MEASURE RIGIDITY OF RICCI CURVATURE LOWER BOUNDS 5 1 FABIOCAVALLETTIANDANDREAMONDINO 0 2 p Abstract. The measure contraction property, MCP for short, is a weak Ricci curvature lower e bound conditions for metric measure spaces. The goal of this paper is to understand which S structural properties such assumption (or even weaker modifications) implies on the measure, onitssupportandonthegeodesics ofthespace. 2 WestartourinvestigationfromtheeuclideancasebyprovingthatifapositiveRadonmeasure 2 m over Rd is such that (Rd,|·|,m) verifies a weaker variant of MCP, then its support spt(m) must be convex and m has to be absolutely continuous with respect to the relevant Hausdorff ] measureofspt(m). ThisresultisthenusedasastartingpointtoinvestigatetherigidityofMCP G inthemetricframework. M We introduce the new notion of reference measure for a metric space and prove that if (X,d,m) is essentially non-branching and verifies MCP, and µ is an essentially non-branching . h MCPreferencemeasurefor(spt(m),d),thenmisabsolutelycontinuouswithrespecttoµ,onthe t setofpointswhereaninversionplanexists. Asaconsequence,anessentiallynon-branchingMCP a reference measure enjoys a weak type of uniqueness, up to densities. We also prove a stability m property for reference measures under measured Gromov-Hausdorff convergence, provided an [ additionaluniformboundholds. Inthefinalpartwepresentconcreteexamplesofmetricspaceswithreferencemeasures,both 2 in smooth and non-smooth setting. The main example will be the Hausdorff measure over an v Alexandrovspace. Thenweprovethatthefollowingarereferencemeasuresoversmoothspaces: 8 the volume measure of a Riemannianmanifold, the Hausdorffmeasure of an Alexandrov space 3 withboundedcurvature,andtheHaarmeasureofthesubRiemannianHeisenberggroup. 3 3 0 . 1 1. Introduction 0 5 The notion of curvature for a smooth space, i.e. a Riemannian n-dimensional manifold (M,g), 1 is one of most basic geometrical concepts and goes back to the work of Gauss and Riemann; the : v idea being to consider suitable combinations of second derivatives of the metric g. If the space M i is non-smooththis approachhas few chancesto be carriedout, so one has to understandwhatare X thefundamentalgeometricconsequencesofthecurvatureandthensettheseasdefiningproperties. r a ThiswastheapproachbyAlexandrovwho,inlastcentury,definedwhatitmeansforametricspace (X,d) to satisfy an upper or lower bound on the sectional curvature. Such non-smooth spaces are nowcalledAlexandrovspacesandthereisahugeliteratureabouttheirproperties(wedonoteven attempttogiveaselectionofpapers;wejustrecallthatAlexandrov,GromovandPerelmanamong othersgavemajorcontributionstothistheoryandwerefertothetextbook[8]formorereferences). While,asjustrecalled,thenotionoflowerboundonthesectionalcurvaturemakesperfectsense for a metric space (X,d), i.e. a set endowed with a distance function, for defining what it means for a non-smooth space to satisfy a lower Ricci curvature bound one also has to fix a measure m on (X,d), thus getting a so called metric measure space (X,d,m), m.m.s. for short. In the last ten years there has been a flourishing of literature about different notions and properties of lower Riccicurvatureboundsform.m.s.: ononehandBakry-E´mery-Ledoux[6,7]developedthesocalled Gamma-calculus based on the fact that, roughly speaking, the Bochner inequality characterizes 1 2 FABIOCAVALLETTIANDANDREAMONDINO lower Ricci bounds; on the other hand there is a parallel theory via optimal transport based on the idea that one can detect Ricci curvature by examining how the volume changes when opti- mally transported along geodesics. A third way to study the non-smooth spaces with lower Ricci bounds, having its origins in the work of Gromov and mainly due to Cheeger-Colding [13, 14, 15] (see also the more recent developments by Colding-Naber [16]), is to concentrate the attention on those m.m.s. arising as limits (in the measured Gromov-Hausdorff sense) of smooth Riemannian manifolds satisfying Ricci curvature lower bounds; such a point of view is very powerful if one is interested in the structure of these limit spaces. In this paper we will focus on the secondapproachvia optimal transportation,but let us mention thatthereisaprecisecorrespondencebetweenthefirsttwo(see[2]fortheinfinitedimensionalcase and [18, 4] for the finite dimensional one) and there are some recent results on the local structure of such spaces even in the abstract framework (see [22, 23, 30]). TheweakestRiccicurvatureconditionforam.m.s. isthesocalledmeasurecontractionproperty, MCP for short, which keeps track of the distortion of the volume of a set when it is transported to a Dirac delta. The quantitative formalization of having Ricci bounded from below by K R and dimension bounded from above by N > 1 via this approach, the so called MCP(K,N∈) condition, is due to Ohta [34] and has its roots in earlier works by Grigor’yan [24], Sturm [36] and Kuwae-Shioya[27]. Such condition is compatible with the smooth counterpart (i.e. a smooth Riemannian manifold of dimension N has Ricci curvature bounded below by K if and only if it satisfies MCP(K,N)) and is stable under pointed measured Gromov-Hausdorff convergence, pmGH for short, so that the limit spaces of Cheeger-Colding are included in this theory. Also finitedimensionalAlexandrovspaceswithlowercurvatureboundsendowedwiththecorresponding Hausdorff measure satisfy the measure contraction property [34] (see also [28] for a subsequent independent proof) as well as the Heisenberg group endowed with the Haar measure and the Carnot-Carath´eodorydistance [25]. Stronger notions of “Ricci bounded from below by K R and dimension bounded above by ∈ N (1, ]” for m.m.s. are the so called CD(K,N)-conditions introduced independently by Lott- Vil∈lani [2∞9] and by Sturm [37, 38], and the evenstronger notions of RCD(K, ) and RCD∗(K,N)- ∞ spaces (the first were introduced in [3] and further investigated in [1, 2], for the second ones see [4, 18, 22, 23, 30, 39]). All these classes include the smooth Riemannian manifolds satisfying the corresponding curvature-dimension bounds and their pmGH-limits, as well as finite dimensional Alexandrov spaces with curvature bounded from below. On the other hand let us recall that the Heisenberg group does not satisfy any CD(K,N) condition but it does verify MCP, see [25]. So MCP is a strictly weaker notion of curvature than CD. The goal of the present paper is to investigate the structural properties (in particular on the measure, on its support, and on the geodesics) forced by MCP. In order to introduce the problem let us start by analyzing the behavior of the euclidean space Rd, endowed with the d-dimensional Lebesgue measure d and the Euclidean distance . It is almost trivial to check that the triple (Rd, , d) verifieLs the measure contraction prop|er·t|y with |·| L curvature K = 0 and dimension N = d, i.e. MCP(0,d). Moreover it is well known that, at the price of changingthe lowerbound on the curvatureand the upper bound onthe dimension, in the triple (Rd, , d) it is possible to replace the Lebesgue measure with some equivalent measure |·| L and still obtain MCP, provided the density verifies some concavity estimates. A first goal of this work is to investigate, and give affermative answer to, the reverse question: MEASURE RIGIDITY OF RICCI CURVATURE LOWER BOUNDS 3 Let m be a positive Radon measure on Rd such that the triple (Rd, ,m) verifies MCP • |·| (or evena weakercondition). Canwe deduce that m is absolutelycontinuous with respect to k, for some k d? L ≤ Moreover,does the support of m have some nice geometric properties? • The next theorem contains an affirmative answer to these natural questions; before stating it let usremarkthatthenon-degeneracy condition forapositiveRadonmeasuremisgiveninDefinition 3.1 and it is ensured by the measure contraction property (it is indeed much weaker since no quantitative or uniform lower bound on the transported measure is assumed). Theorem 1.1. Let m be a positive Radon measure over Rd and denote with Ω its support. If the metric measure space (Ω, ,m) verifies the non-degeneracy condition then there exists a natural |·| number k d so that: ≤ Ω is convex and contained in a k-dimensional affine subspace of Rd, say Vk; • the measure m is absolutely continuous with respect to kxVk. • L Thesamequestioncanbereformulatedinthemetricframeworkonlyoncethechoiceofafavorite measure is done. More precisely, given a m.m.s. (X,d,µ) satisfying MCP and maybe some other structural assumption, one can ask: Is the support of µ convex? • If m is a positive Radon measure on X also satisfying MCP, can we deduce that m is • absolutely continuous with respect to µ? This kind of properties were proved by Cheeger-Colding [15] in the framework of pmGH-limits of Riemannian manifolds satisfying lower Ricci curvature bounds: more precisely they showed that in the limit space the “favorite measure” is the Hausdorff measure of the relevant dimension, and anyotherpossible limit measurehasto be absolutelycontinuouswith respectto it. Thereforethis paper can be seen as an intrinsic-non smooth analogue of the Cheeger-Colding result. Regardingthefirstquestion,inProposition4.5,wewillprovetheaffirmativeanswer: thesupport ofany Radonmeasuresatisfying MCP has to be weaklyconvex(more precisely the supportofany measure satisfying the strong qualitative MCP condition, defined in (4.3), must be weakly convex; i.e. every couple of points of spt(m) is joined by a geodesic entirely contained in spt(m)). Inordertosolvethesecondproblem,wewillgiveaprecisemeaningofwhatisforusa“favorite measure”: this is what we call reference measure (see Definition 5.1) for a complete and separable metric space (X,d). The crucial property being to behave nicely under geodesic inversion with respect to almost every point of the space. Definition 1.2 (Reference measure). A positive Radon measure µ +(X) is a reference mea- ∈M sure for (X,d) provided it is non-zero, and for µ-a.e. z X there exists πz +(X X) so ∈ ∈ M × that (P ) πz =µ, πz(X X H(z))=0, (P ) πz µ, 1 ♯ 2 ♯ × \ ≪ where P :X X X is the projection on the i-th component, for i=1,2 and i × → H(z):= (x,y) X X : d(x,y)=d(x,z)+d(z,y) . { ∈ × } The measure πz will be called inversion plan and those points where an inversion plan exists will be called inversion points and denoted by Ip(µ). Let us stress that this concept is completely new, to best of our knowledge. In Remark 5.2 we will also point out that existence of an inversion plan in a point z is closely related to the regularityofthe ambientspace (X,d) atz. Inparticular if a conicalsingularity happens in z then no inversion plan at z exists. 4 FABIOCAVALLETTIANDANDREAMONDINO Another ingredient in the next theorem will be the essentially non-branching condition (re- called in Section 2.1): it is an important structural assumption on a m.m.s. and it is fulfilled by a large class of geometrically relevant examples, for instance Riemannian manifolds, Alexan- drovspaces with lowercurvaturebounds, pmGH-limits ofRiemannianmanifolds with lowerRicci curvature bounds, metric measure spaces verifying RCD(K, ) or RCD∗(K,N), the Heisenberg ∞ Group endowed with the Carnot-Carath´eodory metric, etc.. Moreover, since in the proof of the next result we do not need the quantitative controls assumed in MCP(K,N), we just assume a weaker qualitative MCP condition (for the precise notion see Definition 4.1). Theorem 1.3. Let (X,d,m) be an essentially non-branching m.m.s. that verifies the qualitative MCP condition (4.1). Assumethe existenceof a reference measureµ for (Ω,d), where Ω=spt(m), so that (Ω,d,µ) verifies the qualitative MCP condition (4.1) and it is essentially non-branching. If m(Ω Ip(µ))=0, then \ m µ. ≪ In particular, if also m is a reference measure and spt(m)=spt(µ) with µ(Ω Ip(m))=0, then µ \ and m are equivalent measures, i.e. µ m and m µ. ≪ ≪ Inotherwords,oncethesupportisfixed,areferencemeasuresatisfyingMCPisuniquelydetermined up to densities. Once observed that metric measure spaces verifying RCD are all essentially non-branching, Theorem 1.3 can be used straightforwardly to obtain the next uniqueness result. Theorem 1.4. Let (X,d,µ) be a m.m.s. that verifies RCD∗(K,N), with N < and X =spt(µ). Assume that µ is a reference measure for (X,d). If (X,d,m) is a m.m.s. verify∞ing RCD∗(K′,N′), possibly for different K′ and N′, with X =spt(m) and m(X Ip(µ))=0, then \ m µ. ≪ In particular, if also m is a reference measure with µ(X Ip(m))=0, then µ and m are equivalent \ measures, i.e. µ m and m µ. ≪ ≪ ThesecondpartofTheorem1.4thereforesaysthatoncethesupportX andametricdarefixed, there is a unique reference measure, up to densities, so that the triple (X,d,µ) verifies RCD∗. BothTheorem1.3andTheorem1.4canberestateddroppingtheassumptionofm(Ω Ip(µ))=0 \ (and the analogous one in the uniqueness part) at the price of assuming regularity properties on the geometry of Ip(µ) (and on Ip(m)): one way (see Theorem 5.5) is to assume that Ip(µ) is µ- connected. FortheprecisemeaningofthisassumptionseeDefinition5.4,hereweonlymentionthat it is inspired by, and resemble, the convexity property of regular points in Alexandrov geometry. Another way is to assume that Ip(µ) forms an open set in X, see Theorem 5.6. Section 6 is devoted to the investigation of general properties of reference measures: we will establish the local-to-global property of reference measures under the non-branching assumption, we will show that the multiplication by a density does not affect the property of being a reference measure and finally we will address the problem of stability of reference measures with respect to measured Gromov-Hausdorff convergence: it will be proved that if the reference measures are probabilitiesandifauniformboundonthesecondmarginaloftheinversionplanholds,thenotion of reference measure is stable. See Theorem 6.6 for the precise statement and Theorem 6.9 for its application to uniqueness of limit reference measures. This last result should be compared with the aforementioned analogous statement for Ricci limits spaces obtained by Cheeger-Colding [15]. MEASURE RIGIDITY OF RICCI CURVATURE LOWER BOUNDS 5 The final part of the paper, that is formed by Section 7 and Section 8, is devoted to discuss examples and applications of the introduced techniques. The main result will be the existence of an inversion plan at almost every regular point of an Alexandrov space with curvature bounded from below, where the almost everywhere and the inversionplanarereferredtothe Hausdorffmeasureofthe rightdimension,seeTheorem7.3. This shows that the Hausdorff measure is a reference measure and permits to apply all the general theorems stated so far to Alexandrov spaces. Inthelastsectionwethenpresentsmoothexamples. Thetermsmoothhereisusedtoemphasize thatinthosespaceseverypointadmitsaninversionplan. Thesmoothexampleswillbe: the stan- dardvolumemeasureonasmoothRiemannianmanifold,theHausdorffmeasureonanAlexandrov spacewithcurvatureboundedfromaboveandbelow(see Corollary8.3)andthe Heisenberggroup endowed with the Carnot-Carath´eodorydistance and the Haar measure (see Corollary 8.9). As a final comment we want to stress that, while the proof of the existence of an inversion plan in a smooth context relies mainly on the bi-Lipschitz regularityof the exponentialmap, such ingredientonageneralAlexandrovspacefailstohold. Henceafineranalysisisneeded. Combining the Lipschitz regularityofthe exponentialmaptogetherwith the factthat itmaps sets ofpositive measuretosetsofpositivemeasure,usingalsoDisintegrationTheorem,onecanconstructa“local” inversion plan. Suitably iterating this construction one can obtain a “global” inversion plan and prove Theorem 7.3. The existence of an inversion plan can be therefore understood as a measure- theoretic reformulation of a bi-Lipschitz control on the behavior of geodesic hinges and, as for a general Alexandrov space not so much can be proved on the regularity of the exponential map, Theorem 7.3 seems to be a novelty also in this direction. We end the introduction with some comments regarding possible future applications of the techniques here introduced. In the class of those metric measure spaces admitting a bi-Lipschitz map with a subset • of a Euclidean space, one of the relevant questions to ask is whether the image of such a metric space has positive Lebesgue measure or not. To this aim, let (X,d,µ) satisfy MCP, φ:X Rd be a bi-Lipschitz map for some positive d and call η :=(φ) µ. Proving that η ♯ → verifiesthe non-degeneracycondition(see Definition 3.1)wouldimply, thanks to Theorem 1.1, that d(φ(X))>0; yielding the rectifiability property of (X,d,µ). IntherecLent[17],CsornyeiandJonesprovedthatifaRadonmeasureµoverRd makesthe • triple (Rd, ,µ) a Lipschitz differentiability space (see [12] for the definition of Lipschitz |·| differentiability space), then µ is absolutely continuous with respect to the d-dimensional Lebesgue measure d. Then Theorem 1.1 could be compared with [17]. In particular a L relevantquestioncouldbe tocheckifa metricmeasurespaceverifyingthe non-degeneracy condition is also a Lipschitz differentiability space. Understandingandcharacterizingthesingularpartofanonsmoothspaceis animportant • andchallengingissue. Asexplainedabovethesetofinversionpointsisstronglylinkedwith theregularset,i.e. thecomplementaryofthesingularset. Itwouldbeinterestingtofurther investigate this link for instance in Alexandrov, or more generally in RCD∗(K,N)-spaces. GivenanRCD∗(K,N)-space (X,d,m), it is aninteresting open problemwhether the mea- • sure m is absolutely continuous with respect to the Hausdorff measure of the relevant dimension. We believe that the techniques here introduced could be useful to attach this question which would be the natural generalization to RCD∗(K,N)-spaces to the afore- mentionedresultofCheeger-Colding[15]intheframeworkofpmGH-limitsofRiemannian manifolds satisfying lower Ricci curvature bounds. 6 FABIOCAVALLETTIANDANDREAMONDINO InSection6 we addressthe problemof stability ofa referencemeasurewith respectto the • pmGH-convergence. As a next step one would like to study measured tangent spaces to pointed m.m.s. endowed with a reference measure. Recall that the set Tan(X,d,µ,x¯) of tangent spaces to (X,d,µ) at x¯ is the collection of all pmGH limits of sequences (X,1/r d,µri,x¯) where µri is a properly rescaled version of µ. i· x¯ x¯ Acknowledgment. A.M. acknowledges the support of the ETH Fellowship. The project started during the conference YEP XI at EURANDOM (Eindhoven, The Netherlands) and continued during the “ERC Research Period on Calculus of Variations and Analysis in Metric Spaces” at ScuolaNormale Superiore(Pisa-Italy),the authorswishto thank the institutions for the excellent working conditions and the organizersof the events for the stimulating atmosphere. Many thanks also to Luigi Ambrosio and Tapio Rajala for carefully reading a preliminary version of the paper, to Anton Petrunin for a fruitful conversation on Alexandrov spaces and to the two anonymous referees for the valuable comments and suggestions. 2. Setting We now recall some terminology and general notation. In what follows we say that a triple (X,d,m) is a metric measure space, m.m.s. for short, if and only if: (X,d) is a complete and separable metric space; • the measure m belongs to +(X), • M where +(X)denotesthespaceofpositiveRadonmeasureoverX. Wewillalsosometimesassume M m to be a probability measure, that is m(X) = 1, but this will be specified at the beginning of each section. Moreover a metric space is a geodesic space if and only if for each x,y X there ∈ exists γ Geo so that γ = x,γ = y. Here we are using the following notation for the space of 0 1 ∈ geodesics: Geo:= γ C([0,1],X):d(γ ,γ )=(s t)d(γ ,γ ),s,t [0,1] . s t 0 1 { ∈ − ∈ } The metric ball is denoted with B (o) := z X : d(z,o)<r . Recall that for complete geodesic r { ∈ } spaceslocalcompactnessisequivalenttoproperness(ametricspaceisproperifeveryclosedballis compact). As we will alwaysdealwith spacesenjoying some type ofmeasure contractionproperty implying an upper bound on the dimension of the space, we directly assume the ambient space (X,d)to be proper. Hence fromnowonwe assume the following: the ambientmetric space(X,d) is geodesic, complete, separable and proper. We will also use quite often the set of geodesics ending in a fixed point of the space: for z X ∈ Geo(z):= γ Geo:γ =z =Geo e−1(z), { ∈ 1 } ∩ 1 where for any t [0,1], e denotes the evaluation map: t ∈ e :Geo X, e (γ):=γ . t t t → A geodesic metric space (X,d) is said to be non-branching if and only if for any γ1,γ2 Geo, it ∈ holds: γ1 =γ2, γ1 =γ2, t¯ (0,1) = γ1 =γ2, s [0,1]. 0 0 t¯ t¯ ∈ ⇒ s s ∀ ∈ We denotewith (X)thespaceofprobabilitymeasureswithfinite secondmomentendowedwith 2 P the L2-Wasserstein distance W defined as follows: for µ ,µ (X) we set 2 0 1 2 ∈P (2.1) W2(µ ,µ )=inf d2(x,y)π(dxdy), 2 0 1 π ZX MEASURE RIGIDITY OF RICCI CURVATURE LOWER BOUNDS 7 where the infimum is taken over all π (X X) with µ and µ as the first and the second 0 1 ∈ P × marginal. Assuming the space (X,d) to be geodesic, also the space ( (X),W ) is geodesic. It 2 2 P turns out that any geodesic (µ ) in ( (X),W ) can be lifted to a measure ν (Geo), so t t∈[0,1] 2 2 P ∈ P that (e ) ν = µ for all t [0,1]. Given µ ,µ (X), we denote by OptGeo(µ ,µ ) the space t ♯ t 0 1 2 0 1 ∈ ∈ P of all ν (Geo) for which (e ,e ) ν realizesthe minimum in (2.1). If (X,d) is geodesic,then the 0 1 ♯ ∈P set OptGeo(µ ,µ ) is non-empty for any µ ,µ (X). 0 1 0 1 2 ∈P It is also convenient to introduce the following closed sets: H := (x,y,z) X3 :d(x,y)=d(x,z)+d(z,y) , ∈ (2.2) H(z) := P(cid:8)12(H (X2 z ))= (x,y) X2 : d(x(cid:9),y)=d(x,z)+d(z,y) , ∩ ×{ } { ∈ } where for any n N, P :Xn X2 is the projection on the ij-th component, for i,j =1,...,n. ij ∈ → 2.1. The essential non-branching property. We recall the following definition. Definition2.1. Ametricmeasurespace(X,d,m)isessentiallynon-branching ifandonlyifforany µ ,µ (X) which are absolutely continuous with respect to m any element of OptGeo(µ ,µ ) 0 1 2 0 1 ∈P is concentrated on a set of non-branching geodesics. If (X,d,m) is essentially non-branching, one can deduce some information on the branching points as follows. Consider the set (2.3) D := (z,w) X X :γ1,γ2 Geo,γ1 =γ2 =w, γ1 =γ2 =z,γ1 =γ2 , { ∈ × ∈ 0 0 1 1 6 } and observe that thanks to the properness assumption, it is σ-compact: it can be written as countable union of compact sets. Observe that also X2 D is σ-compact and D is symmetric: if \ (z,w) D also (w,z) D. From essential non-branching we can deduce that the σ-compact set ∈ ∈ D(z):=P (D ( z X)), 2 ∩ { }× has m-measure zero (see for instance [39], in any case this statement will follow by the arguments below which give a more detailed description of the branching set). We need a more refined property of branching structures that can be obtained observing that any L2 optimal transportation between any measure and a Dirac delta, say in z X, is also an ∈ L1-optimaltransportation. Whatfollowsiscontainedin[10]andisprovedforRCDspacesbutthe same result holds for essentially non-branching spaces. Consider the following closed sets: Γ:= (x,y,z) X3 :d(x,z) d(y,z)=d(x,y) , Γ−1 := (x,y,z) X3 :(y,x,z) Γ , ∈ − ∈ ∈ and R:=(cid:8) Γ Γ−1. We want to analyze possible b(cid:9)ranching struct(cid:8)ures inside X X. We ther(cid:9)efore ∪ × consider the set of forward branching: (2.4) A := (x,z) X X : y,w X,(x,y,z),(x,w,z) Γ, (y,w,z) / R , + { ∈ × ∃ ∈ ∈ ∈ } and the set of backwardbranching: (2.5) A := (x,z) X X : y,w X,(x,y,z),(x,w,z) Γ−1, (y,w,z) / R . − ∈ × ∃ ∈ ∈ ∈ As the space X is(cid:8)proper by assumption, it is easily seen that both A+ and A− ar(cid:9)e σ-compact sets. The information we will use is the following: define A (z):=P ((X z ) A ), A (z):=P ((X z ) A ), + 1 + − 1 − ×{ } ∩ ×{ } ∩ then it follows from Proposition 4.5 of [10] (notice that what is denoted with in [10], in the T present framework coincides with the whole X X) that × (2.6) m(A (z))=m(A (z))=0, + − 8 FABIOCAVALLETTIANDANDREAMONDINO provided (X,d,m) is essentially non-branching. In particular the σ-compact set (z):=X (A (z) A (z)), nb + − T \ ∪ contains all the points, up to a set of m-measure zero, moved via optimal transportation towards the end pointz andthe whole set (z) is coveredby a family of non-branchinggeodesics. It will nb T also convenient to introduce the set :=X X (A A ) and to observe that trivially nb + − T × \ ∪ (z)=P ((X z ) ). nb 1 nb T ×{ } ∩T 2.2. Disintegration of measures. We include here aversionofthe DisintegrationTheorem(for a comprehensive treatment see for instance [19]). Givenameasurablespace(R,R),i.e. R isaσ-algebraofsubsetsofR,andafunctionr :R S, with S general set, we can endow S with the push forward σ-algebra S of R: → Q S r−1(Q) R, ∈ ⇐⇒ ∈ which could be also defined as the biggest σ-algebra on S such that r is measurable. Moreover givenaprobabilitymeasureρon(R,R),defineaprobabilitymeasureη on(S,S)bypushforward via r, i.e. η :=r ρ. ♯ Definition 2.2. A disintegration of ρ consistent with r is a map (with slight abuse of notation still denoted with) ρ:R S [0,1] such that, set ρ (B):=ρ(B,s), the following hold: s × → (1) ρ () is a probability measure on (R,R) for all s S, s (2) ρ(B· ) is η-measurable for all B R, ∈ · ∈ and satisfies for all B R,C S the consistency condition ∈ ∈ ρ B r−1(C) = ρ (B)η(ds). s ∩ ZC (cid:0) (cid:1) A disintegration is strongly consistent with respect to r if for all s we have ρ (r−1(s)) = 1. The s measures ρ are called conditional probabilities. s We recall the following version of the disintegration theorem that can be found in [19, Section 452]. Recall that a σ-algebra is countably generated if there exists a countable family of sets so J that coincide with the smallest σ-algebra containing them. J Theorem 2.3 (Disintegration of measures). Assume that (R,R,ρ) is a countably generated pro- bability space and R= R is a partition of R. Denote with r :R S the quotient map: s∈S s ∪ → s=r(x) x R , s ⇐⇒ ∈ and with (S,S,η) the quotient measure space. Assume (S,S)=(X, (X)) with X Polish space, B where (X) denotes the Borel σ-algebra. Then there exists a unique strongly consistent disinte- B gration s ρ w.r.t. r, where uniqueness is understood in the following sense: if ρ ,ρ are two s 1 2 7→ consistent disintegrations then ρ ()=ρ () for η-a.e. s S. 1,s 2,s · · ∈ MEASURE RIGIDITY OF RICCI CURVATURE LOWER BOUNDS 9 3. Non-degenerate reference measure on Rd Let m be a positive Radon measure over Rd that we will consider to be equipped with the Euclidean distance. Denote with Ω the support of m: Ω:=spt(m). Definition 3.1 (Non-Degenerate measure). Consider the m.m.s. (Ω, ,m). We say that the |·| measure m is non-degenerate if and only if for any compact set A Ω of positive m-measure it ⊂ holds (3.1) m(A )>0, t [0,1), t,o ∀ ∈ where o is any element of Ω and (3.2) A := z Rd :z =(1 t)y+to, y A . t,o { ∈ − ∈ } Remark 3.2. The nondegeneracyconditionmaybe regardedasa veryweakcurvature condition on the m.m.s. (Ω, ,m). Indeed it is implied by MCP(K,n), for any K and n. |·| For the rest of this section we tacitly assume that m is a non-degenerate measure. We start by proving a geometric property of Ω. Lemma 3.3. The set Ω is a convex subset of Rd. Proof. Consider x,y spt(m). Then for any positive radius δ it holds m(B (x) Ω) > 0. Then δ ∈ ∩ (3.1) implies that m((B¯ (x) Ω) )>0, t [0,1). δ t,y ∩ ∀ ∈ Now observe that for any t [0,1), ∈ (B¯ (x) Ω) B (x ), with x :=(1 t)x+ty. δ t,y (1−t)δ t t ∩ ⊂ − It follows that m(B (x )) > 0 for any positive δ, therefore x spt(m) = Ω and the claim (1−t)δ t t follows. ∈ (cid:3) We deduce from Lemma 3.3 that Ω has a well defined dimension: there exists a minimal k N with k d so that Ω is contained in a k-dimensional affine subspace of Rd that we identify w∈ith Rk. ≤ Wecanthereforeconsiderthe relativeinteriorandthe relativeboundaryofΩ inRk: thereexist U,C Rk so that Ω=U C and U C = with U maximal open subset in Rk contained in Ω. ⊂ ∪ ∩ ∅ Proposition 3.4. If k is the dimension of Ω, the measure mx is absolutely continuous with U respect to the k-dimensional Lebesgue measure k. L Proof. Assume by contradiction the existence of a compact set K U so that ⊂ m(K)>0, k(K)=0. L The statement we are proving is local, therefore we can consider x K and δ > 0 so that 0 ∈ B (x ) U and K B (x ). Since the measure m is non-degenerate in (Ω, ,m), for every δ 0 δ 0 ⊂ ⊂ |·| 10 FABIOCAVALLETTIANDANDREAMONDINO o ∂B (x ) and t [0,1] it holds m(K )>0. Therefore we have: δ 0 t,o ∈ ∈ 0 < m(K ) k−1(do) 1(dt) t,o H L Z[0,1]Z∂Bδ(x0) = m k−1 1 (z,o,t) B (x ) ∂B [0,1):z =(1 t)x+to, x K ⊗H ⊗L { ∈ δ 0 × δ(x0)× − ∈ } = k−1x (cid:0) 1x ( (o,t):f (o,t) K )m(dz) (cid:1) H ∂Bδ(x0)⊗L [0,1) { z ∈ } ZBδ(x0) (3.3) = (f ) k−1x 1x (K) m(dz), z ♯ H ∂Bδ(x0)⊗L [0,1) ZBδ(x0) (cid:16) (cid:17) where z to f (o,t):= − , t [0,1). z 1 t ∀ ∈ − Since k−1x 1x is equivalent to kx , the function f can be defined directly on H ∂Bδ(x0)⊗L [0,1) L Bδ(x0) z B (x ). For ease of notation we also assume x =0, then: δ 0 0 z w z w f (w):= − =δ − . z 1 w/δ · δ w −| | −| | Then T 1 δ df (w) =δ(z w) Id z − ∇ δ w − δ w (cid:18) −| |(cid:19) −| | T 1 w δ =δ(z w) Id. − (δ w)2 w − δ w −| | (cid:18)| |(cid:19) −| | Using the following Lemma 3.5, we get (δ w) 1 w T det −| | df (w) =det Id (z w) z (cid:18)− δ (cid:19) − δ−|w| − (cid:18)|w|(cid:19) ! 1 =1 z w,w . − (δ w)w h − i −| | | | Thus det (δ−|w|)df (w) = 0 only when δ w = z,w . Since z,w B (0), it follows that − δ z | | h i ∈ δ det(df (w)(cid:16))=0forallwan(cid:17)dz. Therefore, k(K)=0implies(f ) k−1x 1x (K)= z 6 L z ♯ H ∂Bδ(x0)⊗L [0,1) 0 contradicting (3.3) and the claim is proved. (cid:16) (cid:17) (cid:3) Lemma 3.5. Let Id be the n n identity matrix and let a,b be n 1 column vectors. Then × × det(Id+abT)=1+bT a. Proof. Observe that one can write Id 0 Id+abT a Id 0 Id a = bT 1 · 0 1 · bT 1 0 1+bT a (cid:20) (cid:21) (cid:20) (cid:21) (cid:20)− (cid:21) (cid:20) (cid:21) The proof follows then by the standard product rules of the determinant and of matrices in block forms. (cid:3) To conclude we analyze the boundary. Putting together the next Proposition and the previous results, Theorem 1.1 is proved. Proposition 3.6. If C is the boundary of U for the trace topology of Rk, then m(C)=0.