (First order differentiation formula for distance along local flows) Let T > 0. If (Ft)t∈[0,T ],(Gt)t∈[0,T ] are local flows of (Ut), (Vt) from S 1 and S 2 respectively, then for (m × m)-a.e. (x, y) ∈ S 1 × S 2,d(Ft(x),Gt(y)) ∈ W1,1([0,T ]) and

ddt

d(Ft(x),Gt(y)) = 〈∇dGt(y),Ut〉(Ft(x)) + 〈∇dFt(x),Vt〉(Gt(y)) for a.e. t ∈ [0,T ].

We mention that if (Wt) is defined as in proposition 3.6 from (Ut) and (Vt), then it is straightforward to checkusing the arguments of [BS20] that (Ft,Gt) is a local flow of (Wt) from S 1 × S 2. Again, (Wt) here naturallybelongs in L1([0,T ], L2

loc(T (X×X))) so Definition 3.16 needs to be altered to allow for this. We refer to [GR18]for relevant definitions.

The next proposition gives control on the metric speeds of the curves t 7→ Ft(x) of a local flow F. Aspointed out in [GT18, (A.22)], it follows from a similar argument as in [GIG18, Theorem 2.3.18] after a smalladjustment since we do not a priori assume the absolute continuity of the curves F·(x).

Proposition 3.19. Let V ∈ L1([0,T ], L2(T X)) and let (Ft) be a local flow of (Vt) from S . For m-a.e. x ∈ S thecurve t 7→ Ft(x) is absolutely continuous and its metric speed mst(F·(x)) at time t satisfies

mst(F·(x)) = |Vt|(Ft(x)) for a.e. t ∈ [0,T ].

3.3. Second order interpolation formula. The proof of a second order interpolation formula for the distancefunction (see (8) and (5)) along flows requires the results of [GT18]. The hard work is done there and theirresult immediately implies an analogous second order interpolation formula (Theorem 3.20) for the Wassersteindistance. It is our goal to pass this fromula from Wasserstein distance to distance on the space itself.

For the rest of the subsection we will always be in the setting of some RCD(K,N) space (X, d,m) with K ∈ Rand N ∈ [1,∞). We fix a Borel selection (2.26) of constant speed geodesics γx,y from all x ∈ X to all y ∈ Xparameterized on the unit interval. We denote by γx,y the unit speed reparameterization of γx,y to [0, d(x, y)].We start with the following formulation of the main result from [GT18].

Theorem 3.20. ( [GT18, Theorem 5.13]) Let µ0, µ1 ∈ P2(X) be compactly supported and satisfy µ0, µ1 ≤ Cmfor some C > 0. Let (µt) be the unique W2-geodesic connecting µ0 to µ1. For every t ∈ [0, 1], let φt be anyfunction so that for some s 6= t, s ∈ [0, 1], the function −(s − t)φt is a Kantorovich potential from µt to µs. Forany V ∈ H1,2

C (T X), the map [0, 1] 3 t 7→∫〈V,∇φt〉dµt is in C1([0, 1]) and

ddt

( ∫〈V,∇φt〉dµt

)=

∫(∇V : (∇φt ⊗ ∇φt)) dµt for all t ∈ [0, 1]. (22)

The next lemma follows from the previous theorem.

Lemma 3.21. Let p ∈ X and ν ≤ Cm be a nonnegative, compactly supported measure. For any V ∈ H1,2C (T X),∫

〈V,∇dp〉(x) dν(x) −∫〈V,∇dp〉(γp,x(

12

)) dν(x) =∫ 1

12

( ∫d(p, x)(∇V : (∇dp ⊗ ∇dp))(γp,x(t)) dν(x)

)dt,

(23)

where γx,y is as defined in the beginning of this subsection.23

Note that although ∇dp⊗∇dp is not in L2(T⊗2(X)), it is locally (i.e. it is after multiplication by the character-istic function of any compact set). Therefore, by the locality properties of the objects involved, ∇V : (∇dp⊗∇dp)is well-defined and is in L2(m).

Proof. Fix reprsentatives for 〈V,∇dp〉 and ∇V : (∇dp ⊗ ∇dp) ∈ L2(m).We claim for m-a.e. x ∈ X,

〈V,∇dp〉(x) − 〈V,∇dp〉(γp,x(12

)) =

∫ 1

12

d(p, x)(∇V : (∇dp ⊗ ∇dp))(γp,x(t)) dt. (24)

The right side integral is finite for m-a.e. x by using a Fubini’s theorem argument along with Theorem 2.25(15).

Suppose (24) does not hold for m-a.e. x ∈ X, then without loss of generality we may assume there exists abounded set S with 0 < m(S ) < ∞ so that

〈V,∇dp〉(x) − 〈V,∇dp〉(γp,x(12

)) >∫ 1

12

d(p, x)(∇V : (∇dp ⊗ ∇dp))(γp,x(t)) dt (25)

for each x ∈ S . Let µ := 1m(S ) m|S . Multiplying both sides of (25) by d(p, x) and integrating with respect to

µ, we immediately contradict Theorem 3.20. Therefore, (24) holds and so the lemma follows for any ν withcompact support and bounded density.

To proceed we state the segment inequality for L1(m) functions, first introduced by Cheeger-Colding in[CC96, Theorem 2.11]. This has been established for the metric measure setting in [vR08] but we will give aself-contained proof since the decomposition procedure for the family of geodesics used in the proof will beused again in the proof of Proposition 3.23.

Theorem 3.22. (Segment inequality for L1 functions on RCD spaces) Let (X, d,m) be an RCD(K,N) space withK ∈ R and N ∈ [1,∞). Let R > 0. Let f ∈ L1

loc(X) be nonnegative and µ ≤ A(m ×m) be a nonnegative measureon X × X supported on BR(p) × BR(p) for some 0 < R ≤ R and p ∈ X. Then∫ 1

0

( ∫f (γx,y(t))d(x, y) dµ(x, y)

)dt

≤ AC(K,N, R)R[m(π1(supp(µ))) + m(π2(supp(µ)))]∫

B2R(p)f (z) dm(z),

(26)

where π1, π2 are projections onto the first and the second coordinate respectively and γx,y is as defined in thebeginning of this subsection.

Proof. By Radon-Nikodym theorem, µ = g(m × m) for some compactly supported g ∈ L∞(X × X,m × m).We denote g1

x(·) := g(x, ·) and g2y(·) := g(·, y). By Fubini’s Theorem, ‖ g1

x ‖L∞≤ A for m-a.e. x and similarly,‖ g2

y ‖L∞≤ A for m-a.e. y.For each t ∈ [ 1

2 , 1] and m-a.e. x ∈ π1(supp(µ)), by Theorem 2.25 (15),

(γx,·(t))∗(g1xm) ≤ AC(K,N, R)m|B2R(p). (27)

Similarly for t ∈ [0, 12 ] and m-a.e. y ∈ π2(supp(µ)),

(γ·,y(t))∗(g2ym) ≤ AC(K,N, R)m|B2R(p). (28)

We conclude ∫ 1

0

( ∫f (γx,y(t))d(x, y) dµ(x, y)

)dt

≤ 2R∫ 1

0

( ∫f (γx,y(t)) dµ(x, y)

)dt

24

= 2R( ∫ 1

12

( ∫f (γx,y(t)) dµ(x, y)

)dt +

∫ 12

0

( ∫f (γx,y(t)) dµ(x, y)

)dt

)≤ AC(K,N, R)R[m(π1(supp(µ))) + m(π2(supp(µ)))]

∫B2R(p)

f (z) dm(z),

where (27), (28) and Fubini’s theorem was used in the last line.

We now prove the main interpolation formula of this subsection. The main idea is to use Lemma 3.21 alongwith the decomposition procedure from the proof of the semgent inequality. Notice that the right side of (29)makes sense due to the segment inequality.

Proposition 3.23. (Second order interpolation formula) Let µ ≤ C(m × m) be a nonnegative and compactlysupported measure on X × X. Let V ∈ H1,2

C (T X). Then∫(〈V,∇dx〉(y) + 〈V,∇dy〉(x)) dµ(x, y) =

∫ 1

0

∫d(x, y)(∇V : ∇dx ⊗ ∇dx)(γx,y(t)) dµ(x, y) dt, (29)

where γx,y is as defined in the beginning of this subsection.

Proof. Let µ = g(m×m) for compactly supported g ∈ L∞(X × X,m×m) with g1x(·) := g(x, ·) and g2

y(·) := g(·, y).‖ g1

x ‖L∞≤ C for m-a.e. x and ‖ g2y ‖L∞≤ C for m-a.e. y. We have∫

〈V,∇dx〉(y) dµ(x, y) −∫〈V,∇dx〉(γx,y(

12

)) dµ(x, y)

=

∫ ∫ (〈V,∇dx〉(y) − 〈V,∇dx〉(γx,y(

12

)))

d(g1xm)(y) dm(x) . by Fubini’s theorem

=

∫ ∫ 1

12

( ∫d(x, y)(∇V : (∇dx ⊗ ∇dx))(γx,y(t)) d(g1

xm)(y))

dt dm(x) , by 3.21

=

∫ 1

12

( ∫ (d(x, y)(∇V : (∇dx ⊗ ∇dx))(γx,y(t))

)dµ(x, y)

)dt , by Fubini’s theorem.

(30)

Making the analogous argument on t ∈ [0, 12 ], we obtain∫

〈V,∇dy〉(x) dµ(x, y) −∫〈V,∇dy〉(γx,y(

12

)) dµ(x, y)

=

∫ 12

0

( ∫ (d(x, y)(∇V : (∇dy ⊗ ∇dy))(γx,y(t))

)dµ(x, y)

)dt.

(31)

The claim then follows if∫〈V,∇dx〉(γx,y(

12

)) dµ(x, y) +

∫〈V,∇dy〉(γx,y(

12

)) dµ(x, y) = 0 (32)

and ∫ 12

0

( ∫ (d(x, y)(∇V : (∇dy ⊗ ∇dy))(γx,y(t))

)dµ(x, y)

)dt

=

∫ 12

0

( ∫ (d(x, y)(∇V : (∇dx ⊗ ∇dx))(γx,y(t))

)dµ(x, y)

)dt,

(33)

which we show in Lemma 3.25 and Remark 3.26.

25

The following two lemmas show that, in a measure-theoretic sense, ∇dp and ∇dq in the interior of a geodesicbetween p and q “point in opposite directions”.

Lemma 3.24. Let p, q ∈ X. Let γp,q : [0, 1]→ X be a constant speed geodesic from p to q and let z = γ(t0) forsome t0 ∈ (0, 1). Let f be a locally Lipschitz function, then lip( f + dp)(z) = lip( f − dq)(z).

Proof. By classical Abresch-Gromoll inequality 2.43, for x in a sufficiently small neighbourhood of z, ep,q(x) ≤Cd(x, z)1+α. Therefore, for any such x,

|( f (x) + dp(x)) − ( f (z) + dp(z))|d(x, z)

−|( f (x) − dq(x)) − ( f (z) − dq(z))|

d(x, z)

≤Cd(x, z)1+α

d(x, z)= Cd(x, z)α.

This shows that lip( f + dp)(z) = lip( f − dq)(z) since Cd(x, z)α → 0 as d(x, z)→ 0.

Lemma 3.25. In the notations of 3.23, for any t ∈ (0, 1),∫〈V,∇dx〉(γx,y(t)) dµ(x, y) +

∫〈V,∇dy〉(γx,y(t)) dµ(x, y) = 0.

Proof. Fix t ∈ (0, 1). Let µ = g(m ×m) for compactly supported g ∈ L∞(X × X,m ×m) with g1x(·) := g(x, ·) and

g2y(·) := g(·, y). ‖ g1

x ‖L∞≤ C for m-a.e. x and ‖ g2y ‖L∞≤ C for m-a.e. y.

We first prove the claim for V = ∇ f where f is locally Lipschitz. The general claim then follows byapproximation. We observe the following:

1. By Remark 2.5, for each x ∈ X,

〈∇ f ,∇dx〉 =|∇( f + dx)|2−|∇ f |2−|∇dx|

2

2

=(lip( f + dx))2 − (lip( f ))2 − (lip(dx))2

2m − a.e..

2. Similarly, for each y ∈ X,

〈∇ f ,−∇dy〉 =|∇( f − dy)|2−|∇ f |2−|−∇dy|

2

2

=(lip( f − dy))2 − (lip( f ))2 − (lip(−dy))2

2m − a.e..

3. (γx,·(t))∗(g1xm) is of bounded density w.r.t. m for m-a.e. x by Theorem 2.25 (15).

Lemma 3.24 and Fubini’s Theorem then gives the claim for ∇ f ,∫〈∇ f ,∇dx〉(γx,y(t)) dµ(x, y)

=

∫ (∫〈∇ f ,∇dx〉(γx,y(t)) d(g1

xm)(y))

dm(x)

=

∫ (∫ (lip( f + dx))2 − (lip( f ))2 − (lip(dx))2

2(γx,y(t)) d(g1

xm)(y))

dm(x) , by observations 1 and 3

=

∫ (∫ (lip( f − dy))2 − (lip( f ))2 − (lip(−dy))2

2(γx,y(t)) d(g1

xm)(y))

dm(x) , by 3.24 and observ. 3

=

∫〈∇ f ,−∇dy〉(γx,y(t)) dµ(x, y) , by observations 2 and 3.

26

Remark 3.26. 3.25 implies (33) as well. This is because Lemma 3.21 is true for any interval [a, b] ⊆ (0, 1] andso one can use the same argument as in Proposition 3.23 to say that, for 0 < s < 1

2 ,∫ 12

s

( ∫ (d(x, y)(∇V : (∇dy ⊗ ∇dy))(γx,y(t))

)dµ(x, y)

)dt

=

∫〈V,∇dy〉(γx,y(

12

)) dµ(x, y) −∫〈V,∇dy〉(γx,y(s)) dµ(x, y)

= −

∫〈V,∇dx〉(γx,y(

12

)) dµ(x, y) +

∫〈V,∇dy〉(γx,y(s)) dµ(x, y)

=

∫ 12

s

( ∫ (d(x, y)(∇V : (∇dx ⊗ ∇dx))(γx,y(t))

)dµ(x, y)

)dt.

Taking a limit s→ 0 gives (33).

The second order interpolation formula 3.23 and first order differentiation formula for distance along localflows 3.18 immediately give an integral version of (5). They also give the following related estimate whichwill be used heavily in Section 5. Let U, V ∈ L1([0,T ], L2(T X)) be bounded (see Definition 3.1) and S 1, S 2 bebounded sets of positive measure. Let (Ft)t∈[0,T ], (Gt)t∈[0,T ] be local flows of U, V from S 1, S 2 respectively. Letr > 0 and for each t ∈ [0,T ], define dtF,G

r (t) : S 1 × S 2 → [0, r] the distance distortion on scale r at t by

dtF,Gr (t)(x, y) := min

r, max

0≤τ≤t|d(x, y) − d(Fτ(x),Gτ(y))|

. (34)

Define ΓF,Gr (t) :=

(x, y) ∈ S 1 × S 2 : dtF,G

r (t)(x, y) < r. The terminology and definition of the distance distor-

tion function comes from [KW11], where it was used in a similar way as in this paper to analyze the geometryof gradient flows.

Proposition 3.27. Let W ∈ H1,2C (T X). The map t 7→

∫S 1×S 2

dtF,Gr (t)(x, y) d(m × m)(x, y) is Lipschitz on [0,T ]

and satisfiesddt

∫S 1×S 2

dtF,Gr (t)(x, y) d(m × m)(x, y)

≤

∫Γ

F,Gr (t)

(|Ut −W |(Ft(x)) + |Vt −W |(Gt(y))) d(m × m)(x, y)

+

∫ 1

0

∫Γ

F,Gr (t)

d(Ft(x),Gt(y))|∇W |HS(γFt(x),Gt(y)(s)) d(m × m)(x, y) ds

for a.e. t ∈ [0,T ], where γ·,· is as defined in the beginning of this subsection.

Proof. First fix representatives for all involved measure-theoretic objects. For any (x, y) ∈ S 1×S 2, dtF,Gr (t)(x, y)

is continuous, monontone non-decreasing and bounded between 0 and r as a function of t ∈ [0,T ]. Therefore,dtF,G

r (t)(x, y) is differentiable for a.e. t ∈ [0,T ] and d+

dt dtF,Gr (t)(x, y) = 0 for all t where (x, y) /∈ Γ

F,Gr (t).

Furthermore, by boundedness of U, V and Proposition 3.19, F·(x), G·(y) are uniformly Lipschitz curves form-a.e. x ∈ S 1 and m-a.e. y ∈ S 2 respectively. In particular, the functions [0,T ] 3 t 7→ d(Ft(x),Gt(y)) areuniformly Lipschitz for (m × m)-a.e. (x, y) ∈ S 1 × S 2. The same is true for the functions t 7→ dtF,G

r (t)(x, y).Therefore, t 7→

∫S 1×S 2

dtF,Gr (t)(x, y) d(m × m)(x, y) is Lipschitz on [0,T ] as well.

By Proposition 3.18, for (m × m)-a.e.(x, y) ∈ S 1 × S 2, d(Ft(x),Gt(y)) ∈ W1,1([0,T ]) andddt

d(Ft(x),Gt(y)) = 〈∇dGt(y),Ut〉(Ft(x)) + 〈∇dFt(x),Vt〉(Gt(y)). (35)

27

At any point of differentiability for both d(Ft(x),Gt(y)) and dtF,Gr (t)(x, y), it is clear from definition that d

dt dtF,Gr (t)

(x, y) ≤ ( ddt d(Ft(x),Gt(y)))+

For any t ∈ [0,T ], let ΓF,Gr (t) :=

(x, y) : 〈∇dGt(y),Ut〉(Ft(x)) + 〈∇dFt(x),Vt〉(Gt(y)) ≥ 0

∩ Γ

F,Gr (t). Then

µt := (Ft,Gt)∗((m×m)|Γ

F,Gr (t)) is compactly supported by Proposition 3.19 and has bounded density with respect

to (m × m) by definition 3.16 of a local flow. By the second order interpolation formula 3.23,∫(〈W,∇dy〉(x) + 〈W,∇dx〉(y)) dµt(x, y) =

∫ 1

0

∫d(x, y)(∇W : ∇dx ⊗ ∇dx)(γx,y(s)) dµt(x, y) ds. (36)

Therefore,∫(〈Ut,∇dy〉(x) + 〈Vt,∇dx〉(y)) dµt(x, y)

=

∫(〈Ut −W,∇dy〉(x) + 〈Vt −W,∇dx〉(y)) dµt(x, y)

+

∫ 1

0

∫d(x, y)(∇W : ∇dx ⊗ ∇dx)(γx,y(s)) dµt(x, y) ds , by (36)

≤

∫(|Ut −W |(x) + |Vt −W |(y)) dµt(x, y)

+

∫ 1

0

∫d(x, y)|∇W |HS(γx,y(s)) dµt(x, y) ds , since |∇dx|, |∇dy|= 1 m-a.e.

≤

∫Γ

F,Gr (t)

(|Ut −W |(Ft(x)) + |Vt −W |(Gt(y))) d(m × m)(x, y)

+

∫ 1

0

∫Γ

F,Gr (t)

d(Ft(x),Gt(y))|∇W |HS(γFt(x),Gt(y)(s)) d(m × m)(x, y) ds,

(37)

by definition of µt = (Ft,Gt)∗((m × m)|Γ

F,Gr (t)), Γ

F,Gr (t) ⊆ Γ

F,Gr (t) and the fact that all integrands are positive.

To conlcude, we have for a.e. t ∈ [0,T ], for (m × m)-a.e. (x, y) ∈ S 1 × S 2, d(Ft(x),Gt(y)) and dtF,Gr (t)(x, y)

are both differentiabile in t. For any such t,ddt

∫S 1×S 2

dtF,Gr (t)(x, y) d(m × m)(x, y)

=

∫S 1×S 2

ddt

dtF,Gr (t)(x, y) d(m × m)(x, y) , by DCT since dtr is uniformly Lipschitz for a.e. (x, y)

=

∫Γ

F,Gr (t)

ddt

dtF,Gr (t)(x, y) d(m × m)(x, y)

≤

∫Γ

F,Gr (t)

(ddt

d(Ft(x),Gt(y)))+ d(m × m)(x, y)

=

∫Γ

F,Gr (t)

〈∇dGt(y),Ut〉(Ft(x)) + 〈∇dFt(x),Vt〉(Gt(y)) d(m × m)(x, y) , by (35) and definition of Γ

≤

∫Γ

F,Gr (t)

(|Ut −W |(Ft(x)) + |Vt −W |(Gt(y))) d(m × m)(x, y)

28

+

∫ 1

0

∫Γ

F,Gr (t)

d(Ft(x),Gt(y))|∇W |HS(γFt(x),Gt(y)(s)) d(m × m)(x, y) ds , by (37).

4. Estimates on the heat flow approximations of distance and excess functions

Here we collect estimates on the heat flow approximations of distance and excess functions, all of whichwere established in [CN12]. All their arguments translate directly to the RCD setting due to the availability ofthe improved Bochner inequality, the Li-Yau Harnack and gradient inequalities, and the various estimates ofSubsection 2.10. We record their proofs for the sake of completeness, making minor regularity and measure-theoretic adjustments as needed.

In this section we fix (X, d,m) an RCD(−(N − 1),N) space for N ∈ (1,∞), 0 < δ < 12 , and two points

p, q ∈ X with d(p, q) ≤ 1. Any time we use c it is always a constant depending only on N and δ unless specifiedotherwise. We fix the following notations:

1. dp,q = d(p, q) ≤ 1 and for any ε > 0, dε := εdp,q.2. d−(x) := d(p, x).3. d+(x) := d(p, q) − d(x, q).4. e(x) := d(p, x) + d(x, q) − d(p, q) = d−(x) − d+(x).

We will consider these functions multiplied by some appropriate cut off functions. Let ψ± : X → R be thegood cut off functions as in 2.36 satisfying

ψ− =

1 on A δ8 dp,q,8dp,q

(p)0 on X \ A δ

16 dp,q,16dp,q(p) , ψ+ =

1 on A δ8 dp,q,8dp,q

(q)0 on X \ A δ

16 dp,q,16dp,q(q) .

Let ψ := ψ+ψ−, e0 := ψe and h±0 := ψd±. We denote5. h±t := Ht(h±0 ) and et := Ht(e0).6. Xr,s := Ardp,q,sdp,q(p) ∩ Ardp,q,sdp,q(q).

By definition e0 = e, h±0 = h± on X δ8 ,8

and et = h−t − h+t by uniqueness of heat flow.

We will always take the continuous representative whenever possible. This in particular applies to et, h±t ,∆et,and ∆h±t for t > 0. We remark that since h±t and et are Lipschitz, one can also take the local Lipschitz constant asthe representatives of |∇h±t | and |∇et| by 2.5. These have a sufficiently nice continuity property, see Lemma 4.1,which makes most of our m-a.e. statements about |∇h±t | and |∇et| pointwise and ease certain measure-theoreticdifficulties in the arguments for this section.

Lemma 4.1. Let (X, d,m) be an RCD(K,N) space for K ∈ R and N ∈ [1,∞). Let f : X → R be a Lipschitzfunction. Fix U ⊆ X open and x ∈ U. Then

lip( f )(x) ≤ ess supU

lip( f ).

Proof. For any ε > 0, there exists y ∈ U so that | f (y)− f (x)|d(y,x) ≥ lip( f )(x) − ε

2 . By continuity of f , there exists r > 0

so that Br(y) ⊆ U and for any z ∈ Br(y), | f (y)− f (x)|d(y,x) ≥ lip( f )(x) − ε. Let γz,x : [0, 1] → X be a constant speed

geodesic from z to x. The local Lipschitz constant is an upper gradient of f , see [AGS14a, Remark 2.7], andtherefore, ?

Br(y)| f (z) − f (x)| dm(z) ≤

?Br(y)

∫ 1

0d(z, x) lip( f )(γz,x(s)) ds dm(z).

Since we know>

Br(y)| f (z) − f (x)|≥ lip( f )(x) − ε and for each s < 1, (γ·,x(s))∗(m) is absolutely continuous w.r.t.m by Theorem 2.25, we conclude ess supU lip( f ) ≥ lip( f )(x) − ε.

We proceed with our estimates for et and h±t .

29

Lemma 4.2. There exists a constant c(N, δ) such that for all t > 0,

∆h−t ,−∆h+t ,∆et ≤

c(N, δ)dp,q

. (38)

Proof. We show the claim for et; the proof is analogous for others. By Laplacian comparison theorem for thedistance function 2.29, see also Remark 2.30, and the definition ψ, e0 ∈ D(∆) with

∆e0 = ∆ψem + 〈∇ψ,∇e〉m + ψ∆e ≤c(N, δ)

dp,qm. (39)

We know et(x) =∫

Ht(x, y)e0(y)dm(y). For t > 0, et ∈ D(∆) and

∆et =

∫∆xHt(x, y)e0(y) dm(y)

=

∫∆yHt(x, y)e0(y) dm(y) , by symmetry of Ht

=

∫Ht(x, y)d∆e0(y) , since e0 is compactly supported

≤c(N, δ)

dp,q.

(40)

Lemma 4.3. There exists a constant c(N, δ) such that for all 0 < ε ≤ ε(N, δ) and x ∈ X δ4 ,5

the following holds:

1. |ed2ε(y)|≤ c(ε2dp,q + e(x)) for every y ∈ B10dε (x);

2. |∇ed2ε|(y) ≤ c

(ε +

ε−1e(x)dp,q

)for m-a.e. y ∈ B10dε (x);

3. |∆ed2ε(y)|≤ c

(1

dp,q+

ε−2e(x)d2

p,q

)for every y ∈ B10dε (x);

4.>

Bdε (y)|Hess ed2ε|2HS≤ c

(1

dp,q+

ε−2e(x)d2

p,q

)2for every y ∈ B10dε (x).

Proof. et(x) = e0(x) +t∫

0∆es(x)ds pointwise by definition of the heat flow and the continuity es.

By Lemma 4.2,et(x) ≤ e0(x) +

cdp,q

t = e(x) +c

dp,qt. (41)

Setting s = d2ε , t = 2d2

ε and y ∈ B10dε (x) in the statement of the Li-Yau Harnack inequality, 2.31, we concludeed2

ε(y) ≤ c(N)e2d2

ε(x) , by 2.31 and dp,q ≤ 1

≤ c(e(x) + ε2dp,q) , by (41).(42)

This proves statement 1 of the lemma.To prove 3 of the lemma, first notice that we need only establish a lower bound on ∆ed2

ε(y) since 4.2 already

gives us the desired upper bound. This is an application of the Li-Yau gradient inequality 2.32 and statement 1.The bound holds pointwise even though 2.32 holds only a.e. due to the existence of a continuous representatitiveof ∆et.

Statement 2 of the lemma follows from another application of Li-Yau gradient inequality 2.32 along with thebounds from statements 1 and 3.

30

For the last statement take good cut off function φ supported on B2dε (y) with φ ≡ 1 on Bdε (y). We have∫X

(∆ed2ε)2φdm = −

∫X

〈∇ed2ε,∇(∆ed2

εφ)〉dm

= −

∫X

〈∇ed2ε,∇∆ed2

ε〉φdm −

∫X

〈∇ed2ε,∇φ〉∆ed2

εdm.

(43)

Integrating φ with |Hess ed2ε|2HS and applying the improved Bochner inequality 2.19, we get∫

Bdε

|Hess ed2ε|2HS dm ≤

∫X

|Hess ed2ε|2HSφ dm

≤

∫X

12φd∆|∇ed2

ε|2−

∫X

〈∇ed2ε,∇∆ed2

ε〉φ dm +

∫X

(N − 1)|∇ed2ε|2φ dm , by 2.19

≤

∫X

12

∆φ|∇ed2ε|2dm +

∫X

(∆ed2ε)2φdm +

∫X

|∇ed2ε||∇φ|∆ed2

εdm +

∫X

(N − 1)|∇ed2ε|2φdm , by (43).

(44)

Applying to this computation properties 1 - 3 of the lemma, property 2 of good cut off functions 2.35 andBishop-Gromov volume comparison 2.4, we obtain statement 4 of the lemma.

Next we prove estimates on the heat flow approximation of the distance functions.

Lemma 4.4. There exists c(N, δ) such that for every ε ≤ ε(N, δ) and x ∈ X δ4 ,5

,

|h±d2ε− d±|(x) ≤ c(ε2dp,q + e(x)).

Proof. From the Laplacian bounds in 4.2, for x ∈ X δ4 ,5

,

h−d2ε(x) − d−(x) =

∫ d2ε

0∆h−t (x) dt ≤ cε2dp,q, (45)

and

d+(x) − h+

d2ε(x) =

∫ d2ε

0−∆h+

t (x) dt ≤ cε2dp,q. (46)

To obtain bounds in the other direction, we note that

h−d2ε− d−(x) = h+

d2ε− d+(x) + ed2

ε(x) − e(x).

We conclude using this with the bound |ed2ε(x)|≤ c(ε2dp,q + e(x)) from statement 1 of Lemma 4.3 and bounds

(45) or (46).

We will end up wanting to establish appropriate gradient and Hessian bounds along curves that are close tobeing a geodesic between p and q. This requires the following definition.

Definition 4.5. A unit speed, piecewise geodesic curve σ between p and q is called an ε-geodesic between pand q if ||σ|−dp,q|≤ ε

2dp,q, where |σ| is the length of σ.

Remark 4.6. Notice that x lies on an ε-geodesic iff e(x) ≤ ε2dp,q.

The previous lemma 4.4 can now be restated in terms of ε-geodesics.

Corollary 4.7. There exists c(N, δ) such that for every ε-geodesic between p and q with ε ≤ ε(N, δ), andδ3 ≤ t ≤ 1 − δ

3 ,|h±d2

ε− d±|(σ(t)) ≤ c(ε2dp,q).

31

Proof. This follows from the 4.4 since1. e(σ(t)) ≤ ε2dp,q

2. Since σ is unit speed and an ε-geodesic, as long as ε <

√δ

12, we will have σ(t) ∈ X δ

4 ,5.

We establish an upper bound on the norm of the gradient of h±t for x ∈ X δ5 ,6

.

Lemma 4.8. There exists c(N, δ) such that for ε ≤ ε(N, δ) and m-a.e. x ∈ X δ5 ,6

,

|∇h±d2ε|≤ 1 + cd2

ε .

Proof. By Bakry-Ledoux estimate 2.16, for any t > 0,

|∇h±t |≤ e2(N−1)tHt(|∇h±0 |) m-a.e.. (47)

By definition of h±0 = ψd±, we have the following a.e. bounds on |∇h±0 |.1. |∇h±0 |= 1 in X δ

8 ,82. |∇h±0 |= 0 in X\X δ

16 ,163. In X δ

16 ,16\X δ8 ,8,

|∇h±0 | = |∇ψ||d±|+|ψ||∇d±| ≤

c(N)δdp,q

|d±|+1 ≤ c(N, δ). (48)

Finally, for any x ∈ X δ2 ,4

,

Ht(|∇h±0 |)(x) =

∫X

Ht(x, y)|∇h±0 |(y) dm(y)

=

∫X δ

16 ,16\X δ8 ,8

Ht(x, y)|∇h±0 |(y) dm(y) +

∫X δ

4 ,8

Ht(x, y)|∇h±0 |(y) dm(y)

≤ c(N, δ)∫

X δ16 ,16\X δ

8 ,8

∇Ht(x, y) dm(y) + 1 , by (48)

≤ c(N, δ)(δ

8)−2t + 1,

(49)

where the last lines uses statement 2 of the heat kernel bounds 2.38.The lemma follows by combining (47) and (49), with t = d2

ε for small ε.

We will now establish some integral bounds on |∇h±t |. Roughly, we want to apply the L1-Harnack inequality2.39 to |∇h±|. To this effect, we give some regularity of the heat flow in the time parameter.

Lemma 4.9. Let (X, d,m) be an RCD(K,N) space for some K ∈ R, N ∈ [1,∞). Let f ∈ L2(m).1. Ht( f ) ∈ C1((0,∞), L2(m)) and

ddt

Ht( f ) = ∆Ht( f ) ∀t > 0; (50)

2. Ht( f ) ∈ C0((0,∞),W1,2(X));3. Ht( f ) ∈ C1((0,∞),W1,2(X)) and in particular |∇Ht( f )|2∈ C1((0,∞), L1(m)) with

ddt|∇Ht( f )|2= 2〈∇Ht( f ),∇∆Ht( f )〉 ∀t > 0. (51)

If f or |∇ f | is in L∞(X,m), then |∇Ht( f )|2∈ C1((0,∞), L2(m)) and the same formula holds.

32

Proof. Since Ht( f ) = Ht′( f ) +∫ t

t′ ∆Hs( f )ds for t > t′ > 0 and ∆Hs( f ) ∈ C0((0,∞), L2(m)), statement 1 followsby fundamental theorem of calculus.

For t > 0,

lims→0

∫X

|∇Ht+s( f ) − ∇Ht( f )|2 = lims→0

∫X

(Ht+s( f )∆Ht+s( f )) + 2(Ht+s( f )∆Ht( f )) − (Ht( f )∆Ht( f ))

= 0 , since all terms involved are continuous from s→ L2(m).

We already know Ht( f ) ∈ C0((0,∞), L2(X)), so statement 2 follows.Applying statement 2 to ∆Hε( f ) for arbitrarily small positive ε, we see that ∆Ht( f ) ∈ C0((0,∞), W1,2(X)).

The first part of statement 3 then follows by applying fundamental theorem of calculus to Ht( f ) = Ht′( f ) +∫ tt′ ∆Hs( f )ds viewed as a W1,2(X)-valued Bochner integral. The second part follows by a direct computation.

Notice that if f or |∇ f | is in L∞(X,m), then |∇ht( f )|∈ L∞ by L∞-to-Lipschitz regularization (14) or by Bakry-Ledoux estimate 2.16.

Lemma 4.10. Let φ ∈ D(∆) be nonngative, compactly supported, time independent with |φ|, |∇φ|, |∆φ|≤ K1. Ifh is the heat flow of some h0 ∈ L2(m) ∩ L∞(m) and |∇h|≤ K2 on φ > 0, then ( ∂∂t − ∆)[φ2|∇h|2] ≤ c(N,K1,K2)weakly in (0,∞) × X as in Definition 2.33.

Proof. Let t > 0, ht ∈ TestF(X) by the L∞-to-Lipschitz regularization property of Ht (14). Let f ∈ TestF(X).By [GIG18, Proposition 3.3.22], 〈∇ht,∇ht〉 ∈ W1,2(X) and

〈∇〈∇ht,∇ht〉,∇ f 〉 = 2 Hess(ht)(∇ht,∇ f ) m-a.e..

Therefore, |∇〈∇ht,∇ht〉∇ f |≤ 2|Hess(ht)|HS|∇ht||∇ f |m-a.e..We then have, for any ε > 0 and m-a.e.,

4φ|〈∇|∇ht|2,∇φ〉| ≤ 8φ|Hess(ht)|HS|∇ht||∇φ| , by above and the density of TestF(X) in W1,2(X).

≤ 4εφ2|Hess(ht)|2HS|∇ht|2+

4ε|∇φ|2.

Choosing ε > 0 small so that 4εK22 < 2,

∆(φ2|∇ht|2) = φ2∆|∇ht|

2+(2〈∇|∇ht|2,∇φ2〉 + |∇ht|

2∆φ2)m

≥ (2φ2|Hess(ht)|2HS+2φ2〈∇∆ht,∇ht〉 − 2(N − 1)φ2|∇ht|2+4φ〈∇|∇ht|

2,∇φ〉 + |∇ht|2∆φ2)m

≥ (2φ2〈∇∆ht,∇ht〉 − c)m,

where the improved Bochner inequality 2.19 is used for line 2 and the previous estimate with ε was used forline 3.

Finally, by Lemma 4.9, ddtφ

2|∇ht|2= 2φ2〈∇∆ht,∇h〉. This lets us conclude.

Theorem 4.11. There exists a constant c(N, δ) such that for all ε ≤ ε(N, δ),1. if x ∈ X δ

2 ,3with e(x) ≤ ε2dp,q then

>B10dε (x)||∇h±

d2ε|−1|≤ cε;

2. if σ is an ε-geodesic connecting p and q, then(1− δ2 )dp,q∫δ2 dp,q

>B10dε (σ(s))||∇h±

d2ε|−1|≤ cε2dp,q.

Proof. We prove the theorem in the case of h−t . The h+t case is similar. We will take the local Lipschitz constant

representatitves for |∇h±t |. All statements made will be for ε sufficiently small depending on N and δ so we willforgo repeating this.

By Lemma 4.8, we choose c′(N, δ) so that |∇h−t |≤ 1 + c′t2 for all x ∈ X δ5 ,6

and t ≤ ε′(N, δ)2d2p,q. This means

there exists c′′(N, δ) so thatwt := 1 + c′′t − |∇h−t |

2≥ 0 on X δ5 ,6.

33

Let φ = φ+φ−, where φ± are annular good cutoff functions (2.36) around p and q respectively so that φ = 1 onX δ

4 ,5and φ = 0 on X\X δ

5 ,6. By Lemma 4.10,

(∂t − ∆)|φ2wt|≥ −c weakly in (0, d2ε′) × X.

Applying the L1-Harnack inequality 2.39, we have, for x ∈ X δ3 ,4

,?B10dε (x)

wd2ε≤ c[ess inf

B10dε (x)w2d2

ε+ d2

ε ]. (52)

We will show that the right side is sufficiently small. Let e(x) ≤ ε2dp,q and let γ(t) be a unit speed geodesicfrom x to p. By Corollary 4.7,

|h−2d2ε(x) − h−2d2

ε(γ(10dε)) − 10dε |

≤ |h−2d2ε(x) − d−(x)|+|h−2d2

ε(γ(10dε)) − d−(γ(10dε))|+|d−(x) − d−(γ(10dε)) − 10dε |

≤ cε2dp,q.

(53)

The local Lipschitz constant |∇h−2d2

ε| is an upper gradient, [AGS14a, Remark 2.7]. Therefore,

|h−2d2ε(σ) − h−2d2

ε(γ(10dε))|≤

∫ 10dε

0|∇h−2d2

ε|(γ(s)) ds. (54)

We have ∫ 10dε

0w2d2

ε(γ(s)) ds =

∫ 10dε

0(1 + cd2

ε − |∇h−2d2ε|2(γ(s))) ds

≤ 10dε + cd3ε −

110dε

(∫ 10dε

0|∇h−2d2

ε|(γ(s)) ds)2 , by Cauchy-Schwarz

≤ 10dε + cd3ε −

110dε

(h−2d2ε(x) − h−2d2

ε(γ(10dε)))2 , by (54)

≤ 10dε + cd3ε −

110dε

(10dε − cε2dp,q)2 , by (53)

≤ cε , if ε < 1

(55)

In particular, there exists s ∈ [0, 10dε] so that w2d2ε(γ(s)) ≤ cε. Applying Lemma 4.1 to |∇h±t | and γ(s) ∈

B10dε (x), we conclude ess infB10dε (x)

w2d2ε≤ cε and so statement 1 is proved by (52).

By Corollary 4.7, arguing as in (53) ,

|h−2d2ε

(σ((1 −

δ

2)dp,q)

)− h−2d2

ε

(σ(δ

2dp,q)

)− (1 − δ)dp,q| ≤ cε2dp,q. (56)

Arguing as in (54) and (55), ∫ (1− δ2 )dp,q

δ2 dp,q

w2d2ε(σ(s)) ds ≤ cε2dp,q. (57)

34

Finally, ∫ (1− δ2 )dp,q

δ2 dp,q

(?B10dε (σ(s))

||∇h−d2ε|2−1|

)ds ≤

∫ (1− δ2 )dp,q

δ2 dp,q

(?B10dε (σ(s))

wd2ε

+ cε2dp,q)

ds

≤ c∫ (1− δ2 )dp,q

δ2 dp,q

(ess inf

B10d2ε

(σ(s))w2d2

ε+ cε2dp,q

)ds , by (52)

≤ c∫ (1− δ2 )dp,q

δ2 dp,q

(w2d2

ε(σ(s)) + cε2dp,q

)ds , by Lemma 4.1

≤ cε2dp,q.

(58)

We now prove the main Hessian estimate for h±t .

Theorem 4.12. There exists a constant c(N, δ) such that for any 0 < ε ≤ ε(N, δ), any x ∈ X δ2 ,3

with e(x) ≤ ε2dp,q,

or any ε-geodesic σ connecting p and q, there exists r ∈ [ 12 , 2] with

1. |h±rd2ε− d±|≤ cε2dp,q;

2.>

Bdε (x)||∇h±rd2ε|2−1|≤ cε;

3.∫ (1− δ2 )dp,qδ2 dp,q

( >Bdε (σ(s))||∇h±

rd2ε|2−1|

)ds ≤ cε2dp,q;

4.∫ (1− δ2 )dp,qδ2 dp,q

( >Bdε (σ(s))|Hess h±

rd2ε|2)

ds ≤ cd2

p,q.

Proof. Statement 1 follows from Lemma 4.4 and statements 2 and 3 follow from Theorem 4.11 with Bishop-Gromov. Note that any r ∈ [ 1

2 , 2] works in the first 3 statements.Using 2.35, we fix, for each s ∈ ( δ2 dp,q, (1 − δ

2 )dp,q), good cut off function φ with φ ≡ 1 on Bdε (σ(s)),vanishing outside of B3dε (σ(s)), and dε |∇φ|, d2

ε |∆φ|≤ c(N). Similarly, fix α(t) a smooth function in time so that0 ≤ α(t) ≤ 1, α(t) ≡ 1 for t ∈ [ 1

2 d2ε , 2d2

ε ], vanishing for t outside of [ 14 d2

ε , 4d2ε ], and satisfying |α′|≤ 10d−2

ε .Applying the improved Bochner inequality 2.19 to h±t , we obtain, for each s, t∫

α(t)φ|Hess h±t |2 dm ≤

∫α(t)φd(∆|∇h±t |

2) + 2∫

α(t)φ((N − 1)|∇h±t |

2−〈∇h±t ,∇∆h±t 〉)

dm

=

∫α(t)(|∇h±t |

2−1)∆(φ) dm + 2(N − 1)∫

α(t)φ|∇h±t |2 dm −

∫α(t)φ∂t(|∇h±t |

2) dm.(59)

In the last line, we used the definition of the Laplacians along with the fact that∫

∆φdm = 0 for the first termand Lemma 4.9 for the third term. Integrating in time using integration by parts and

∫ ∞0 α′(t)dt = 0 on the third

term of the previous line,∫ ∞

0

∫α(t)φ|Hess h±t |

2 dm dt

≤

∫ ∞

0

( ∫α(t)(|∇h±t |

2−1)∆(φ) dm + 2(N − 1)∫

α(t)φ|∇h±t |2 dm +

∫α′(t)φ(|∇h±t |

2−1) dm)

dt.(60)

35

Using what we know about φ and α and using Bishop-Gromov in line 2 of the following, we obtain∫ 2d2ε

12 d2

ε

?Bdε (σ(s))

|Hess h±t |2 dm dt

≤

∫ 4d2ε

14 d2

ε

(?B3dε (σ(s))

((|∇h±t |

2−1)∆(φ) + 2(N − 1)|∇h±t |2+α′(t)(|∇h±t |

2−1))

dm)

dt

≤

∫ 4d2ε

14 d2

ε

(?B3dε (σ(s))

2(N − 1) + cd−2ε ||∇h±t |

2−1| dm)

dt.

(61)

Integrating across σ for s ∈ [ δ2 dp,q, (1 − δ2 )dp,q],∫ 2d2

ε

12 d2

ε

( ∫ (1− δ2 )dp,q

δ2 dp,q

?Bdε (σ(s))

|Hess h±t |2 dm ds

)dt

≤ cd−2ε

∫ 4d2ε

14 d2

ε

( ∫ (1− δ2 )dp,q

δ2 dp,q

?B3dε (σ(s))

cd2ε + ||∇h±t |

2−1| dm ds)

dt

≤ cε2dp,q , by statement 2 of Theorem 4.11.

(62)

Therefore, statement 4 holds for some r ∈ [ 12 , 2] and t = rd2

ε .

Lemma 4.13. Let ε ≤ ε(N, δ). Let γx,p be any unit speed geodesic from x ∈ X to p. Then for m-a.e. x ∈ X δ2 ,3

and any 0 ≤ t1 < t2 ≤ dx,p −δ2 , the following estimates hold:

1.∫ dx,p−

δ2

0 ||∇h−d2ε|2−1|(γx,p(s)) ds ≤ c(N,δ)

dp,q(e(x) + d2

ε );

2.∫ dx,p−

δ2

0 |〈∇h−d2ε,∇d−〉 − 1|(γx,p(s)) ds ≤ c(N,δ)

dp,q(e(x) + d2

ε );

3.∫ t2

t1|∇h−

d2ε− ∇d−|(γx,p(s)) ds ≤ c(N,δ)

√t2 − t1√

dp,q

(√

e(x) + dε).

Proof. The bounds on (|∇h−d2ε|2−1)+ and (〈∇h−

d2ε,∇d−〉 − 1)+ for statements 1 and 2 come from Lemma 4.8,

Fubini’s theorem and statement 2 of Theorem 3.15. The bound on the negative part comes from an estimatelike (56) combined with Corollary 3.14 for statement 2, and then an additional application Cauchy-Schwarzfor statement 1. We note that if one traces the proof of (56) back to Lemma 4.4, it is clear that one can obtainbounds where the excess is not related to the heat flow time as they have been for the past several claims.

For statement 3,

|∇h−d2ε− ∇d−|2= |∇h−d2

ε|2+1 − 2〈∇h−d2

ε,∇d−〉 ≤ ||∇h−d2

ε|2−1|+2|〈∇h−d2

ε,∇d−〉 − 1| m-a.e..

Therefore, statement 1, 2, Cauchy-Schwarz and an argument by Fubini’s theorem using statement 2 of Theorem3.15 gives statement 3.

5. Gromov-Hausdorff approximation

This section will be divided into three subsections. The main lemma proved in the first subsection gives away of overcoming the lack of start of induction in the arguments of [CN12] generalized to the RCD setting. Inthe second subsection we use the main lemma to construct geodesics with nice properties in its interior. Finally,we prove the main theorem in the third subsection. To be precise, we prove a slightly weaker version of themain theorem analagous to the main result of [CN12], which will be used to prove non-branching in Section 6and, subsequently, the main theorem.

36

Fix (X, d,m), an RCD(−(N − 1),N) metric measure space for N ∈ (1,∞) and p, q ∈ X with d(p,q)=1. Fix0 < δ < 0.1. For any x1, x2 ∈ X, we fix a constant speed geodesic from x1 to x2 parameterized on [0, 1] anddenote it γx1,x2 . By Remark 2.26, we may assume the map X × X × [0, 1] 3 (x1, x2, t) 7→ γx1,x2(t) is Borel. Theunit speed reparameterizations of γx1,x2 to the interval [0, d(x1, x2)] will be denoted γx1,x2 . γ will denote γp,q.For each x ∈ X, define Ψ : X × [0,∞)→ X by

(x, s) 7→ Ψs(x) =

γx,p(s) if d(x, p) ≥ s,p if d(x, p) < s.

(63)

Similarly, define Φ : X × [0,∞)→ X by

(x, s) 7→ Φs(x) =

γx,q(s) if d(x, q) ≥ s,q if d(x, q) < s.

(64)

By integral Abresch-Gromoll inequality 2.42, for any sufficiently small r ≤ r(N, δ) and any δ ≤ t0 ≤ 1 − δ,?Br(γ(t0))

e ≤ c0(N, δ)r2.

Therefore, there exists a subset S ⊆ Br(γ(t0)) so that1. m(S )

m(Br(γ(t0)) ≥ 1 − V(1,10)3 (see 2.4 for the definiton of V := V−(N−1),N)

2. ∀z ∈ S , e(z) ≤ c1(N, δ)2r2.

We fix such a c1 for the rest of this section and assume in addition c1 > 100.In all subsections the letter c will be used to represent different constants which only depend on N and δ.

Any constant which will be used repeatedly will be given a subscript. We will continue using the notations ofSection 4.

5.1. Proof of main lemma.

Lemma 5.1. (Main lemma) There exists ε1(N, δ) > 0 and r1(N, δ) > 0 so that for all r ≤ r1 and δ ≤ t0 ≤ 1 − δ,there exists z ∈ Br(γ(t0)) so that

1. V(1, 100) ≤ m(Br(Ψs(z)))m(Br(z)) ≤ 1

V(1,100) for any s ≤ ε1.2. There exists A ⊆ Br(z) with m(A) ≥ (1 − V(1, 10))m(Br(z)) and Ψs(A) ⊆ B2r(Ψs(z)) for any s ≤ ε1.3. e(z) ≤ c2

1r2.

Proof. Fix δ ≤ t0 ≤ 1 − δ and a scale r ≤ r1(N, δ). r1 need only be chosen smaller than the radius boundsrequired for the application of various theorems in the proof; most notably Theorem 2.42 and the estimates ofSection 4. It will be clear that all the required radius bounds only depend on N and δ so we will not addressthis each time for the sake of brevity. In addition, we assume r1 ≤

δ10 .

By Bishop-Gromov volume comparison 2.4, integral Abresch-Gromoll inequality 2.42, and the fact that Ψ isdefined using unit speed geodesics, it is clear there exist ε depending on N, δ and r, and z ∈ Br(γ(t0)) satisfying

1. V(1, 100) ≤ m(Br(Ψs(z)))m(Br(z)) ≤ 1

V(1,100) for any s ≤ ε.2. There exists A ⊆ Br(z) with m(A) ≥ (1 − V(1, 10))m(Br(z)) and Ψs(A) ⊆ B2r(Ψs(z)) for any s ≤ ε.3. e(z) ≤ c2

1r2.We will remove the dependence of ε on r.

To this effect, we will show that if 1, 2 and 3 hold for some z ∈ Br(γ(t0)) and all s ≤ ε less than or equalto some ε1(N, δ) to be fixed later, then in fact we can choose z′ ∈ Br(γ(t0)) satisfying 3 which significantlyimproves the estimates in 1 and 2 for s ≤ ε. To be precise, we find z′ ∈ Br(γ(t0)) satisfying

1’. 2V(1, 100) ≤ m(Br(Ψs(z′)))m(Br(z′)) ≤ 1

2V(1,100) for any s ≤ ε.2’. There exists A′ ⊆ Br(z′) with m(A′) ≥ (1 − V(1, 10))m(Br(z′)) and Ψs(A′) ⊆ B 3r

2(Ψs(z′)) for any s ≤ ε.

3’. e(z′) ≤ c21r2.

37

We a priori assume ε1 ≤δ

10 and impose more bounds on ε1 as the proof continues. Let z satisfy 1, 2 and 3 forsome ε ≤ ε1.

Let w be the midpoint of z and γ(t0). We know B r2(w) ⊆ Br(γ(t0)) ∩ Br(z). By Bishop-Gromov and since

r ≤ r which was assumed to be less than 0.01,

m(Br(γ(t0)) ∩ Br(z))m(Br(z))

≥m(B r

2(w))

m(Br(z))≥

m(B r2(w))

m(B 3r2

(w))≥ V(

r2,

3r2

) > V(12,

32

).

Using m(A) ≥ (1 − V(1, 10))m(Br(z)) > (1 − V( 12 ,

32 )

3 )m(Br(z)) and the previous estimate,

m(A ∩ Br(γ(t0)))m(Br(z))

>23

V(12,

32

).

Therefore,m(A ∩ Br(γ(t0)))

m(Br(γ(t0)))=

m(A ∩ Br(γ(t0)))m(Br(z))

m(Br(z))m(Br(γ(t0)))

>23

V(12,

32

)V(r, 2r) >23

V(12,

32

)V(32, 3) >

23

V(1, 10).(65)

Define the setD1 := A ∩ Br(γ(t0)) ∩

e(x) ≤ c2

1r2, (66)

where c1 is as fixed earlier. We will choose a z′ satisfyings propereties 1’ - 3’ from D1. From (65) and thedefinition of c1,

m(D1)m(Br(γ(t0)))

>13

V(1, 10).

Therefore, by Bishop-Gromov,m(D1)

m(Br(z))≥ c(N). (67)

Since e(z) ≤ c21r2 by property 3 of z, the curve traversing γz,p in reverse and then γz,q is a c1r-geodesic from

p to q. Fix h− ≡ h−ρ(c1r)2 satisfying statement 4 of Theorem 4.12 for the balls of radius c1r along this curve,

where ρ ∈ [ 12 , 2].

Since z has low excess, by integral Abresch-Gromoll, there exists B2r(z)′ ⊆ B2r(z) so that

e(x) ≤ c(N, δ)r2 ∀x ∈ B2r(z)′ andm(B2r(z)′)m(B2r(z))

≥ 1 −12

V(1, 10)2. (68)

For all s ∈ [0, ε] and (x, y) ∈ X × X, define

dt1(s)(x, y) := minr, max

0≤τ≤s|d(x, y) − d(Ψτ(x),Ψτ(y))|

(69)

andU s

1 := (x, y) ∈ D1 × B2r(z)′|dt1(s)(x, y) < r. (70)Consider

∫D1×B2r(z)′ dt1(s)(x, y) d(m × m)(x, y) for 0 ≤ s ≤ ε. Since r ≤ r1 ≤

δ10 , ε ≤ ε1 ≤

δ10 , and t0 ≥ δ,

(Ψs)s∈[0,ε] is a local flow of −∇dp from both D1 and B2r(z)′. Therefore, s 7→∫

D1×B2r(z)′ dt1(s)(x, y) d(m×m)(x, y)

38

is Lipschitz anddds

∫D1×B2r(z)′

dt1(s)(x, y) d(m × m)(x, y)

≤

∫U s

1

(|∇h− − ∇dp|(Ψs(x)) + |∇h− − ∇dp|(Ψs(y))

)d(m × m)(x, y)

+

1∫0

∫U s

1

d(Ψs(x),Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) d(m × m)(x, y) dτ,

(71)

for a.e. s ∈ [0, ε] by Proposition 3.27.For any s ∈ [0, ε] and (x, y) ∈ U s

1,1. d(x, y) < 3r since D1 ⊆ Br(z) and B2r(z)′ ⊆ B2r(z);2. d(Ψs(x),Ψs(z)) < 2r since D1 ⊆ A by definition (66);3. dt1(s)(x, y) < r by definition of U s

1 (70).Therefore, Ψs(y) ∈ B6r(Ψs(z)) by triangle inequality and so (Ψs,Ψs)(U s

1) ⊆ B c12 r(Ψs(z)) × B c1

2 r(Ψs(z)) since weassumed c1> 100. Therefore,

1∫0

∫U s

1

d(Ψs(x),Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) d(m × m)(x, y) dτ

≤ c(N, δ)

1∫0

∫(Ψs,Ψs)(U s

1)

d(x, y)|Hess h−|HS(γx,y(τ)) d(m × m)(x, y) dτ , by Theorem 3.15, 2

≤ c(N, δ)rm(B c12 r(Ψs(z)))

∫Bc1r(Ψs(z))

|Hess h−|HS dm , by segment inequality 3.22

≤ c(N, δ)rm(Br(z))2?

Bc1r(Ψs(z))

|Hess h−|HS dm , by Bishop-Gromov and property 1 of z.

(72)

Integrating in s ∈ [0, ε],ε∫

0

( 1∫0

∫U s

1

d(Ψs(x),Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) d(m × m)(x, y) dτ)

ds

≤ crm(Br(z))2

ε∫0

?Bc1r(Ψs(z))

|Hess h−|HS dm ds

≤ c(N, δ)rm(Br(z))2 √ε,

(73)

where the last line follows from the definition of h−, statement 4 of Theorem 4.12, and Cauchy-Schwarz.

39

By statement 3 of 4.13, the excess bound on the elements of D1 (66) (and therefore also on the elementsΨs(D1)), and Bishop-Gromov,

ε∫0

∫U s

1

|∇h− − ∇dp|(Ψs(x)) d(m × m)(x, y) ds ≤ c(N, δ)rm(Br(z))2 √ε. (74)

Similarly by the excess bounds on the elements of B2r(z)′ (68),ε∫

0

∫U s

1

|∇h− − ∇dp|(Ψs(y)) d(m × m)(x, y) ds ≤ c(N, δ)rm(Br(z))2 √ε. (75)

Combining (73) - (75) with the bound (71) on dds

∫D1×B2r(z)′

dt1(s)(x, y), we obtain∫D1×B2r(z)′

dt1(ε)(x, y) d(m × m)(x, y) =

∫ ε

0[

dds

∫D1×B2r(z)′

dt1(s)(x, y) d(m × m)(x, y)] ds

≤ c(N, δ)rm(Br(z))2 √ε.

(76)

Since D1 takes a non-trivial portion of the measure of Br(z) by (67),?D1

∫B2r(z)′

dt1(ε)(x, y) dm(y) ≤ c(N, δ)rm(Br(z))√ε.

In particular, there exists z′ ∈ D1 so that∫B2r(z)′

dt1(ε)(z′, y) dm(y) ≤ crm(Br(z))√ε.

By definition of D1, property 3’ of z′ is satisfied.We next check property 2’ is satisfied for the chosen z′ as well if ε is sufficiently small. Define Br(z′)′ :=

Br(z′) ∩ B2r(z)′. By the previous estimate and Bishop-Gromov,∫Br(z′)′

dt1(ε)(z′, y) dm(y) ≤ crm(Br(z))√ε ≤ c(N, δ)rm(Br(z′))

√ε. (77)

Using this, we bound ε1 sufficiently small depending on N and δ so that for ε ≤ ε1,∫Br(z′)′

dt1(ε)(z′, y) dm(y) ≤14

rm(Br(z′))V(1, 10). (78)

For example, ε1 ≤ ( V(1,10)4c )2 suffices, where c is the last one from (77). Moreover, B2r(z)′ takes significant mass

in B2r(z) from (68) and so by Bishop-Gromov,

m(Br(z′)′)m(Br(z′))

≥ 1 −(m(B2r(z)′)

m(B2r(z))m(B2r(z))m(Br(z′))

)≥ 1 −

(12

(V(1, 10)2)V(1, 3)

)≥ 1 −

12

V(1, 10). (79)

Combining (78) and (79), we conclude there exists A′ ⊆ Br(z′)′ so thatm(A′)

m(Br(z′))≥ 1 − V(1, 10) and dt1(ε)(z′, y) ≤

12

r ∀y ∈ A′. (80)

The latter implies Ψs(A′) ⊆ B 3r2

(Ψs(z′)) for any s ≤ ε and so property 2’ of z′ is satisfied.

40

This also gives one direction of the bound in property 1’ for z′. For each s ≤ ε,

m(Br(Ψs(z′)))m(Br(z′))

≥ V(1,32

)m(B 3r

2(Ψs(z′)))

m(Br(z′)), by Bishop-Gromov

≥ V(1,32

)m(Ψs(A′))m(Br(z′))

≥ V(1,32

)(1 + c(N, δ)s)−N m(A′)m(Br(z′))

, by Theorem 3.15, 2

≥ V(1,32

)(1 + c(N, δ)s)−N(1 − V(1, 10)).

We bound ε1 sufficiently small depending on N and δ so that for s ≤ ε ≤ ε1, the last line is greater than2V(1, 100).

The other direction of the bound in property 1’ of z’ will be proved similarly by sending a sufficiently largeportion of Br(Ψs(z′)) close to z′ (in fact z) using a flow which does not decrease measure significantly and thenusing Bishop-Gromov. To do this, we first use the RLF associated to −∇h− to send a portion of Br(z′) close toΨs(z′). We then use the inverse flow (i.e. the RLF associated to ∇h−) on the image of that portion to make surea large enough portion of Br(Ψs(z′)) indeed ends up close to z′ under the inverse flow.|∇h−0 |∈ L∞(m) by (48) and so |∇h−|,∆h− ∈ L∞(m) by Bakry-Ledoux estimate 2.16 and h− ∈ W2,2(X) by

Corollary 2.20 of the improved Bochner inequality. Therefore, the time-independent vector fields −∇h− and∇h− are bounded and satisfy the conditions of the existence and uniqueness of RLFs Theorem 3.4. Let (Ψt)t∈[0,1]and (Ψ−t)t∈[0,1] be the associated RLFs of −∇h− and ∇h− for t ∈ [0, 1] respectively. The choice of notation isdue to Proposition 3.12, which says Ψ−t and Ψt are m-a.e. inverses of each other.

Since e(z′) ≤ c21r2 and m(A′) ≥ (1 − V(1, 10))m(Br(z′)), integral Abresch-Gromoll gives A′′ ⊆ A′ so that

e(x) ≤ c(N, δ)r2 ∀x ∈ A′′ andm(A′′)

m(Br(z′))≥ 1 − 2V(1, 10). (81)

For all s ∈ [0, ε] and (x, y) ∈ X × X, define

dt2(s)(x, y) := minr, max

0≤τ≤s|d(x, y) − d(Ψτ(x), Ψτ(y))|

(82)

andU s

2 := (x, y) ∈ A′′ × Br(z′)|dt2(s)(x, y) < r. (83)Consider

∫A′′×Br(z′) dt2(s)(x, y) d(m × m)(x, y) for 0 ≤ s ≤ ε. By Proposition 3.27, for a.e. s ∈ [0, ε],

dds

∫A′′×Br(z′)

dt2(s)(x, y) d(m × m)(x, y)

≤

∫U s

2

|∇h− − ∇dp|(Ψs(x)) d(m × m)(x, y)

+

1∫0

∫U s

2

d(Ψs(x), Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) d(m × m)(x, y) dτ.

(84)

For any s ∈ [0, ε] and (x, y) ∈ U s2,

1. d(x, y) < 2r since A′′ ⊆ Br(z′);2. d(Ψs(x),Ψs(z)) ≤ d(Ψs(x),Ψs(z′)) + d(Ψs(z′),Ψs(z)) < 7

2 r by definition of A′ (80) and z′ ∈ D1 ⊆ A(66);

3. dt2(s)(x, y) < r by definition of U s2 (84).

41

Therefore, Ψs(y) ∈ B 13r2

(Ψs(z)) by triangle inequality and so (Ψs, Ψs)(U s2) ⊆ B c1

2 r(Ψs(z)) × B c12 r(Ψs(z)) since

c1 > 100. Therefore,1∫

0

∫U s

2

d(Ψs(x), Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) d(m × m)(x, y) dτ

≤ c(N, δ)

1∫0

∫(Ψs,Ψs)(U s

2)

d(x, y)|Hess h−|HS(γx,y(τ)) d(m × m)(x, y) dτ , by 3.15 2, 4.2 and 3.4 (18)

≤ c(N, δ)rm(B c12 r(Ψs(z)))

∫Bc1r(Ψs(z))

|Hess h−|HS dm , by segment inequality 3.22

≤ c(N, δ)rm(Br(z))2?

Bc1r(Ψs(z))

|Hess h−|HS dm , by Bishop-Gromov and property 1 of z.

(85)

Integrating in s ∈ [0, ε],ε∫

0

( 1∫0

∫U s

2

d(Ψs(x), Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) d(m × m)(x, y) dτ)

ds

≤ crm(Br(z))2

ε∫0

?Bc1r(Ψs(z))

|Hess h−|HS dm ds

≤ c(N, δ)rm(Br(z))2 √ε,

(86)

where the last line follows from the definition of h−, statement 4 of Theorem 4.12, and Cauchy-Schwarz.By statement 3 of 4.13, the excess bound on the elements of A′′ (81), and Bishop-Gromov,

ε∫0

∫U s

2

|∇h− − ∇dp|(Ψs(x)) d(m × m)(x, y) ds ≤ c(N, δ)rm(Br(z))2 √ε. (87)

Combining (86), (87) with the bound (84) we obtain,∫A′′×Br(z′)

dt2(ε)(x, y) d(m × m)(x, y) =

∫ ε

0[

dds

∫A′′×Br(z′)

dt2(s)(x, y) d(m × m)(x, y)] ds

≤ c(N, δ)rm(Br(z))2 √ε.

(88)

A′′ is comparable in measure to Br(z′) by (81) and hence also to Br(z) by Bishop-Gromov. Therefore, thereexists z1 ∈ A′′ so that ∫

Br(z′)

dt2(ε)(z1, y) dm(y) ≤ c(N, δ)rm(Br(z))√ε.

By Bishop-Gromov, m(Br(z))m(Br(z′)) ≤ c(N) and so?

Br(z′)

dt2(ε)(z1, y) dm(y) ≤ c(N, δ)r√ε.

42

Using this, we bound ε1 sufficiently small depending on N and δ so that there exists D2 ⊆ Br(z′) withm(D2)

m(Br(z′))≥ 1 − V(1, 10) and dt2(ε)(z1, y) ≤

r2∀y ∈ D2. (89)

For each y ∈ D2 and s ∈ [0, ε],1. d(z1, y) < 2r;2. d(Ψs(z1),Ψs(z)) ≤ d(Ψs(z1),Ψs(z′)) + d(Ψs(z′),Ψs(z)) < 7

2 r;3. dt2(ε)(z1, y) ≤ r

2 ,and so

Ψs(D2) ⊆ B4r(Ψs(z′)) ⊆ B6r(Ψs(z)). (90)Moreover, Ψs(D2) is non-trivial in measure compared to Br(z).

m(Ψs(D2))m(Br(z))

≥ e−c(N,δ) δ10

m(D2)m(Br(z))

, by 4.2, 3.4 (18), and ε1 ≤δ

10≥ c(N, δ) , by definition of D2 and Bishop-Gromov.

(91)

We will now flow Ψs(D2) back by Ψ−t and use that to control the flow of Br(Ψs(z′)) under Ψ−t. Fix s ∈ [0, ε].By Proposition 3.12, we may assume, up to choosing a full measure subset, that D2 satisfies

Ψ−t(Ψs(x)) = Ψs−t(x) ∀t ∈ [0, s] and ∀x ∈ D2. (92)

For all t ∈ [0, s] and (x, y) ∈ X × X, define

dt3(t)(x, y) := minr, max

0≤τ≤t|d(x, y) − d(Ψ−τ(x), Ψ−τ(y))|

(93)

andU t

3 := (x, y) ∈ Ψs(D2) × Br(Ψs(z′))|dt3(t)(x, y) < r. (94)We note that U t

3 implicitly depends on s. Consider∫Ψs(D2)×Br(Ψs(z′))

dt3(t)(x, y) d(m × m)(x, y) for 0 ≤ t ≤ s. Byproposition 3.27, for a.e. t ∈ [0, s],

ddt

∫Ψs(D2)×Br(Ψs(z′))

dt3(t)(x, y) d(m × m)(x, y)

≤

1∫0

∫U t

3

d(Ψ−t(x), Ψ−t(y))|Hess h−|HS(γΨ−t(x),Ψ−t(y)(τ)) d(m × m)(x, y) dτ,

(95)

For any t ∈ [0, s], ω ∈ [0, t] and (x, y) ∈ U t3,

1. d(x, y) < 5r since Ψs(D2) ⊆ B4r(Ψs(z′)) by (90);2. d(Ψ−ω(x),Ψs−ω(z)) = d(Ψs−ω(x′),Ψs−ω(z)) < 6r for some x′ ∈ D2 by (90) and (92);3. dt3(t)(x, y) < r by definition of U t

3 (94).Hence,

Ψ−ω(y) ∈ B12r(Ψs−ω(z)) (96)by triangle inequality. Therefore, (Ψ−ω, Ψ−ω)(U t

3) ⊆ B c12 r(Ψs−ω(z)) × B c1

2 r(Ψs−ω(z)) for all ω ∈ [0, t] since c1

> 100. For any (x, y) ∈ U t3,

∆h−(Ψ−ω(x)) = ∆h+(Ψ−ω(x)) + ∆e(Ψ−ω(x))

≥ −c(N, δ) , by Lemma 4.2 and Lemma 4.3 3, using e(z) ≤ c21r2,

(97)

43

where h+, e are heat flow approximations of h+0 and e0 respectively up to the same time as h−. We have the

same bound for ∆h−(Ψ−ω(y)). Therefore,1∫

0

∫U t

3

d(Ψ−t(x), Ψ−t(y))|Hess h−|HS(γΨ−t(x),Ψ−t(y)(τ)) d(m × m)(x, y) dτ

≤ c(N, δ)

1∫0

∫(Ψ−t ,Ψ−t)(U t

3)

d(x, y)|Hess h−|HS(γx,y(τ)) d(m × m)(x, y) dτ, by (97) and Remark 3.5

≤ c(N, δ)rm(B c12 r(Ψs−t(z)))

∫Bc1r(Ψs−t(z))

|Hess h−|HS dm , by segment inequality 3.22

≤ c(N, δ)rm(Br(z))2?

Bc1r(Ψs−t(z))

|Hess h−|HS dm , by Bishop-Gromov and property 1 of z.

(98)

Integrating in t ∈ [0, s],s∫

0

( 1∫0

∫U t

3

d(Ψ−t(x), Ψ−t(y))|Hess h−|HS(γΨ−t(x),Ψ−t(y)(τ)) d(m × m)(x, y) dτ)

dt

≤ crm(Br(z))2

s∫0

?Bc1r(Ψs−t(z))

|Hess h−|HS dm ds

≤ c(N, δ)rm(Br(z))2 √s,

(99)

where the last line follows from the definition of h−, statement 4 of Theorem 4.12, and Cauchy-Schwarz.Therefore, ∫

Ψs(D2)×Br(Ψs(z′))

dt3(s)(x, y) d(m × m)(x, y) =

∫ s

0[

ddt

∫Ψs(D2)×Br(Ψs(z′))

dt3(t)(x, y) d(m × m)(x, y)] dt

≤ c(N, δ)rm(Br(z))2 √s.

(100)

We previously computed that Ψs(D2) is non-trivial in measure compared to Br(z) in (91) and so there existsz2 ∈ Ψs(D2) with ∫

Br(Ψs(z′))

dt3(s)(z2, y) dm(y) ≤ c(N, δ)rm(Br(z))√

s.

By Bishop-Gromov, m(Br(Ψs(z′)))m(Br(Ψs(z))) ≥ c(N) and so by property 1 of z,?

Br(Ψs(z′))

dt3(s)(z2, y) dm(y) ≤ c(N, δ)r√

s.

Using this, we bound ε1 sufficiently small depending on N and δ so there exists D3 ⊆ Br(Ψs(z′)) withm(D3)

m(Br(Ψs(z′)))≥ 1 − V(1, 10) and dt3(s)(z2, y) ≤

12

r ∀y ∈ D3. (101)

For each y ∈ D3,1. d(z2, y) < 5r by (90);

44

2. d(Ψ−s(z2), z′) = d(z′2, z′) < r, where z′2 ∈ D2 ⊆ Br(z′) is so that Ψs(z′2) = z2;

3. dt3(s)(z2, y) ≤ 12 r,

and so Ψ−s(D3) ⊆ B7r(z′). Notice also for the next calculation that Ψ−t(D3) ⊆ B12r(z) for any t ∈ [0, s] by thecaulations of (96). Therefore, one has the same lower bound for the Laplacian of h− on Ψ−t(D3) as in (97).

We estimatem(Br(Ψs(z′)))

m(Br(z′))≤

1V(1, 7)

m(Br(Ψs(z′)))m(B7r(z′))

, by Bishop-Gromov

≤1

V(1, 7)1

1 − V(1, 10)m(D3)

m(B7r(z′)), by property (101) of D3

≤1

V(1, 7)1

1 − V(1, 10)ec(N,δ)s m(Ψ−s(D3))

m(B7r(z′)), by Remark 3.5

≤1

V(1, 7)1

1 − V(1, 10)ecs.

We bound ε1 sufficiently small depending on N and δ so that for s ≤ ε ≤ ε1, the last line is less than 12V(1,100) .

All this imply that if 1, 2 and 3 hold for z ∈ Br(γ(t0)) and ε ≤ ε1(N, δ), then there exists z′ ∈ Br(γ(t0))satisfying 1’, 2’ and 3’ for the same ε. By Bishop-Gromov volume comparison and the fact that Ψ is definedusing unit speed geodesics, there is some T (N, δ, r) > 0 so that 1 and 2 hold for z′, A′, and 0 ≤ s ≤ ε + T .Combining this with the existence of some z and ε depending on N, δ, and r satisfying 1, 2 and 3 mentioned atthe beginning of the proof, we conclude there exists z ∈ Br(γ(t0)) so that 1, 2 and 3 hold for ε = ε1.

5.2. Construction of limit geodesics. In this section we construct a geodesics between p and q which hasproperties 1 and 2 of Lemma 5.1 on the geodesic itself. Roughly, this means we construct γ so that small ballscentered on γ between δ and 1 − δ stay close to the geodesic itself for a short amount time under the flows Ψ

and Φ.We start by showing that for any fixed scale r we can find points z arbitrarily close to γ(t0) which have the

properties 1 - 3 as in Lemma 5.1. In order to do this we will prove the following lemma which will form ourinduction step.

Lemma 5.2. There exists ε2(N, δ) > 0 and r2(N, δ) > 0 so that for any r ≤ r2, δ ≤ t0 ≤ 1 − δ, if there existsz ∈ Br(γ(t0)) and ε ≤ ε2 so that

1. V(1, 100) ≤ m(Br(Ψs(z)))m(Br(z)) ≤ 1

V(1,100) for any s ≤ ε;2. There exists Ar ⊆ Br(z) with m(Ar) ≥ (1 − V(1, 10))m(Br(z)) and Ψs(Ar) ⊆ B2r(Ψs(z)) for any s ≤ ε;3. e(z) ≤ c2

1r2,then for the same z and any r′ ∈ [4r, 16r],

i. V(1, 100) ≤ m(Br′ (Ψs(z)))m(Br′ (z)) ≤ 1

V(1,100) for any s ≤ ε;ii. There exists Ar′ ⊆ Br′(z) with m(Ar′) ≥ (1−V(1, 10))m(Br′(z)) and Ψs(Ar′) ⊆ B2r′(Ψs(z)) for any s ≤ ε;

iii. e(z) ≤ c21r2 < c2

1(r′)2.

Proof. Fix δ ≤ t0 ≤ 1 − δ. We assume r2 ≤δ

1000 and ε2 ≤δ

1000 to begin with but will impose more bounds onboth depending on N and δ as the proof continues. We will not keep track of r2 for the sake of brevity. Fix ascale r ≤ r2 and r′ ∈ [4r, 16r]. Fix z, Ar, and ε ≤ ε2 so that 1, 2 and 3 hold.

Since e(z) ≤ c21r2, by integral Abresch-Gromoll, there exists Br′(z)′ ⊆ Br′(z) so that

e(x) ≤ c(N, δ)r2 ∀x ∈ Br′(z)′ andm(Br′(z)′)m(Br′(z))

≥ 1 −12

V(1, 10). (102)

45

Similarly, there exists A′ ⊆ Ar so that

e(x) ≤ c(N, δ)r2 ∀x ∈ A′ andm(A′)

m(Br(z))≥ 1 − 2V(1, 10). (103)

As in the previous lemma, the curve traversing γz,p in reverse and then γz,q is a c1r-geodesic from p to q.Fix h− ≡ h−

ρ(c1r)2 satisfying statement 4 of Theorem 4.12 for the balls of radius c1r along this curve, where

ρ ∈ [ 12 , 2].

For all s ∈ [0, ε] and (x, y) ∈ X × X, define

dt1(s)(x, y) := minr, max

0≤τ≤s|d(x, y) − d(Ψτ(x),Ψτ(y))|

(104)

andU s

1 := (x, y) ∈ A′ × Br′(z)′|dt1(s)(x, y) < r. (105)Consider

∫A′×Br′ (z)′ dt1(s)(x, y) d(m × m)(x, y) for 0 ≤ s ≤ ε. For any s ∈ [0, ε] and (x, y) ∈ U s

1,

1. d(x, y) < r′ + r;2. d(Ψs(x),Ψs(z)) < 2r by definition of A′ ⊆ Ar;3. dt1(s)(x, y) < r.

Therefore, Ψs(y) ∈ Br′+4r(Ψs(z)) ⊆ B c12 r(Ψs(z)) since c1> 100.

Using exactly the same type of computation as the first part of the proof of the main lemma, by interpolatingbetween the two local flows of Ψs from A′ and Br′(z)′ with ∇h−, we obtain∫

A′×Br′ (z)′

dt1(ε)(x, y) d(m × m)(x, y) ≤ c(N, δ)rm(Br(z))2 √ε. (106)

Since A′ takes a significant portion of the measure of Br(z) by (103),?A′

∫Br′ (z)′

dt1(ε)(x, y) dm(y) ≤ c(N, δ)rm(Br(z))√ε,

and so there exists z′ ∈ A′ so that ∫Br′ (z)′

dt1(ε)(z′, y) dm(y) ≤ crm(Br(z))√ε.

Therefore, by the fact that m(Br′ (z)′)m(Br′ (z)) ≥ 1 − 1

2 V(1, 10) and Bishop-Gromov,?Br′ (z)′

dt1(ε)(z′, y) dm(y) ≤ c(N, δ)r√ε.

Using this, we bound ε2 sufficiently small depending on N and δ so that there exists Ar′ ⊆ Br′(z)′ withm(Ar′)

m(Br′(z))≥ 1 − V(1, 10) and dt1(ε)(z′, y) ≤

r2∀y ∈ Ar′ . (107)

For each y ∈ Ar′ and s ∈ [0, ε],1. d(z′, y) < r′ + r;2. d(Ψs(z′),Ψs(z)) < 2r;3. dt1(ε)(z′, y) ≤ r

2 ,

46

and soΨs(Ar′) ⊆ Br′+ 7r

2(Ψs(z)) ⊆ B2r′(Ψs(z)). (108)

This proves property ii.This also gives one direction of the bound in property i. For each s ≤ ε,

m(Br′(Ψs(z)))m(Br′(z))

≥ V(1, 2)m(B2r′(Ψs(z)))

m(Br′(z′)), by Bishop-Gromov

≥ V(1, 2)m(Ψs(Ar′))m(Br′(z′))

≥ V(1, 2)(1 + c(N, δ)s)−N m(Ar′)m(Br′(z′))

, by Theorem 3.15, 2

≥ V(1, 2)(1 + c(N, δ)s)−N(1 − V(1, 10)).

We bound ε2 sufficiently small depending on N and δ so that for s ≤ ε ≤ ε2, the last line is greater thanV(1, 100).

To obtain the other direction of the bound in property i, we employ the same strategy as the proof of the mainlemma as well. Let (Ψt)t∈[0,1] and (Ψ−t)t∈[0,1] be the RLFs of the time-independent vector fields −∇h− and ∇h−

respectively as before.For all s ∈ [0, ε] and (x, y) ∈ X × X, define

dt2(s)(x, y) := minr, max

0≤τ≤s|d(x, y) − d(Ψτ(x), Ψτ(y))|

(109)

andU s

2 := (x, y) ∈ A′ × Br(z)|dt2(s)(x, y) < r. (110)Consider

∫A′×Br(z) dt2(s)(x, y) d(m × m)(x, y) for 0 ≤ s ≤ ε. For any s ∈ [0, ε] and (x, y) ∈ U s

2,

1. d(x, y) < 2r;2. d(Ψs(x),Ψs(z)) < 2r by definition of A′ ⊆ Ar;3. dt2(s)(x, y) < r.

Therefore, Ψs(y) ∈ B5r(Ψs(z)) ⊆ B c12 r(Ψs(z)).

Using exactly the same type of computation as the second part of the proof of the main lemma,∫A′×Br(z)

dt2(ε)(x, y) d(m × m)(x, y) ≤ c(N, δ)rm(Br(z))2 √ε. (111)

A′ takes a significant portion of the measure of Br(z) by (103). The same considerations as before gives theexistence of D1 ⊆ Br(z) so that

m(D1)m(Br(z))

≥ 1 − V(1, 10) and Ψs(D1) ⊆ B5r(Ψs(z)), (112)

for any s ∈ [0, ε] after we bound ε2 sufficiently small depending only on N and δ. Moreover, Ψs(D1) is non-trivial in measure compared to Br(z).

m(Ψs(D1))m(Br(z))

≥ e−c(N,δ) δ1000

m(D1)m(Br(z))

, by 4.2, 3.4 (18), and ε2 ≤δ

1000≥ c(N, δ) , by definition of D1.

(113)

Fix s ∈ [0, ε]. By Proposition 3.12, we may assume, up to choosing a full measure subset, that D1 satisfies

Ψ−t(Ψs(x)) = Ψs−t(x) ∀t ∈ [0, s] and ∀x ∈ D1. (114)

47

For all t ∈ [0, s] and (x, y) ∈ X × X, define

dt3(t)(x, y) := minr, max

0≤τ≤t|d(x, y) − d(Ψ−τ(x), Ψ−τ(y))|

(115)

andU t

3 := (x, y) ∈ Ψs(D1) × Br′(Ψs(z))|dt3(t)(x, y) < r. (116)Consider

∫Ψs(D1)×Br′ (Ψs(z)) dt3(t)(x, y) d(m×m)(x, y) for 0 ≤ t ≤ s. For any t ∈ [0, s], ω ∈ [0, t], and (x, y) ∈ U t

3,

1. d(x, y) < r′ + 5r by (112);2. d(Ψ−ω(x),Ψs−ω(z)) = d(Ψs−ω(x′),Ψs−ω(z)) < 5r for some x′ ∈ D1 by (112) and (114);3. dt3(t)(x, y) < r.

Hence,Ψ−ω(y) ∈ Br′+11r(Ψs−ω(z)) (117)

by triangle inequality. Therefore, (Ψ−ω, Ψ−ω)(U t3) ⊆ B c1

2 r(Ψs−ω(z)) × B c12 r(Ψs−ω(z)) for any ω ∈ [0, t] since c1

> 100.Using exactly the same type of computation as the third part of the proof of the main lemma,∫

Ψs(D1)×Br′ (Ψs(z))

dt3(ε)(x, y) d(m × m)(x, y) ≤ c(N, δ)rm(Br(z))2 √ε. (118)

By (113), Ψs(D1) is non-trivial in measure compared to Br(z). By Bishop-Gromov and property 1 of z,Br′(Ψs(z)) is also non-trivial in measure compared to Br(z). The same considerations as before gives the exis-tence of D2 ⊆ Br′(Ψs(z)) so that

m(D2)m(Br′(Ψs(z)))

≥ 1 − V(1, 10) and Ψ−s(D2) ⊆ Br′+7r(z) ⊆ B3r′(z), (119)

for any s ∈ [0, ε] after we bound ε2 sufficiently small depending only on N and δ.We estimate

m(Br′(Ψs(z)))m(Br′(z))

≤1

V(1, 3)m(Br′(Ψs(z)))

m(B3r′(z)), by Bishop-Gromov

≤1

V(1, 3)1

1 − V(1, 10)m(D2)

m(B3r′(z)), by property (119) of D2

≤1

V(1, 3)1

1 − V(1, 10)ec(N,δ)s m(Ψ−s(D2))

m(B3r′(z)), by Remark 3.5

≤1

V(1, 7)1

1 − V(1, 10)ecs,

where for the third inequality we used the fact that Ψ−t(D2) ⊆ B27r(Ψs−t(z)) for 0 ≤ t ≤ s by the calcluationsof (117); On these sets ∆h− is bounded below by −c(N, δ) by the same argument as (97). Using this, we boundε2 sufficiently small depending on N and δ so that for s ≤ ε ≤ ε2, the last line is less than V(1, 100). This showsthe other half of the bound in property i. Property iii is obvious and so we conclude.

Combined with the main lemma, this gives

Lemma 5.3. There exists ε3(N, δ) > 0 and r3(N, δ) > 0 so that for any r ≤ r3 and δ ≤ t0 ≤ 1 − δ, there existsz ∈ Br(γ(t0)) so that for any r ≤ r′ ≤ r3:

1. V(1, 100) ≤ m(Br′ (Ψs(z)))m(Br′ (z)) ≤ 1

V(1,100) for any s ≤ ε3;2. There exists A ⊆ Br′(z) with m(A) ≥ (1 − V(1, 10))m(Br′(z)) and Ψs(A) ⊆ B2r′(Ψs(z)) for any s ≤ ε3;3. e(z) ≤ c2

1r2.

48

Proof. Choose ε3 = min ε1, ε2 and r3 := min r1, r2. Apply 5.1 to find some z ∈ B r4(γ(t0)) which satisfies

properties 1 - 3 on the scale of r4 and then use 5.2 repeatedly to conclude.

The following corollary immediately follows from Lemma 5.3 by a limiting argument.

Corollary 5.4. There exists ε3(N, δ) > 0 and r3(N, δ) > 0 so that if γ is the unique geodesic between p and q,then for all r ≤ r3 and δ ≤ t0 ≤ 1 − δ,

1. V(1, 100) ≤ m(Br(γ(t0−s)))m(Br(γ(t0))) ≤

1V(1,100) for any 0 ≤ s ≤ ε3.

2. There exists A ⊆ Br(γ(t0)) with m(A) ≥ (1 − V(1, 10))m(Br(γ(t0))) and Ψs(A) ⊆ B2r(γ(t0 − s)) for any0 ≤ s ≤ ε3.

We can use the corollary directly in the proof of Theorem 5.10 to prove the main result for all uniquegeodesics. Taking Theorem 2.25 and Remark 2.26 into account, this is already enough for several applications.Nevertheless, we will next prove the existence of a geodesic between any p, q ∈ X with Holder continuity onthe geometry of small radius balls in its interior, which is the full result of [CN12]. The desired geodesic willbe constructed using multiple limiting and gluing arguments.

Lemma 5.5. There exists ε4(N, δ) > 0 and r4(N, δ) > 0 so that for any unit speed geodesic γ from p to q, thereexists a unit speed geodesic γδ from p to q with γδ ≡ γ on [1− δ, 1] so that for all r ≤ r4 and δ ≤ t0 ≤ t1 ≤ 1− δ,if t1 − t0 ≤ ε4 then,

1. V(1, 100)4 ≤m(Br(γδ(t1)))m(Br(γδ(t0))) ≤

1V(1,100)4 ;

2. There exists A ⊆ Br(γδ(t1)) so that m(A) ≥ (1 − V(1, 10))m(Br(γδ(t1))) and Ψs(A) ⊆ B2r(γδ(t1 − s)) forall s ∈ [0, t1 − t0].

Proof. Let ε3, r3 be from Lemma 5.3. We begin by assuming r4 ≤ r3 and ε4 ≤ε32 but will impose more bounds

on both as the proof continues. We will not keep track of r4 for the sake of brevity.Partition [δ, 1 − δ] by σ0 = δ, σ1, ... , σk = 1 − δ for some k ∈ N so that all subintervals have the same

width equal to some ω ∈ [ ε32 , ε3]. This is always possible since the width of the original interval is 1 − 2δ > 0.8

and ε3 is much smaller than 0.4. We construct γδ inductively as follows• Define γδ ≡ γ for t ∈ [1 − δ, 1].• Let 1 ≤ i ≤ k. Assume γδ has been constructed on the interval [σi, 1]. Fix a sequence of r j → 0. For

each j, choose z j ∈ Br j(γδ(σi)) as in the statement of Lemma 5.3. Use Arzela-Ascoli Theorem to take

a limit, after passing to a subsequence, of the unit speed geodesics from z j to p. This limit γδ,i is a unitspeed geodesic from γδ(σi) to p. For t ∈ [σi−1, σi], define γδ(t) = γδ,i(σi − t).• For t ∈ [0, δ], define γδ to be any geodesic from p to γδ(σ0).

For any 1 ≤ i ≤ k, σi − σi−1 ≤ ε3 so it follows from the construction that γδ satisfies, for any r ≤ r4,

i. V(1, 100) ≤ m(Br(γδ(σi−s)))m(Br(γδ(σi)))

≤ 1V(1,100) for any 0 ≤ s ≤ ω;

ii. There exists A ⊆ Br(γδ(σi)) with m(A) ≥ (1 − V(1, 10))m(Br(γδ(σi))) and Ψs(A) ⊆ B2r(γδ(σi − s)) forany 0 ≤ s ≤ ω.

Fix δ ≤ t0 ≤ t1 ≤ 1 − δ with t1 − t0 ≤ ε4 and a scale r ≤ r4. Since ε4 ≤ε32 is no greater than the widths of the

subintervals of the partition ω ≥ ε32 , t0 and t1 must be contained in [σi−2, σi] for some 2 ≤ i ≤ k. Statement 1 of

the lemma then follows trivially from property i of γδ.t0 and t1 must then be either contained in a single subinterval or two neighbouring subintervals of the par-

tition. We will assume the second case; the first case follows from a similar and simpler argument. Lett1 ∈ (σi−1, σi] and t0 ∈ (σi−2, σi−1] for some i. By property ii of γδ and Abresch-Gromoll, there existsA1 ⊆ B r

16(γδ(σi)) so that

1. m(A1)m(B r

16(γδ(σi)))

≥ 1 − 2V(1, 10);

49

2. Ψs(A1) ⊆ B r8(γδ(σi − s)) ∀s ∈ [0, ω];

3. e(x) ≤ c(N, δ)r2 ∀x ∈ A1.Simiarly, there exists A2 ⊆ B r

16(γδ(σi−1)) so that

1. m(A2)m(B r

16(γδ(σi−1))) ≥ 1 − 2V(1, 10);

2. Ψs(A2) ⊆ B r8(γδ(σi−1 − s)) ∀s ∈ [0, ω];

3. e(x) ≤ c(N, δ)r2 ∀x ∈ A2.The plan is as follows: first we show that a significant portion of A1 can be flowed by Ψ a non-trivial amount

of time past γδ(σi−1) while staying close γδ, then we use the flow of A1 under Ψ to control the flow of Br(γδ(t1))under Ψ.

Fix h− ≡ h−ρ(8r)2 satisfying statement 4 of Theorem 4.12 for the balls of radius 8r along γδ, where ρ ∈ [ 1

2 , 2].For all s ∈ [0, ω] and (x, y) ∈ X × X, define

dt(s)(x, y) := minr, max

0≤τ≤s|d(x, y) − d(Ψτ(x),Ψτ(y))|

(120)

andU s

1 := (x, y) ∈ A2 × A1|dt(s)(x,Ψω(y)) < r. (121)Consider

∫A2×A1

dt(s)(x,Ψω(y)) d(m × m)(x, y) for 0 ≤ s ≤ ω.For any s ∈ [0, ω] and (x, y) ∈ U s

1,

1. d(x,Ψω(y)) < 3r16 ;

2. d(Ψs(x), γδ(σi−1 − s)) < r8 ;

3. dt(s)(x,Ψω(y)) < r.Therefore, Ψs(Ψω(y)) ∈ Br+ 5r

16(γδ(σi−1 − s)) and so (Ψs,Ψs Ψω)(U s

1) ⊆ B4r(γδ(σi−1 − s)) × B4r(γδ(σi−1 − s)).By Remark 2.26, we have Ψs Ψω = Ψs+ω m-a.e.. Since we may always choose subsets of full measure wherethe equality is satisfied, we will replace the former with the latter freely.

We have,1∫

0

∫U s

1

d(Ψs(x),Ψs+ω(y)))|Hess h−|HS(γΨs(x),Ψs+ω(y)(τ)) d(m × m)(x, y) dτ

≤ c(N, δ)

1∫0

∫(Ψs,Ψs+ω)(U s

1)

d(x, y)|Hess h−|HS(γx,y(τ)) d(m × m)(x, y) dτ , by Theorem 3.15, 2

≤ c(N, δ)rm(B4r(γδ(σi−1 − s)))∫

B8r(γδ(σi−1−s))

|Hess h−|HS dm , by segment inequality 3.22

≤ c(N, δ)rm(Br(γδ(σi)))2?

B8r(γδ(σi−1−s))

|Hess h−|HS dm , by Bishop-Gromov and property i of γδ.

(122)

50

Integrating in s ∈ [0, ω′] for some ω′ ∈ (0, ω] to be fixed later, we haveω′∫

0

( 1∫0

∫U s

1

d(Ψs(x),Ψs+ω(y))|Hess h−|HS(γΨs(x),Ψs+ω(y)(τ)) d(m × m)(x, y) dτ)

ds

≤ crm(Br(γδ(σi)))2

ω′∫0

?B8r(γδ(σi−1−s))

|Hess h−|HS dm ds

≤ c(N, δ)rm(Br(γδ(σi)))2√ω′,

(123)

where the last line follows from the definition of h−, statement 4 of Theorem 4.12, and Cauchy-Schwarz.By statement 3 of 4.13, the excess bound on the elements of A2 and property i of γδ,

ω′∫0

∫U s

1

|∇h− − ∇dp|(Ψs(x)) d(m × m)(x, y) ds ≤ c(N, δ)rm(Br(γδ(σi)))2√ω′. (124)

Similarly by the excess bounds on the elements of A1,ω′∫

0

∫U s

1

|∇h− − ∇dp|(Ψs+ω(y)) d(m × m)(x, y) ds ≤ c(N, δ)m(Br(γδ(σi)))2r√ω′. (125)

By Proposition 3.27, ∫A2×A1

dt(ω′)(x,Ψτ(y)) ≤ c(N, δ)m(Br(γδ(σi)))2r√ω′. (126)

Arguing as in the proof of Lemma 5.1 and using property i of γδ, we can then fixω′ sufficiently small dependingonly on N and δ so that there exists z ∈ A2 and A′1 ⊆ A1 with

m(A′1)

m(B r16

(γδ(σi)))≥ 1 − 3V(1, 10) and dt(ω′)(z,Ψω(y)) ≤

r16∀y ∈ A′1. (127)

The latter impliesΨs+ω(A′1) ⊆ B 3r

8(γδ(σi−1 − s)) ∀s ∈ [0, ω′]. (128)

Notice by definiton of A1, Ψs(A′1) ⊆ B r8(γδ(σi − s)) for any s ∈ [0, ω].

We now compare the flow of Br(γδ(t1)) to that of Ψσi−t1(A′1) under Ψs for s ∈ [0, ω′]. By integral Abresch-Gromoll there exists Br(γδ(t1))′ ⊆ Br(γδ(t1)) so that

e(x) ≤ c(N, δ)r2 ∀x ∈ Br(γδ(t1))′ andm(Br(γδ(t1))′)m(Br(γδ(t1)))

≥ 1 −12

V(1, 10). (129)

For all s ∈ [0, ω′], define

U s2 := (x, y) ∈ A′1 × Br(γδ(t1))′|dt(s)(Ψσi−t1(x), y) < r. (130)

Consider∫

A′1×Br(γδ(t1))′ dt(s)(Ψσi−t1(x), y) d(m × m)(x, y) for 0 ≤ s ≤ ω′. For any s ∈ [0, ω′] and (x, y) ∈ U s2,

1. d(Ψσi−t1(x), y) < r + r8 by definition of A1;

2. d(Ψs(Ψσi−t1(x)), γδ(t1 − s)) < 3r8 by (128) and the line below it;

3. dt(s)(Ψσi−t1(x), y) < r.

51

Therefore, Ψs(y) ∈ B 5r2

(γδ(t1 − s)) and so (Ψs Ψσi−t1 ,Ψs)(U s2) ⊆ B4r(γδ(t1 − s)) × B4r(γδ(t1 − s)).

By the same type of computations as the first part of this proof, for some ω′′ ∈ (0, ω′] to be fixed later,∫A′1×Br(γ(t1))′

dt(ω′′)(Ψσi−t1(x), y) ≤ c(N, δ)m(Br(γδ(σi)))2r√ω′′. (131)

Arguing as in the proof of Lemma 5.1 and using property i of γδ, we can then fixω′′ sufficiently small dependingonly on N and δ so that there exists z′ ∈ A′1 and A ⊆ Br(γ(t1))′ with

m(A)m(Br(γδ(t1)))

≥ 1 − V(1, 10) and dt(ω′′)(Ψσi−t1(z′), y) ≤r2∀y ∈ A. (132)

The latter impliesΨs(A) ⊆ B2r(γδ(t1 − s)) ∀s ∈ [0, ω′′]. (133)

We bound ε4 ≤ ω′′ and so statement 1 of the lemma is proved.

We may also apply Lemma 5.5 in the other direction of γ towards q. However, there is no guarantee the twogeodesics we end up with in the two applications of the lemma are the same geodesics. Therefore, we will showthat a geodesic which has the properties of 5.5 necessarily has the same properties going in the other direction.The reason for this is Lemma 4.3, which roughly implies the local flows of h+ and h− are close to each othernear a geodesic on the scale of r.

Lemma 5.6. There exists ε5(N, δ) > 0 and r5(N, δ) > 0 so that if there exists a unit speed geodesic γ from p toq, ε ≤ ε5 and r ≤ r5 which satisfy, for all r ≤ r and δ ≤ t0 ≤ t1 ≤ 1 − δ with t1 − t0 ≤ ε,

1. V(1, 100)4 ≤m(Br(γ(t1)))m(Br(γ(t0))) ≤

1V(1,100)4 ;

2. There exists A1 ⊆ Br(γ(t1)) so that m(A1) ≥ (1 − V(1, 10))m(Br(γ(t1))) and Ψs(A1) ⊆ B2r(γ(t1 − s)) forall s ∈ [0, t1 − t0],

then for the same geodesic γ, ε and r, for all r ≤ r and δ ≤ t0 ≤ t1 ≤ 1 − δ with t1 − t0 ≤ ε, there existsA2 ⊆ Br(γ(t0)) so that

m(A2)m(Br(γ(t0)))

≥ 1 − V(1, 10) and Φs(A2) ⊆ B2r(γ(t0 + s)) ∀s ∈ [0, t1 − t0]. (134)

Proof. As a reminder, Φ is defined by (64) and is the local flow of −∇dq, at least from the sets and for the timeinterval we are concerned with.

We assume r5 ≤δ

10 and ε5 ≤δ

10 to begin with but will impose more bounds on both as the proof continues.We will not keep track of r5 for the sake of brevity. Fix γ, ε ≤ ε5 and r ≤ r5 which satisfy conditions 1 and 2.Fix r ≤ r and δ ≤ t0 ≤ t1 ≤ 1 − δ with t1 − t0 ≤ ε.

Fix h− ≡ h−ρ(8r)2 satisfying statement 4 of Theorem 4.12 for the balls of radius 8r along γ, where ρ ∈ [ 1

2 , 2].

Let (Ψt)t∈[0,1] and (Ψ−t)t∈[0,1] be the RLFs of the time-independent vector fields −∇h− and ∇h− respectively asbefore.

The plan is as follows: first we use property 2 to make sure a significant portion of B r16

(γ(t1)) stays close toγ from t1 to t0 under the flow of Ψs, then we reverse flow the image of this portion under Ψ−s and use it to makesure a significant portion of Br(γ(t0)) stays close to γ from t0 to t1 under Φs.

By condition 2 and integral Abresch-Gromoll, there exists A1 ⊆ B r16

(γ(t1)) so that

1. m(A1)m(B r

16(γ(t1))) ≥ 1 − 2V(1, 10);

2. Ψs(A1) ⊆ B r8(γ(t1 − s)) ∀s ∈ [0, t1 − t0];

3. e(x) ≤ c(N, δ)r2 ∀x ∈ A1.

52

For all s ∈ [0, t1 − t0] and (x, y) ∈ X × X, define

dt1(s)(x, y) := minr, max

0≤τ≤s|d(x, y) − d(Ψτ(x), Ψτ(y))|

(135)

andU s

1 := (x, y) ∈ A1 × B r16

(γ(t1))|dt1(s)(x, y) < r. (136)

Consider∫

A1×B r16

(γ(t1)) dt1(s)(x, y) d(m × m)(x, y) for 0 ≤ s ≤ t1 − t0. For any s ∈ [0, t1 − t0] and (x, y) ∈ U s1,

1. d(x, y) < r8 ;

2. d(Ψs(x), γ(t1 − s)) < r8 by definition of A1;

3. dt1(s)(x, y) < r.Therefore, Ψs(y) ∈ B 5r

4γ(t1 − s) by triangle inequality and so (Ψs, Ψs)(U s

1) ⊆ B4r(Ψs(z)) × B4r(Ψs(z)).Using exactly the same type of computation as the second part of the proof of the main lemma,∫

A1×B r16

(γ(t1))

dt1(t1 − t0)(x, y) d(m × m)(x, y) ≤ c(N, δ)rm(Br(γ(t1)))2 √t1 − t0. (137)

A1 takes a significant portion of the measure of Br(γ(t1)) by definition. The same considerations as in the proofof Lemma 5.1 gives the existence of z ∈ A1 and D1 ⊆ Br(γ(t1)) so that

m(D1)m(B r

16(γ(t1)))

≥ 1 − V(1, 10) and dt1(t1 − t0)(z, y) ≤r

16∀y ∈ D1, (138)

after we bound ε5 sufficiently small depending only on N and δ. The latter implies

Ψs(D1) ⊆ B 5r16

(γ(t1 − s)) ∀s ∈ [0, t1 − t0]. (139)

By Proposition 3.12, we may assume in addition that D1 satisfies

Ψ−s(Ψt1−t0(x)) = Ψt1−t0−s(x) ∀s ∈ [0, t1 − t0] and ∀x ∈ D1. (140)

Furthermore, Ψs(D1) is non-trivial in measure compared to Br(γ(t1)) for all s ∈ [0, t1 − t0].

m(Ψs(D1))m(Br(γ(t1)))

≥ e−c(N,δ) δ10

m(D1)m(Br(z))

, by 4.2, 3.4 (18), and ε5 ≤δ

10≥ c(N, δ) , by definition of D1 and Bishop-Gromov.

(141)

By integral Abresch-Gromoll, there exists Br(γ(t0))′ ⊆ Br(γ(t0)) so that

e(x) ≤ c(N, δ)r2 ∀x ∈ Br(γ(t0))′ andm(Br(γ(t0))′)m(Br(γ(t0)))

≥ 1 −12

V(1, 10). (142)

For all s ∈ [0, t1 − t0] and (x, y) ∈ X × X, define

dt2(s)(x, y) := minr, max

0≤τ≤s|d(x, y) − d(Ψ−τ(x),Φτ(y))|

(143)

andU s

2 := (x, y) ∈ Ψt1−t0(D1) × Br(γ(t0))′|dt2(s)(x, y) < r. (144)

53

Consider∫Ψt1−t0 (D1)×Br(γ(t0))′ dt2(s)(x, y) d(m × m)(x, y) for 0 ≤ s ≤ t1 − t0. By proposition 3.27, for a.e. s ∈

[0, t1 − t0],dds

∫Ψt1−t0 (D1)×Br(γ(t0))′

dt2(s)(x, y) d(m × m)(x, y)

≤

∫U s

2

|∇h− + ∇dq|(Φs(y)) d(m × m)(x, y)

+

1∫0

∫U s

2

d(Ψs(x),Φs(y))|Hess h−|HS(γΨs(x),Φs(y)(τ)) d(m × m)(x, y) dτ.

(145)

For any s ∈ [0, t1 − t0], s′ ∈ [0, s], and (x, y) ∈ U s2,

1. d(x, y) < r + 5r16 by (139);

2. d(Ψ−s′(x), γ(t0 + s′)) = d(Ψt1−t0−s′(x′), γ(t0 + s′)) < 5r16 for some x′ ∈ D1 by (139) and (140);

3. dt2(s)(x, y) < r by definition of U s2 (144).

Hence,Φs′(y) ∈ B2r+ 5r

8(γ(t0 + s′)) (146)

by triangle inequality. Therefore, (Ψ−s′ ,Φs′)(U s2) ∈ B4r(γ(t0 + s′)) × B4r(γ(t0 + s′)) for all s′ ∈ [0, s]. For any

(x, y) ∈ U s2,

∆h−(Ψ−s′(x)) = ∆h+(Ψ−s′(x)) + ∆e(Ψ−s′(x))≥ −c(N, δ) , by Lemma 4.2 and Lemma 4.3 3,

(147)

where h+, e are heat flow approximations of h+0 and e0 respectively up to the same time as h−. Therefore,

1∫0

∫U s

2

d(Ψ−s(x),Φs(y))|Hess h−|HS(γΨ−s(x),Φs(y)(τ)) d(m × m)(x, y) dτ

≤ c(N, δ)

1∫0

∫(Ψ−s,Φs)(U s

2)

d(x, y)|Hess h−|HS(γx,y(τ)) d(m × m)(x, y) dτ, by 3.15 2, (147) and 3.5

≤ c(N, δ)rm(B4r(γ(t0 + s)))∫

B8r(γ(t0+s))

|Hess h−|HS dm , by segment inequality 3.22

≤ c(N, δ)rm(Br(γ(t1)))2?

B8r(γ(t0+s))

|Hess h−|HS dm , by Bishop-Gromov and property 1 of γ.

(148)

54

Integrating in s ∈ [0, t1 − t0],t1−t0∫0

( 1∫0

∫U s

2

d(Ψ−s(x),Φs(y))|Hess h−|HS(γΨ−s(x),Φs(y)(τ)) d(m × m)(x, y) dτ)

dt

≤ crm(Br(γ(t1)))2

t1−t0∫0

?B8r(γ(t0+s))

|Hess h−|HS dm ds

≤ c(N, δ)rm(Br(γ(t1)))2 √t1 − t0,

(149)

where the last line follows from the definition of h−, statement 4 of Theorem 4.12, and Cauchy-Schwarz.We have

t1−t0∫0

∫U s

2

|∇h+ + ∇dq|(Φs(y)) d(m × m)(x, y) ds ≤ c(N, δ)rm(Br(γ(t1)))2 √t1 − t0, (150)

where the first inequality is from statement 3 of Lemma 4.13 and the excess bound on the elements of Br(γ(t0))′

(142), and the second inequality is from property 1 of γ and Bishop-Gromov. Similarly, using statement 3 ofLemma 4.3 with (146),

t1−t0∫0

∫U s

2

|∇h− − ∇h+|(Φs(y)) d(m × m)(x, y) ds ≤ c(N, δ)rm(Br(γ(t1)))2(t1 − t0)

≤ crm(Br(γ(t1)))2 √t1 − t0.

(151)

Combining (148) - (150) with the bound (145) on dds

∫Ψt1−t0 (D1)×Br(γ(t0))′

dt2(s)(x, y), we obtain

∫Ψt1−t0 (D1)×Br(γ(t0))′

dt2(t1 − t0)(x, y) d(m × m)(x, y)

=

∫ t1−t0

0[

dds

∫Ψt1−t0 (D1)×Br(γ(t0))′

dt2(s)(x, y) d(m × m)(x, y)] ds

≤ c(N, δ)rm(Br(γ(t1)))2 √t1 − t0.

(152)

Both Ψt1−t0(D1) and Br(γ(t0))′ are non-trivial in measure compared to Br(γ(t1)) by (141) and property 1respectively. The same considerations as in the proof of Lemma 5.1 gives the existence of z′ ∈ Ψt1−t0(D1) andA2 ⊆ Br(γ(t0))′ so that

m(A2)m(Br(γ(t0)))

≥ 1 − V(1, 10) and dt2(t1 − t0)(z′, y) ≤3r8∀y ∈ A2, (153)

after we bound ε5 sufficiently small depending only on N and δ. The latter implies

Φs(A2) ⊆ B2r(γ(t0 + s)) ∀s ∈ [0, t1 − t0]. (154)

This finishes the proof of the lemma.

Lemmas 5.5 and 5.6 give

55

Corollary 5.7. For any δ ∈ (0, 0.1), there exists ε6(N, δ) > 0 and r6(N, δ) > 0 so that for any unit speed geodesicγ from p to q, there exists a unit speed geodesic γδ from p to q with with γδ ≡ γ on [1 − δ, 1] so that for allr ≤ r6 and δ ≤ t0 ≤ t1 ≤ 1 − δ, if t1 − t0 ≤ ε6, then

1. V(1, 100)4 ≤m(Br(γδ(t1)))m(Br(γδ(t0))) ≤

1V(1,100)4 ;

2. There exists A1 ⊆ Br(γδ(t1)) so that m(A1) ≥ (1 − V(1, 10))m(Br(γδ(t1))) and Ψs(A1) ⊆ B2r(γδ(t1 − s))for all s ∈ [0, t1 − t0];

3. There exists A2 ⊆ Br(γδ(t0)) so that m(A2) ≥ (1 − V(1, 10))m(Br(γδ(t0))) and Φs(A2) ⊆ B2r(γδ(t0 + s))for all s ∈ [0, t1 − t0].

We would like to now construct a geodesic that has this behaviour for all δ by taking a limit of γδ as δ → 0.To have the properties pass over to the limit, we need to make sure each γδ also satisfies the above propertiesfor any δ′ > δ. For all δ ∈ (0, 0.1), we fix the constants ε6(N, δ) and r6(N, δ) which come from taking theminimimum of their respective counterparts in lemmas 5.5 and 5.6.

Lemma 5.8. Let δ ∈ (0, 0.1) and assume some unit speed geodesic γδ from p to q satisfies properties 1 - 3 forδ, ε6(N, δ) and r6(N, δ). Then for any δ′ ∈ (δ, 0.1), γδ also satisfies properties 1 - 3 for δ′, ε6(N, δ′) and r6(N, δ′)

Proof. Fix δ′ ∈ (δ, 0.1). Use γδ to construct some γδ′

with the construction of Lemma 5.5. γδ′

satisfiesproperties 1 - 3 for δ′, ε6(N, δ′) and r6(N, δ′) by lemmas 5.5 and 5.6. Furthermore, γδ(t) = γδ

′

(t) for allt ∈ [1 − δ′, 1] by construction. We will show γδ(t) = γδ

′

(t) for all t ∈ [δ′, 1] which will allow us to conclude.Assume this is not the case. Define s0 := mins ∈ [δ′, 1 − δ′] : γδ(t) = γδ

′

(t)∀ t ∈ [s, 1]. By assumption,δ′ < s0 ≤ 1− δ′. Therefore, there exists t0 ∈ [δ′, s0) so that t1 − t0 ≤ minε6(N, δ), ε6(N, δ′) and γδ(t0) 6= γδ

′

(t0).

Choose any r < minr6(N, δ), r6(N, δ′), d(γδ(t0),γδ′(t0))

4 and consider Br(γδ(t1)). On one hand, most of Br(γδ(t1))needs to end up in B2r(γδ(t0)) under Ψt1−t0 by property 2 for γδ. On the other hand, most of Br(γδ(t1)) needsto end up in B2r(γδ

′

(t0)) under Ψt1−t0 by property 2 for γδ′

. These two balls are disjoint and so we have acontradiction.

This immediately gives the existence of a geodesic with the desired properties for any δ ∈ (0, 0.1).

Theorem 5.9. There exists a unit speed geodesic γ from p to q so that for any δ ∈ (0, 0.1), there existsε6(N, δ) > 0 and r6(N, δ) > 0 so that for all r ≤ r6 and δ ≤ t0 ≤ t1 ≤ 1 − δ, if t1 − t0 ≤ ε6, then

1. V(1, 100)4 ≤m(Br(γ(t1)))m(Br(γ(t0))) ≤

1V(1,100)4 ;

2. There exists A1 ⊆ Br(γ(t1)) so that m(A1) ≥ (1 − V(1, 10))m(Br(γ(t1))) and Ψs(A1) ⊆ B2r(γ(t1 − s)) forall s ∈ [0, t1 − t0];

3. There exists A2 ⊆ Br(γ(t0)) so that m(A2) ≥ (1 − V(1, 10))m(Br(γ(t0))) and Φs(A2) ⊆ B2r(γ(t0 + s)) forall s ∈ [0, t1 − t0].

Proof. Fix any unit speed geodesic γ from p to q and then use the construction of Lemma 5.5 to obtain a γδ

for each δ ∈ (0, 0.1). By Arzela-Ascoli theorem, we can take a limit γ of γδ as δ → 0 after passing to asubsequence. By Lemma 5.8 and Bishop-Gromov, γ will have the desired properties.

5.3. Proof of main theorem. We now prove the Holder continuity in pointed Gromov-Hausdorff distance ofsmall balls along the interior of any geodesic between p and q constructed in Theorem 5.9 using essentially thesame argument as in [CN12].

Theorem 5.10. There exists a unit speed geodesic γ between p and q so that for any δ ∈ (0, 0.1), there existsε(N, δ) > 0, r(N, δ) > 0 and C(N, δ) so that for any r ≤ r and t0, t1 ∈ [δ, 1 − δ], if |t1 − t0|≤ ε then

dpGH((Br(γ(t0)), γ(t0)), (Br(γ(t1)), γ(t1))

)≤ Cr|t1 − t0|

12N(1+2N) 1.

1We note that the Holder exponent is slightly different to that of [CN12] due to a minor error in the first line of equation (3.38).

56

Proof. Let γ, ε6 and r6 be as in Theorem 5.9 and let γ = γ. Fix δ ∈ (0, 0.1). We begin by assuming ε ≤ ε6and r ≤ r6 but will impose more bounds as the proof continues. Fix r ≤ r and δ ≤ t0 ≤ t1 ≤ 1 − δ withω := t1 − t0 ≤ ε.

We begin by showing that a large portion of Br(γ(t1)) is maped by Ψω close (on the scale of r) to γ(t0), wherethe closeness and the relative size of the portion are both Holder dependent on ω. This also shows that themeasure of Br(γ) is Holder along γ|[δ,1−δ] as a consequence.

Define η := ωN

2(1+2N) and µ := η1N = ω

12(1+2N) . By property 2 of γ from Theorem 5.9 and integral Abresch-

Gromoll, there exists Bµr(γ(t1))′ ⊆ Bµr(γ(t1)) so that

1. m(Bµr(γ(t1))′)m(Bµr(γ(t1))) ≤ 1 − 2V(1, 10);

2. Ψs(Bµr(γ(t1))′) ⊆ B2µr(γ(t1 − s)) ∀s ∈ [0, ω];3. e(x) ≤ c(N, δ)µ2r2 ≤ cr2 ∀x ∈ Bµr(γ(t1))′.

By integral Abresch-Gromoll, there exists Br(γ(t1))′ ⊆ Br(γ(t1)) so that

e(x) ≤ c(N, δ)1η

r2 ∀x ∈ Br(γ(t1))′ andm(Br(γ(t1))′)m(Br(γ(t1)))

≥ 1 − η. (155)

Fix h− ≡ h−ρ(4r)2 satisfying statement 4 of Theorem 4.12 for the balls of radius 4r along γ, where ρ ∈ [ 1

2 , 2].For all s ∈ [0, ω] and (x, y) ∈ X × X, define

dt(s)(x, y) := min4r, max

0≤τ≤s|d(x, y) − d(Ψτ(x),Ψτ(y))|

. (156)

Note for any (x, y) ∈ Bµr(γ(t1))′ × Br(γ(t1))′, Ψs(x),Ψs(y) ∈ B2r(γ(t1 − s)) and so dt(s)(x, y) < 4r.By Proposition 3.27, for a.e. s ∈ [0, ω],

dds

∫Bµr(γ(t1))′×Br(γ(t1))′

dt(s)(x, y) d(m × m)(x, y)

≤

∫Bµr(γ(t1))′×Br(γ(t1))′

(|∇h− − ∇dp|(Ψs(x)) + |∇h− − ∇dp|(Ψs(y))

)d(m × m)(x, y)

+

1∫0

∫Bµr(γ(t1))′×Br(γ(t1))′

d(Ψs(x),Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) d(m × m)(x, y) dτ.

(157)

We estimate1∫

0

∫Bµr(γ(t1))′×Br(γ(t1))′

d(Ψs(x),Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) d(m × m)(x, y) dτ

≤ c(N, δ)

1∫0

∫(Ψs,Ψs)(Bµr(γ(t1))′×Br(γ(t1))′)

d(x, y)|Hess h−|HS(γx,y(τ)) d(m × m)(x, y) dτ , by Theorem 3.15, 2

≤ c(N, δ)rm(B2r(γ(t1 − s)))∫

B4r(γ(t1−s))

|Hess h−|HS dm , by segment inequality 3.22

≤ c(N, δ)rm(Br(γ(t1)))2?

B4r(γ(t1−s))

|Hess h−|HS dm , by Bishop-Gromov and property 1 of γ.

(158)

57

Integrating in s ∈ [0, ω],ω∫

0

( 1∫0

∫Bµr(γ(t1))′×Br(γ(t1))′

d(Ψs(x),Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) d(m × m)(x, y) dτ)

ds

≤ crm(Br(γ(t1)))2

ω∫0

?B4r(γ(t1−s))

|Hess h−|HS dm ds

≤ c(N, δ)m(Br(γ(t1)))2 √ωr,

(159)

where the last line follows from the definition of h−, statement 4 of Theorem 4.12. and Cauchy-Schwarz.By statement 3 of 4.13 and the excess bound on the elements of Bµr(γ(t1))′,

ω∫0

∫Bµr(γ(t1))′×Br(γ(t1))′

|∇h− − ∇dp|(Ψs(x)) d(m × m)(x, y) ds ≤ c(N, δ)m(Br(γ(t1)))2 √ωr. (160)

Similarly by the excess bound on the elemnts of Br(γ(t1))′ (155),ω∫

0

∫Bµr(γ(t1))′×Br(γ(t1))′

|∇h− − ∇dp|(Ψs(y)) d(m × m)(x, y) ds

≤ c(N, δ)1√η

m(Br(γ(t1)))m(Bµr(γ(t1)))√ωr.

(161)

Combining (159) - (161) with (157) we immediately obtain∫Bµr(γ(t1))′×Br(γ(t1))′

dt(ω)(x, y) d(m × m)(x, y)

=

∫ ω

0[

dds

∫Bµr(γ(t1))′×Br(γ(t1))′

dt(s)(x, y) d(m × m)(x, y)] ds

≤ c(N, δ)m(Br(γ(t1)))(m(Br(γ(t1))) +

1√η

m(Bµr(γ(t1))))√ωr.

(162)

By Bishop-Gromov, m(Bµr(γ(t1)))m(Br(γ(t1))) ≥ C(N)µN = cη. Therefore, there exists z ∈ Bµr(γ(t1))′ so that∫

Br(γ(t1))′

dt(ω)(z, y) dm(y) ≤ c(N, δ)m(Br(γ(t1)))(1η

+1√η

)√ωr ≤ c(N, δ)m(Br(γ(t1)))

1η

√ωr.

This means there exists D ⊆ Br(γ(t1))′ so thatm(D)

m(Br(γ(t1)))≥ 1 − 2η and dt(ω)(z, y) ≤ c(N, δ)

1η2

√ωr ∀y ∈ D. (163)

Since ω = (η2µ)2, the latter combined with z ∈ Bµr(γ(t1))′ implies

Ψω(D) ⊆ B(1+c(N,δ)µ)r(γ(t0)). (164)

58

We have the following estimate on the volume of Br(γ(t1)) compared to volume of Br(γ(t0)) after possiblyconstraining ε further depending on N and δ.

m(Br(γ(t1)))m(Br(γ(t0)))

≤ (1 + cη)m(D)

m(Br(γ(t0))), by (163)

≤ (1 + cη)(1 + c(N, δ)ω)N m(Ψω(D))m(Br(γ(t0)))

, by Theorem 3.15, 2

≤ (1 + cη)(1 + cω)N(1 + c(N, δ)µ)N m(Ψω(D))m(B(1+cµ)r(γ(t0)))

, by Bishop-Gromov

≤ 1 + c(N, δ)µ = 1 + cω1

2(1+2N) .

(165)

Making the same calculation with Φ in the other direction as well, we obtain the following Holder estimate onvolume ∣∣∣∣m(Br(γ(t1)))

m(Br(γ(t0)))− 1

∣∣∣∣ ≤ c(N, δ)|t1 − t0|1

2(1+2N) . (166)

We now show the required Bishop-Gromov approximation can be constructed by using Ψω on a cµr-densesubset of Br(γ(t1)).

Fix representatitves for |Hess h−|HS and |∇h− − ∇dp|. Using the same calculation as before,ω∫

0

( 1∫0

∫D×D

d(Ψs(x),Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) d(m × m)(x, y) dτ)

ds

≤ c(N, δ)m(Br(γ(t1)))2 √ωr,

(167)

and so by Fubini’s theorem, there exists A ⊆ D so that1. m(A)

m(Br(γ(t1))) ≥ 1 − 3η;

2.ω∫

0

( 1∫0

∫D

d(Ψs(x),Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) dm(y) dτ)

ds ≤ c 1ηm(Br(γ(t1)))

√ωr for all x ∈ A.

For each x ∈ A, there exists Ax ⊆ D so that1. m(Ax)

m(Br(γ(t1))) ≥ 1 − 3η;

2.ω∫

0

( 1∫0

d(Ψs(x),Ψs(y))|Hess h−|HS(γΨs(x),Ψs(y)(τ)) dτ)

ds ≤ c 1η2

√ωr for all y ∈ Ax.

Since A and each Ax are contained in Br(γ(t1))′, their elements have an excess bound of c 1ηr2 by (155). By

statement 3 of 4.13, for m-a.e. x ∈ A,ω∫

0

|∇h− − ∇dp|(Ψs(x))ds ≤ c(N, δ)1√η

√ωr. (168)

Similarly, for each x ∈ A and m-a.e. y ∈ Ax,ω∫

0

|∇h− − ∇dp|(Ψs(y))ds ≤ c(N, δ)1√η

√ωr. (169)

By a Fubini’s theorem argument, it is clear that the inequality in Proposition 3.27 holds pointwise for (m×m)-a.e. (x, y). We first replace A with a full measure subset so that in addition the inequality in Proposition 3.27

59

holds for all x ∈ A and m-a.e. y ∈ Ax. We then replace each Ax with a full measure subset so that the sameinequality holds for all x ∈ A and all y ∈ Ax. Therefore, for all x ∈ A, y ∈ Ax,

dt(ω)(x, y) ≤ c(N, δ)( 1η2 +

1√η

)√ωr ≤ c(N, δ)

1η2

√ωr ≤ cµr.

For any x, y ∈ A, Ax ∩ Ay is c(N)η1n r-dense in Br(γ(t1)) and so there exists some z ∈ Ax ∩ Ay where

d(x, z) < cη1n r = cµr. Therefore,

|d(Ψω(x),Ψω(y)) − d(x, y))|≤ |d(Ψω(x),Ψω(y)) − d(Ψω(z),Ψω(y)|+|d(Ψω(z),Ψω(y)) − d(z, y)|+|d(z, y) − d(x, y)|≤ c(N, δ)µr.

(170)

Moreover, we have the following estimate on the volume of Ψω(A) ⊆ B(1+cµ)r(γ(t0)) after possibly constrainingε further depending on N and δ.

m(Ψω(A))m(B1+cµr(γ(t0)))

≥1

(1 + c(N, δ)ω)N

m(A)m(B(1+cµ)r(γ(t0)))

, by Theorem 3.15, 2

≥1

(1 + cω)N

1(1 + c(N, δ)µ)N

m(A)m(Br(γ(t0)))

, by Bishop-Gromov

≥1

(1 + cω)N

1(1 + cµ)N

11 + c(N, δ)µ

m(A)m(Br(γ(t1)))

, by (165)

≥1

(1 + cω)N

1(1 + cµ)N

11 + cµ

(1 − 3η) ≥ 1 − c(N, δ)µ.

(171)

To summarize, A ⊆ Br(γ(t1)) is so that1. Ψω(A) ⊆ B(1+c(N,δ)µ)r(γ(t0));2. ∀x, y ∈ A, |d(Ψω(x),Ψω(y)) − d(x, y)|≤ c(N, δ)µr;3. A is c(N)µr-dense in Br(γ(t1));4. Ψω(A) is c(N, δ)µ

1N r-dense in B(1+c(N,δ)µ)r(γ(t0)).

Moreover, there exists c(N) so that m(Bc(N)µr(γ(t1))) ≥ 2η1−V(1,10) m(Br(γ(t1))) by Bishop-Gromov. By property 2

of γ, there exists Bcµr(γ(t1))′ ⊆ Bcµr(γ(t1)) so that

m(Bcµr(γ(t1))′)m(Bcµr(γ(t1)))

≥ 1 − V(1, 10) and Ψω(Bcµr(γ(t1))′) ⊆ B2cµr(γ(t0)).

Therefore, Bc(N)µr(γ(t1))′ ∩ A is non-empty by measure considerations. In other words, there is an element in Awhich is c(N)µr close to γ(t1) and is mapped 2c(N)µr close to γ(t0) under Ψω.

These facts about A allow for the construction of a c(N, δ)µ1N r pointed Gromov-Hausdorff approximation

which finishes the proof.

Before we prove Theorem 1.1, we will first prove that RCD(K,N) spaces are non-branching in the nextsection using the construction we have developed so far. A corollary then follows which immediately givesTheorem 1.1.

6. Applications

6.1. Non-branching. In this subsection, we prove that RCD(K,N) spaces are non-branching. The use of theessentially non-branching property of RCD(K,N) spaces in the proof was pointed out to the author by VitaliKapovitch.

60

Proof of Theorem 1.3. Assume otherwise. By zooming in and cutting off geodesics if necessary, we may as-sume (X, d,m) is an RCD(−(N − 1),N) space for some N ∈ (1,∞) and we have two unit speed geodesicsγ1, γ2 : [0, 1]→ X with

1. γ1(0) = γ2(0) = p for p ∈ X;2. γ1(1) = q1 and γ2(1) = q2 for q1, q2 ∈ X;3. maxt ∈ [0, 1] : γ1(s) = γ2(s) ∀s ∈ [0, t] = 0.5.

Let p′ = γ1(0.5) and Ψs be as in (63) towards p.Since (X, 2d,m) is again an RCD(−(N −1),N) space and 2d(p, p′) = 1, we may apply Theorem 5.9 to obtain

a 2d-unit speed geodesic γ : [0, 1] → X between p and p′. Reparameterize γ to γ : [0, 0.5] → X so that γ is ad-unit speed geodesic.

Fix any δ ∈ (0, 0.1), use Corollary 5.7 to construct a unit speed geodesic γδ1 : [0, 1] → X from p to q1 withγδ1(t) = γ(t) for all t ∈ [0, δ]. Therefore, the proof of Theorem 5.10 passes for γδ1 for the same δ and in particularwe have the estimates (163) - (166) for γδ1. As a reminder, for δ ≤ s1 < s2 ≤ 1 − δ and sufficiently small r,

• (163) and (164) imply a portion of Br(γδ1(s2)) is sent to a ball of radius slightly larger than r aroundγδ1(s1) by Ψs2−s1 , where the relative size of the portion and the increase in radius on the scale of r canbe both made Holder dependent on s2 − s1 and go uniformly to 1 and 0 respectively as s2 − s1 → 0.• (166) implies the ratio between the measures of Br(γδ1(s1)) and Br(γδ1(s2)) is Holder dependent on s2−s1

and in particular goes uniformly to 1 as s2 − s1 → 0We show that γδ1(t) = γ(t) for all t ∈ [0, 0.5]. Suppose not, let t0 := maxt ∈ [0, 0.5] : γδ1(s) = γ(s) ∀s ∈

[0, t] and so t0 ∈ [δ, 0.5).We claim there exists t1 ∈ (t0, 0.5) and r > 0 so that for any r ≤ r, there exists A1 ⊆ Br(γδ1(t1)) and

A2 ⊆ Br(γ(t1)) so that1. γδ1(t1) 6= γ(t1).2. Ψt1−t0(A1) ⊆ Br(γδ1(t0)) and Ψt1−t0(A2) ⊆ Br(γ(t0));

3.m(Ψt1−t0 (A1))m(Br(γδ1(t0)))

> 12 and

m(Ψt1−t0 (A2))m(Br(γ(t0))) > 1

2 .

We can choose t1 arbitrarily close to t0 so that statement 1 holds by definition of t0. Statements 2 and 3 thenfollow from (163) - (166) for γδ1, the same for γ, Bishop-Gromov and statement 2 of Theorem 3.15 to controlthe volume distortion of Ψ, as soon as t1 is chosen close enough to t0. Choosing r ≤ minr, d(γδ1(t1), γ(t1))/4,it is straightforward to check using the triangle inequality that the two points from Br(γδ1(t1)) and Br(γ(t0))respectively cannot lie on the same geodesic towards p. As such, any point which lies in Ψt1−t0(A1)∩Ψt1−t0(A2)must be so that a geodesic from p to that point can be extended to two branching geodesics. It is knownby [CAV14, Proposition 4.5] that the subset of points x ∈ X where a geodesic from p to x can be extended totwo branching geodesics has measure 0 and so m(Ψt1−t0(A1) ∩ Ψt1−t0(A2)) = 0. We now have a contradictionwith properties 2 and 3 of A1 and A2.

Therefore, γδ1(t) = γ(t) for all t ∈ [0, 0.5]. Since this is true for all δ ∈ (0, 0.1), taking δ → 0 and usingArzela-Ascoli Theorem, after possibly passing to a subsequence, we obtain a geodesic γ1 satisfying Theorem5.9 with γ1 ≡ γ on [0, 0.5] and γ1(1) = q1. The same construction for γ2 gives γ2 satisfying Theorem 5.9 withγ2 ≡ γ on [0, 0.5] and γ2(1) = q2. Applying the previous argument again for γ1 and γ2 shows that they cannotsplit, which is a contradiction.

As a corollary, we have the following improvement of Theorem 5.10.

Corollary 6.1. Let (X, d,m) be an RCD(−(N − 1),N) space for some N ∈ (1,∞) and p, q ∈ X with d(p, q) = 1.For any δ ∈ (0, 0.1), there exists ε(N, δ) > 0, r(N, δ) > 0 and C(N, δ) so that for any unit speed geodesic γbetween p and q, r ≤ r, and t0, t1 ∈ [δ, 1 − δ], if |t1 − t0|≤ ε then

dpGH((Br(γ(t0)), γ(t0)), (Br(γ(t1)), γ(t1)))

)≤ Cr|t1 − t0|

12N(1+2N) .

61

Proof. Fix any s ∈ [0, 0.5]. Since (X, 2d,m) is again an RCD(−(N − 1),N) space and 2d(γ(s), γ(s + 0.5)) = 1,we may use Theorem 5.10 to construct some 2d-unit speed geodesic γs between γ(s) and γ(s + 0.5). Since X isnon-branching, γs and γ must coincide between γ(s) and γ(s + 0.5). Since this is true for all s ∈ [0, 0.5] and allγs have the Holder properties of Theorem 5.10 on (X, 2d,m). The same is true for γ on (X, d,m) with slightlyworse constants.

Theorems 1.1 now follows immediately by rescaling.

6.2. Dimension and weak convexity of the regular set. In this subsection we will extend to the RCD(K,N)setting the results of [CN12] on regular sets. All proofs translate directly from [CN12]. We mention again thatTheorem 6.2 has already been established using a new argument involving the Green’s function in [BS20].

Theorem 6.2. (Constancy of the dimension) Let (X, d,m) be an RCD(K,N) m.m.s. for some K ∈ R andN ∈ (1,∞). Assume X is not a point. There exists a unique n ∈ N, 1 ≤ n ≤ N so that m(X\Rn) = 0.

Proof. Let A1, A2 ⊆ X × X be the sets of (x, y) ∈ X × X so that geodesics from x to y are extendible past x and yrespectively. For each x ∈ X, let Ax be the set of y ∈ X so that geodesics from x to y are extendible past y. Usingthe arguments of [CAV14, Section 4], A1, A2 are (m × m)-measurable and Ax is m-measurable for all x ∈ X.m(X\Ax) = 0 for any x ∈ X by a standard argument using Bishop-Gromov. Let A := A1 ∩ A2, Fubini’s theoremthen gives (m × m)((X × X)\A) = 0.

Let γx,y : [0, 1] → X be a Borel selection (2.26) of constant speed geodesics from any x ∈ X to any y ∈ X.Since m(S) = 0, it follows from applying the segment inequality to the characteristic function of S that for(m × m)-a.e. (x, y) ∈ X × X, γx,y ∩ Rreg has full measure, and therefore is also dense, in [0, 1]. By Theorem1.1, for any geodesic γ and k, γ ∩ Rk is closed relative to the interior of γ. Combining these with the fact thatalmost every γx,y is extendible, we obtain for (m × m)-a.e. (x, y) ∈ X × X, there exists k ∈ N with 1 ≤ k ≤ Nso that γx,y ⊆ Rk. This leads to a contradiction if there are two regular sets of different dimension with positivemeasure.

Definition 6.3. (m-a.e. convexity) Let (X, d,m) be a m.m.s.. Let S be an m-measurable set in X. S is m-a.e.convex iff for (m × m) almost every pair (x, y) ∈ S × S , there exists a minimizing geodesic γ ⊆ S connecting xand y.

Definition 6.4. (weak convexity) Let (X, d) be a metric space. S ⊆ X is weakly convex iff for all (x, y) ∈ S × Sand ε > 0, and there exists an ε-geodesic (see Definition 4.5) γ ⊆ S connecting x and y.

Theorem 6.5. (m-a.e. and weak convexity of the regular set) Let Rn be as in Theorem 6.2, then1. Rn is m-a.e. convex;2. Rn is weakly convex.

In particular, Rn is connected.

Proof. Statement 1 is contained in the proof of Theorem 6.2. The proof of statement 2 follows verbatimfrom [CN12, Theorem 1.20].

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