Ricci Curvature on Alexandrov spaces and Rigidity Theorems
aa r X i v : . [ m a t h . DG ] S e p RICCI CURVATURE ON ALEXANDROV SPACES AND RIGIDITYTHEOREMS
HUI-CHUN ZHANG AND XI-PING ZHU
Abstract.
In this paper, we introduce a new notion for lower bounds of Ricci curvature onAlexandrov spaces, and extend Cheeger–Gromoll splitting theorem and Cheng’s maximaldiameter theorem to Alexandrov spaces under this Ricci curvature condition. Introduction
Alexandrov spaces with curvature bounded below generalize successfully the concept oflower bounds of sectional curvature from Riemannian manifolds to singular spaces. Theseminal paper [BGP] and the 10th chapter in the text book [BBI] provide excellent in-troductions to this field. Many important theorems in Riemannian geometry had beenextended to Alexandrov spaces, such as Synge’s theorem [Pet1], diameter sphere theorem[Per1], Toponogov splitting theorem [Mi], etc.However, many fundamental results in Riemannian geometry (for example, Bishop–Gromov volume comparison theorem, Cheeger–Gromoll splitting theorem and Cheng’s max-imal diameter theorem) assume only the lower bounds on
Ricci curvature, not on sectionalcurvature. Therefore, it is a very interesting question how to generalize the concept of lowerbounds of Ricci curvature from Riemannian manifolds to singular spaces.Perhaps the first concept of lower bounds of Ricci curvature on singular spaces was givenby Cheeger and Colding (see Appendix 2 in [CC2.I]). They, in [CC1, CC2], studied Gromov–Hausdorff limit spaces of Riemannian manifolds with Ricci curvature (uniformly) boundedbelow. Among other results in [CC1], they proved the following rigidity theorem:
Theorem 1.1. (Cheeger–Colding)Let M i be a sequence of Riemannian manifolds and M i converges to X in sense ofGromov–Hausdorff.(1) If X contains a line and Ric ( M i ) > − ǫ i with ǫ i → , then X is isometric to a directproduct R × Y over some length space Y .(2) If Ric ( M i ) > n − and diameter of M i diam ( M i ) → π, then X is isometric to aspherical suspension [0 , π ] × sin Y over some length space Y . In [Pet4], Petrunin considered to generalize the lower bounds of Ricci curvature for sin-gular spaces via subharmonic functions.Recently, in terms of L − Wasserstein space and optimal mass transportation, Sturm [S1,S2] and Lott–Villani [LV1, LV2] have given a generalization of “Ricci curvature has lowerbounds” for metric measure spaces , independently. They call that curvature-dimensionconditions, denoted by CD ( n, k ) with n ∈ (1 , ∞ ] and k ∈ R . For the convenience of readers,we repeat their definition of CD ( n, k ) in the Appendix of this paper. On the other hand,Sturm in [S2] and Ohta in [O1] introduced another definition of “Ricci curvature boundedbelow” for metric measure spaces, the measure contraction property M CP ( n, k ), which is A metric measure space is a metric space equipped a Borel measure. a slight modification of a property introduced earlier by Sturm in [S3] and in a similar formby Kuwae and Shioya in [KS3, KS4]. The condition
M CP ( n, k ) is indeed an infinitesimalversion of the Bishop–Gromov relative volume comparison condition. For a metric measurespace, Sturm [S2] proved that CD ( n, k ) implies M CP ( n, k ) provided it is non-branching .Note that any Alexandrov space with curvature bounded below is non-branching. Recently,Petrunin [Pet2] proved that any n -dimensional Alexandrov space with curvature > CD ( n,
0) and claimed the general statement that the condition curvature > k (forsome k ∈ R ) implies the condition CD ( n, ( n − k ) can be also proved along the same lines.Let M be a Riemannian manifold with Riemannian distance d and Riemannian volume vol . Lott, Villani in [LV1] and von Renesse, Sturm in [RS, S4] proved that ( M, d, vol )satisfies CD ( ∞ , k ) if and only if Ric ( M ) > k . Indeed, they proved a stronger weightedversion (see Theorem 7.3 in [LV1] and Theorem 1.1 in [RS], Theorem 1.3 in [S4]). Let φ be a smooth function on M with R M e − φ dvol = 1. Lott and Villani in [LV2] proved that( M, d, e − φ · vol ) satisfies CD ( n, k ) if and only if weighted Ricci curvature Ric n ( M ) > k (seeDefinition 4.20– the definition of Ric n – and Theorem 4.22 in [LV2]). A similar result wasproved by Sturm in [S2] (see Theorem 1.7 in [S2]). In particular, they proved that ( M, d, vol )satisfies CD ( n, k ) if and only if Ric ( M ) > k and dim( M ) n. If dim( M ) = n, Ohta in[O1] and Sturm in [S2] proved, independently, that M satisfies M CP ( n, k ) is equivalent to Ric ( M ) > k .Nevertheless, since n -dimensional norm spaces ( V n , k · k p ) satisfy CD ( n,
0) for every p > CD ( n,
0) for general metric measure spaces. Furthermore, it was shown byOhta in [O3] that on a Finsler manifolds M , the curvature-dimension condition CD ( n, k ) isequivalent to the weighted Finsler Ricci curvature condition Ric n ( M ) > k (see also [O4] or[OSt], refer to [O4] for the definition Ric n in Finsler manifolds). That says, the curvature-dimension condition is somewhat a Finsler geometry character. Seemly, it is difficult to showthe rigidity theorems, such as Cheng’s maximal diameter theorem and Obata’s theorem,under CD ( n, n −
1) for general metric measure spaces.As a compensation, Watanabe [W] proved that if a metric measure space M satisfies CD ( n,
0) or
M CP ( n,
0) then M has at most two ends. Ohta [O2] proved that a non-branching compact metric measure space with M CP ( n, n −
1) and diameter = π is home-omorphic to a spherical suspension.Alexandrov spaces with curvature bounded below have richer geometric information thangeneral metric measure spaces. In particular, a finite dimensional norm space with curvaturebounded below must be an inner-product space. Naturally, one would expect that Cheeger–Gromoll splitting theorem still holds on Alexandrov spaces with suitable nonnegative “Riccicurvature condition”.Recently in [KS1], Kuwae and Shioya proved the following topological splitting theoremfor Alexandrov spaces under the M CP ( n,
0) condition:
Theorem 1.2. (Kuwae–Shioya)Let M n be an n -dimensional Alexandrov space. Assume that M n contains a line.(1) If M satisfies M CP ( n, , then M n is homeomorphic to a direct product space R × Y over some topological space Y .(2) If the singular set of M n is closed and the non-singular set is an (incomplete) C ∞ Riemannian manifold of
Ric > , then M n is isometric to a direct product space R × Y over some Alexandrov space Y . A geodesic space is called non-branching if for any quadruple points z, x , x , x with z being the midpointof x and x as well as the midpoint of x and x , it follows that x = x . ICCI CURVATURE ON ALEXANDROV SPACES 3
We remark that Kuwae and Shioya actually obtained a more general weighted measureversion of the above theorem in [KS2].In the following, inspired by Petrunin’s second variation of arc length [Pet1], we willintroduce a new notion of the Ricci curvature bounded below for Alexandrov spaces.Let M be an n − dimensioal Alexandrov space of curvature bounded from below locally without boundary. It is well known in [PP] or [Pet3] that, for any p ∈ M and ξ ∈ Σ p ,there exists a quasi-geodesic starting at p along direction ξ. (See [PP] or section 5 in [Pet3]for the definition and properties of quasi-geodesics.) According to [Pet1], the exponentialmap exp p : T p → M is defined as follows. For any v ∈ T p , exp p ( v ) is a point on somequasi-geodesic of length | v | starting point p along v/ | v | ∈ Σ p . If the quasi-geodesic is notunique, we take one of them as the definition of exp p ( v ) . Let γ : [0 , ℓ ) → M be a geodesic. Without loss of generality, we may assume that aneighborhood U γ of γ has curvature > k for some k < T γ ( t ) at an interior point γ ( t ) ( t ∈ (0 , ℓ ))can be split into a direct metric product. We denote L γ ( t ) = { ξ ∈ T γ ( t ) | ∠ ( ξ, γ + ( t )) = ∠ ( ξ, γ − ( t )) = π/ } , Λ γ ( t ) = { ξ ∈ Σ γ ( t ) | ∠ ( ξ, γ + ( t )) = ∠ ( ξ, γ − ( t )) = π/ } . In [Pet1], Petrunin proved the following second variation formula of arc-length.
Proposition 1.3. (Petrunin)Given any two points q , q ∈ γ , which are not end points, and any positive numbersequence { ε j } ∞ j =1 with ε j → , there exists a subsequence { e ε j } ⊂ { ε j } and an isometry T : L q → L q such that | exp q ( e ε j u ) , exp q ( e ε j T v ) | | q q | + | uv | | q q | · e ε j − k · | q q | · (cid:0) | u | + | v | + h u, v i (cid:1) · e ε j + o ( e ε j ) for any u, v ∈ L q . We remark that for a 2 − dimensional Alexandrov space, Cao, Dai and Mei in [CDM]improved the second variation formula such that the above inequality holds for all { ε j } ∞ j =1 . But for higher dimensions, to the best of our knowledge, we don’t know whether the paralleltranslation T in the above second variation formula can be chosen independent of thesequences { ε j } . Based on this second variation formula, we can propose a condition which resembles thelower bounds for the radial curvature along the geodesic γ .Let { g γ ( t ) }
Theorem 1.7. (Splitting theorem)Let M be an n -dimensional complete non-compact Alexandrov space with nonnegativeRicci curvature and ∂M = ∅ . If M contains a line, then M is isometric to a direct metricproduct R × N for some Alexandrov space N with nonnegative Ricci curvature. Theorem 1.8. (Maximal diameter theorem)Let M be an n -dimensional compact Alexandrov space with Ricci curvature bounded belowby n − and ∂M = ∅ . If the diameter of M is π , then M is isometric to a sphericalsuspension over an Alexandrov space with curvature > . ICCI CURVATURE ON ALEXANDROV SPACES 5
An open question for the curvature-dimension condition CD ( n, k )( k = 0) is “from localto global” (See, for example, the 30th chapter in [V]). In particular, given a metric measurespace which admits a covering and satisfies CD ( n, k ) ( k = 0), we don’t know if the coveringspace with pullback metric still satisfies CD ( n, k ).One advantage of our definition of the Ricci curvature bounded below on Alexandrovspaces is that the definition is purely local. In particular, any covering space of an n -dimensional Alexandrov space with Ricci curvature bounded below by ( n − K still satisfiesthe condition Ric > ( n − K. Meanwhile, we note that Bishop–Gromov volume comparisontheorem also holds on an Alexandrov space with Ricci curvature bounded below (see Corol-lary A.3 in Appendix). Consequently, the same proofs as in Riemannian manifold case (see[A] and, for example, page 275-276 in [P]) give the following estimates on the fundamentalgroup and the first Betti number.
Corollary 1.9.
Let M be a compact n -dimensional Alexandrov space with nonnegativeRicci curvature and ∂M = ∅ . Then its fundamental group has a finite index Bieberbachsubgroup. Corollary 1.10.
Let M be an n -dimensional Alexandrov space with nonnegative Riccicurvature and ∂M = ∅ . Then any finitely generated subgroup of π ( M ) has polynomialgrowth of degree n . If some finitely generated subgroup of π ( M ) has polynomial growthof degree = n , then M is compact and flat. Corollary 1.11.
Let M be an n -dimensional Alexandrov space with ∂M = ∅ .(1) If Ric ( M ) > ( n − K > , then its fundamental group is finite.(2) If Ric ( M ) > ( n − K and diameter of M D , then b ( M ) C ( n, K · D ) for some function C ( n, K · D ) .Moreover, there exists a constants κ ( n ) > such that if K · D > − κ ( n ) , then b ( M ) n. The paper is organized as follows. In Section 2, we recall some necessary materials forAlexandrov spaces. In Section 3, we will define a new representation of Laplacian alonga geodesic and will prove the comparison theorem for the newly-defined representation ofLaplacian (see Theorem 3.3). In Section 4, we will discuss the rigidity part of the comparisontheorem. The maximal diameter theorem and the splitting theorem will be proved in Section5 and 6, respectively. In the appendix, we give a modification of Petrunin’s proof in [Pet2] toshow that the condition on Ricci curvature bounded below implies the curvature-dimensioncondition (see Proposition A.2).
Acknowledgements
We would like to thank Dr. Qintao Deng for helpful discussions.We are also grateful to the referee for helpful comments on the second variation formula.The second author is partially supported by NSFC 10831008 and NKBRPC 2006CB805905.2.
Preliminaries
A metric space ( X, |· , ·| ) is called a length space if for any two point p, q ∈ X , the distancebetween p and q is given by | pq | = inf γ,γ connect p,q Length ( γ ) . A length space X is called a geodesic space if for any two point p, q ∈ X , there exists acurve γ connecting p and q such that Length ( γ ) = | pq | . Such a curve is called a shortestcurve. A geodesic is a unit-speed shortest curve. HUI-CHUN ZHANG AND XI-PING ZHU
Recall that a length space X has curvature > k in an open set U ⊂ X if for any quadruple( p ; a, b, c ) ⊂ U , there holds e ∠ k apb + e ∠ k bpc + e ∠ k cpa π, where e ∠ k apb, e ∠ k bpc, and e ∠ k cpa are the comparison angles in the k − plane. A length space M is called an Alexandrov space with curvature bounded from below locally (for short, wesay M to be an Alexandrov space ), if it is locally compact and any point in M has an openneighborhood U ⊂ M such that M has curvature > k U in U , for some k U ∈ R .Let M be an Alexandrov space without boundary and U ⊂ M be an open set. A locallyLipschitz function u on U is said to be λ − concave on U if for any geodesic γ ⊂ U , theone-variable function u ◦ γ ( t ) − λt / u on M is said to be semi-concave if for any point x ∈ M there is aneighborhood U x ∋ x and a real number λ x such that the restriction u | U x is λ x -concave.Let ψ : R → R be a continuous function. A function u on M called ψ ( u ) − concave if for any point x ∈ M and any ε > U x ∋ x such that u | U x is( ψ ◦ u ( x ) + ε )-concave.If M has curvature > k in U , then it is well-known that the function u = ̺ k ◦ dist p is(1 − ku ) − concave in U \{ p } , where ̺ k ( υ ) = k (cid:0) − cos( √ kυ ) (cid:1) if k > , υ if k = 0 , k (cid:0) cosh( √− kυ ) − (cid:1) if k < , (see, for example, Section 1 in [Pet3]).Let u be a semi-concave function on M . For any point p ∈ M , there exists a u − gradientcurve starting at p . Hence u generates a gradient flow Φ tu : M → M , which is a locallyLipschitz map. (Actually, it is just a semi-flow, because backward flow Φ − tu is not alwayswell-defined.) Particularly, if u is concave, the gradient flow is a 1-Lipschitz map. We referto Section 1 and 2 in [Pet3] for the details on semi-concave functions, gradient curves andgradient flows. 3. Laplacian comparison theorem
Let M be an n -dimensional Alexandrov space without boundary. A canonical Dirichletform E is defined by E ( u, v ) := Z M h∇ u, ∇ v i d vol , for u, v ∈ W , ( M ) . (see [KMS]). The Laplacian associated to the canonical Dirichlet form is given as follows.Let u : U ⊂ M → R be a λ − concave function. The (canonical) Lapliacian of u as asign-Radon measure is defined by Z M φd ∆ u = −E ( u, φ ) = − Z M h∇ φ, ∇ u i d volfor all Lipschitz function φ with compact support in U. In [Pet2], Petrunin proved∆ u nλ · vol , in particular, the singular part of ∆ u is non-positive. If M has curvature > K , then anydistance function dist p ( x ) := d ( p, x ) is cot K ◦ dist p − concave on M \{ p } , where the function ICCI CURVATURE ON ALEXANDROV SPACES 7 cot K ( s ) is defined by cot K ( s ) = √ K · cos( √ Ks )sin( √ Ks ) if K > , s if K = 0 , √− K · cosh( √− Ks )sinh( √− Ks ) if K < . It is a solution of the ordinary differential equation χ ′ ( s ) = − K − χ ( s ). Therefore the aboveinequality ∆ u nλ · vol gives a Laplacian comparison theorem for the distance function onAlexandrov spaces.In [KS1], by using the DC − structure (see [Per2]), Kuwae–Shioya defined a distributionalLaplacian for a distance function dist p by∆ dist p = D i (cid:0)q det ( g ij ) g ij ∂ j dist p (cid:1) on a local chart of M \S ǫ for sufficiently small positive number ǫ , where S ǫ := { x ∈ M : vol(Σ x ) vol( S n − ) − ǫ } and D i is the distributional derivative. Note that the union of all S ǫ has zero measure.One can view the distributional Laplacian ∆ dist p as a sign-Radon measure. In [KMS],Kuwae, Machigashira and Shioya proved that the distributional Laplacian is actually arepresentation of the previous (canonical) Laplacian on M \S ǫ . Moreover in [KS1], Kuwaeand Shioya extended the Laplacian comparison theorem under the weaker condition BG ( k ).Both of the above canonical Laplacian and its DC representation (i.e. the distributionalLaplacian) make sense up to a set which has zero measure.In Riemannian geometry, according to Calabi, the Laplacian comparison theorem holdsin barrier sense, not just in distribution sense. In this section, we will try to give a newrepresentation of the above canonical Laplacian of a distance function, which makes sensein W p , the set of points z ∈ M such that the geodesic pz can be extend beyond z . Wewill also prove a comparison theorem for the new representation under our Ricci curvaturecondition.Let M denote an n -dimensional complete Alexandrov space without boundary. Fix ageodesic γ : [0 , ℓ ) → M with γ (0) = p and denote f = dist p . Let x ∈ γ \{ p } and L x , Λ x be as above in Section 1. Clearly, we may assume that M has curvature > k (for some k <
0) in a neighborhood U γ of γ .Perelman in [Per2] defined a Hessian for a semi-concave function u on almost all point x ∈ M , denoted by Hess x u . It is a bi-linear form on T x (= R n ). But for the given geodesic γ , we can not insure that the Hessian is well defined along γ .We now define a version of Hessian and Laplacian for the distance function f along thegeodesic γ as follows. Note that the tangent space at an interior point x ∈ γ can be split to L x × R and f ◦ γ is linear. So we only need to define the Hessian on the set of orthogonaldirections Λ x .Throughout this paper, S will always denote the set of all sequences { θ j } ∞ j =1 with θ j → j → ∞ and θ j +1 θ j . Definition . Let x ∈ γ \{ p } . Given a sequence θ := { θ j } ∞ j =1 ∈ S , we define a function H θx f : Λ x → R by H θx f ( ξ ) def = lim sup s → , s ∈ θ f ◦ exp x ( s · ξ ) − f ( x ) s / HUI-CHUN ZHANG AND XI-PING ZHU and ∆ θ f ( x ) def = ( n − · I Λ x H θx f ( ξ ) . Since U γ has curvature > k , we know that f is cot k ( | px | ) − concave and dist γ ( ℓ ) iscot k ( | xγ ( ℓ ) | ) − concave near x , which imply(3.1) H θx f cot k ( | px | )for any sequence θ ∈ S , and | γ ( ℓ ) exp x ( s · ξ ) | | xγ ( ℓ ) | + cot k ( | xγ ( ℓ ) | ) · s / o ( s )for any ξ ∈ Λ x . Then by triangle inequality, we have(3.2) H θx f > − cot k ( | xγ ( ℓ ) | ) . Thus H θx f is well defined and bounded. It is easy to see that H θx f is measurable on Λ x andthus it is integrable.If there exists Perelman’s Hessian of f at a point x (see [Per2]), then H θx f ( ξ ) = Hess x f ( ξ, ξ )for all ξ ∈ Λ x and θ ∈ S .Denote by Reg f the set of points z ∈ M such that there exists Perelman’s Hessian of f at z . If we write the Lebesgue decomposition of the canonical Laplacian ∆ f = (∆ f ) sing +(∆ f ) ac · vol, with respect to the n -dimension Hausdorff measure vol, then (∆ f ) ac ( x ) =Tr Hess x f = ∆ θ f ( x ) for all x ∈ W p ∩ Reg f and θ ∈ S . It was shown in [OS, Per2] that Reg f ∩ W p has full measure in M . Thus ∆ θ f ( x ) is actually a representation of the absolutelycontinuous part of the canonical Laplacian ∆ f on W p .Note from the definition that if θ ⊂ θ , then H θ x f H θ x f and ∆ θ f ( x ) ∆ θ f ( x ) . The following lemma is a discrete version of the propagation equation of the Hessian of f along the geodesic γ . Lemma 3.2.
Let f = dist p . Given ǫ > , a continuous functions family { g γ ( t ) }
0. By definition, we have(3.5) f (exp y ( δ ′ j · lη )) f ( y ) + ( lδ ′ j ) H δy f ( η ) + o ( δ ′ j ) . Note that(3.6) f ( z ) − f ( y ) = | yz | ICCI CURVATURE ON ALEXANDROV SPACES 9 and(3.7) f (exp z ( δ ′ j ξ )) − f (exp y ( δ ′ j · lη )) | exp z ( δ ′ j ξ ) , exp y ( δ ′ j · lη ) | . By combining (3.4)–(3.7) and using (1.1) with l = l, l = 1, we have δ ′ j (cid:0) H δz f ( ξ ) − l · H δy f ( η ) (cid:1) + o ( δ ′ j ) δ ′ j · (cid:16) ( l − | yz | − g r ( ξ ) − ǫ · | yz | · ( l + l + 1) (cid:17) + o ( δ ′ j ) , for any l >
0. Hence H δz f ( ξ ) − l · H δy f ( η ) ( l − | yz | − l + l + 13 · | yz | · ( g z − ǫ ) . This completes the proof of the lemma. (cid:3)
The following result is the comparison for the above defined representation of Laplacian.
Theorem 3.3.
Let f = dist p and x ∈ γ \{ p } . If M has Ricci > ( n − K along the geodesic γ ( t ) , then, given any sequence { θ j } ∞ j =1 ∈ S , there exists a subsequence δ = { δ j } of { θ j } suchthat ∆ δ f ( x ) ( n − · cot K ( | px | ) . (If K > , we add assumption | px | < π/ √ K ).Proof. Arbitrarily fix two constants ǫ > K ′ < K with 10 ǫ < K − K ′ . We can choose a point y ∈ px such that | py | > ǫ and(3.8) cot k ( | py | ) cot K ′ ( | py | − ǫ ) . By our definition of Ricci curvature > ( n − K along γ , there exists a continuous functionfamily { g γ ( t ) }
For all 0 a N − , we have I Λ xa H δx a f cot K ′ ( | px a | − ǫ ) , as ω is sufficiently small.We will prove the claim by induction argument with respect to a .Firstly, we know from (3.8) that the case a = 0 is held.Set q = x a , r = x a +1 , µ = | x a x a +1 | and T = T a +1 . Now we suppose that the claim isheld for the case a , i.e., I Λ q H δq f cot K ′ ( | pq | − ǫ ) . We need to show the claim is also held for the case a + 1 . Consider the functions on Λ r (3.9) F l ( ξ ) = l · H δq f ( T ( ξ )) + ( l − µ − l + l + 13 · µ · (cid:0) g r ( ξ ) − ǫ (cid:1) . ¿From Lemma 3.2 above, we have(3.10) H δr f F l for any l > . On the other hand, from (3.9), I Λ r F l = l · I Λ r H δq f ◦ T + ( l − µ − l + l + 13 · µ · (cid:0) I Λ r g r ( ξ ) − ǫ (cid:1) l · (cid:0) cot K ( | pq | − ǫ ) (cid:1) + ( l − µ − l + l + 13 · µ · ¯ K (3.11)for any l > , where ¯ K = K − ǫ .By setting C = max | py | t | px | | cot ′′ K ′ ( t − ǫ ) | , we have(3.12) cot K ′ ( | pq | − ǫ ) cot K ′ ( | pr | − ǫ ) + µ (cid:0) K ′ + cot K ′ ( | pq | − ǫ ) (cid:1) + C µ . Thus by combining (3.11) and (3.12), we get(3.13) I Λ r F l cot K ′ ( | pr | − ǫ ) + A µ ( l ) , where A µ ( l ) = µ (cid:0) K ′ + cot K ′ ( | pq | − ǫ ) (cid:1) + C µ + ( l −
1) cot K ′ ( | pq | − ǫ )+ ( l − µ − l + l + 13 · µ · ¯ K. Denote by B = 1 /µ − µ ¯ K/ K ′ ( | pq | − ǫ ) . Note thatcot K ′ ( | px | − ǫ ) cot cot K ′ ( | py | − ǫ ) . ICCI CURVATURE ON ALEXANDROV SPACES 11
Since ω is small and µ ω , we can assume that cot + B >
0. Choose e l = − ( B + µ ¯ K/ / (cot + B ) . Then we get A µ ( e l ) = − (cid:0) B + µ ¯ K/ (cid:1) + (cid:16) µ (cid:0) K ′ + cot (cid:1) + C µ − cot + B (cid:17) · (cid:0) cot + B (cid:1) cot + B K ′ − ¯ K + C µ + C µ cot + B , where C , C are positive constants independent of µ, ω (may depending on ǫ, K ′ , x and y ).Using µ ω , we get A µ ( e l ) K ′ − ¯ K + C ω + C ω cot + B ω is sufficiently small. Hence, by combining (3.10), (3.13) and A µ ( e l )
0, we get I Λ r H δr f I Λ r F ( e l ) cot K ′ ( | pr | − ǫ ) . This completes the proof of the claim. In particular, we have I Λ x H δx f cot K ′ ( | px | − ǫ ) . Thus by the arbitrariness of ǫ and K ′ and a standard diagonal argument, we obtain asubsequence of δ , denoted again by δ , such that∆ δ f ( x ) ( n − · cot K ( | px | ) . Therefore, we have completed the proof of the theorem. (cid:3) Rigidity estimates
We continue to consider an n -dimensional complete Alexandrov space M without bound-ary. Fix a geodesic γ : [0 , ℓ ) → M with γ (0) = p and denote f = dist p .Let x ∈ γ \{ p } and L x , Λ x be as above. We still assume that a neighborhood U γ of γ hascurvature > k (for some constant k < Lemma 4.1.
Assume M has Ricci > ( n − K along the geodesic γ ( t ) . Let f = dist p and x be an interior point on the geodesic γ ( t ) . Given a sequence θ = { θ j } ∞ j =1 ∈ S , if (4.1) ∆ θ ′ f ( x ) = ( n − · cot K ( | px | ) for any subsequence θ ′ = { θ ′ j } of θ , then there exists a subsequence δ = { δ j } of θ such that (4.2) H δx f ( ξ ) = cot K ( | px | ) almost everywhere ξ ∈ Λ x .(If K > , we add assumption | px | < π/ √ K ).Proof. At first, we will prove the following claim:
Claim:
For any ǫ >
0, we can find a subsequence { δ j } of θ and an integrable function h onΛ x such that H δx f h and I Λ x (cid:0) h − cot K ( | px | ) (cid:1) (cid:0) | cot K ( | px | ) | (cid:1) ǫ. By our definition of Ricci curvature > ( n − K along γ , there exists a continuous functionfamily { g γ ( t ) }
0, we get (cid:16) /µ − µ ( g x − ǫ ) / H δz f ◦ T (cid:17) · h − ( g x − ǫ ) + H δz f ◦ T · (cid:16) /µ − µ ( g x − ǫ ) / (cid:17) + (cid:0) µ ( g x − ǫ ) (cid:1) / . That is,(4.9) (cid:16) /µ − D (cid:17) · ( h − H δz f ◦ T ) − ( g x − ǫ ) − h + (cid:0) µ ( g x − ǫ ) (cid:1) / , where D = µ ( g x − ǫ ) / − h .Denote that C = max | D | = max | h + µ ( g x − ǫ ) / | , which is independent of µ. Thus weget I Λ x ǫ − g x /u − D = I Λ x ( ǫ − g x ) + /u − D − I Λ x ( ǫ − g x ) − /u − D H Λ x ( ǫ − g x ) + /u − C − H Λ x ( ǫ − g x ) − /u + C = 1 /µ H Λ x ( ǫ − g x ) + C H Λ x | g x − ǫ | /µ − C . (4.10)By (4.4), (4.10) and the Ricci curvature condition that H Λ x g x > K − ǫ , we have(4.11) I Λ x ǫ − g x /u − D µ (2 ǫ − K ) + C µ , where constant C is independent on µ .¿From (4.9) and (4.4), we get(4.12) I Λ x h − I Λ x H δz f ◦ T µ (2 ǫ − K ) + C µ − H Λ x h /µ + C + ( C + ǫ ) µ /µ − C . By combining (4.8), (4.12) and noting that T is an isometry, we have I Λ x h cot K ( | px | ) + 2 ǫ + C µ, where constant C is independent on µ . Therefore,(4.13) I Λ x h cot K ( | px | ) + 3 ǫ as µ suffices small.Note that (4.12) implies I Λ x h I Λ x H δz f ◦ T + C µ, where constant C is independent on µ . Using (4.5) and noting that T is an isometry, wehave I Λ x h cot K ( | pz | ) + C µ cot K ( | px | ) + µ (cid:0) K + cot K ( | pz | ) (cid:1) + C µ. Since | px | / < | pz | < | px | , we have I Λ x h cot K ( | px | ) + C µ, where constant C is independent on µ . Thus, when µ is sufficiently small, we get(4.14) I Λ x h cot K ( | px | ) + ǫ. By combining (4.7) and (4.14), we obtain(4.15) cot K ( | px | ) · I Λ x h > cot K ( | px | ) − ǫ · | cot K ( | px | ) | . Hence, by (4.13) and (4.15), we have I Λ x (cid:0) h − cot K ( | px | ) (cid:1) (cid:0) | cot K ( | px | ) | (cid:1) · ǫ. This completes the proof of the claim.Now let us continue the proof of the lemma.Given any ǫ >
0, the above claim implies that the measure ν (cid:0) { ξ ∈ Λ x : H δx f > cot K + ǫ } (cid:1) ν (cid:0) { ξ ∈ Λ x : (cid:12)(cid:12) h − cot K ( | px | ) (cid:12)(cid:12) > ǫ } (cid:1) (cid:0) | cot K ( | px | ) | (cid:1) ǫ/ǫ . Letting ǫ → + , by a standard diagonal argument, we can obtain a subsequence of δ , stilldenoted by δ , such that ν (cid:0) { ξ ∈ Λ x : H δx f > cot K + ǫ } (cid:1) = 0 . By the arbitrariness of ǫ , after a further diagonal argument, we obtain a subsequence of δ ,denoted by δ again, such that ν (cid:0) { ξ ∈ Λ x : H δx f > cot K } (cid:1) = 0 . Thus we have H δx f cot K ( | px | )almost everywhere in Λ x .Finally , by combining (4.1) and the definition of ∆ δ f , we conclude that H δx f = cot K ( | px | )almost everywhere in Λ x . Therefore we have completed the proof of the lemma. (cid:3) In order to deal with the zero-measure set in the above Lemma, we need the following segment inequality of Cheeger and Colding [CC1]. See also [R] for a statement that isstronger than the following proposition.
ICCI CURVATURE ON ALEXANDROV SPACES 15
Proposition 4.2. (Segment inequality)Let M be an n -dimensional Alexandrov space with curvature > k , ( for some constant k < ). Let A , A ⊂ M be two open sets, and let γ y ,y be a geodesic from y to y witharc-parametrization. Assume W ⊂ M is an open set with [ y ∈ A , y ∈ A γ y ,y ⊂ W. If e be a non-negative integrable function on W , then (4.16) Z A × A Z | y y | e ( γ y ,y ( s )) ds C ( n, k , D ) · D · (cid:0) vol ( A ) + vol ( A ) (cid:1) Z W e, where D = sup y ∈ A , y ∈ A | y y | and C ( n, k , D ) = (cid:0) sinh( p − k D ) / sinh( p − k D/ (cid:1) n − . We now define the upper Hessian of f , Hess x f : T x → R ∪ {−∞} by(4.17) Hess x f ( v, v ) def = lim sup s → f ◦ exp x ( s · v ) − f ( x ) − d x f ( v ) · ss / v ∈ T x .Clearly, this definition also works for any semi-concave function on M . If u is a λ − concavefunction, then its upper Hessian Hess x u ( ξ, ξ ) λ for any ξ ∈ Σ x .For a semi-concave function u , we denote its regular set Reg u by Reg u := (cid:8) x ∈ M : there exists Perelman ′ s Hessian of u at x (cid:9) . It was showed in [Per2] that
Reg u has full measure for any semi-concave function u . It isclear that Hess x u = Hess x u for any x ∈ Reg u . Definition . Let p ∈ M . The cut locus of p , denoted by Cut p , is defined to be the set allof points x in M such that geodesic px , from p to x , can not be extended.It was shown in [OS] that Cut p has zero (Hausdorff) measure (see also [Ot]).Set W p = M \ ( { p } ∪ Cut p ) . For any two points x, y ∈ M with x = y , a direction from x to y is denoted by ↑ yx .The following two lemmas are concerned with the rigidity part of Theorem 3.3. Lemma 4.4.
Let M be an n -dimensional Alexandrov space with Ricci curvature > ( n − K and let f = dist p . Suppose that B p ( R ) \{ p } ⊂ W p for some < R π/ √ K (if K ,we set π/ √ K to be + ∞ ). Assume that for each x ∈ B p ( R ) \{ p } , there exists a sequence θ := { θ j } ∞ j =1 ∈ S such that ∆ θ ′ f ( x ) = ( n − · cot K ( | px | ) for any subsequence θ ′ ⊂ θ .Then the function ̺ K ◦ f is (1 − K · ̺ K ◦ f ) − concave in B p ( R ) \{ p } .Proof. It suffices to show one variable function h p := ̺ K ◦ f ◦ γ ( s ) satisfies that h ′′ p − Kh p for any geodesic γ ( s ) ⊂ B p ( R ) \{ p } . Let χ ( s ) be an continuous function on an open interval( a, b ). Here and in the sequel we write χ ′′ ( s ) B for s ∈ ( a, b ) if χ ( s + τ ) χ ( s ) + A · τ + B · τ / o ( τ ) for some A ∈ R . χ ′′ ( s ) < + ∞ means that χ ′′ ( s ) B for some B ∈ R . If χ isanother continuous function on ( a, b ), then χ ′′ χ means χ ′′ ( s ) χ ( s ) for all s ∈ ( a, b ) . Fix a geodesic γ ⊂ B p ( R ) \{ p } . Let x = γ (0), y = γ ( l ) . Without loss of generality, wecan assume that γ is the unique geodesic from x to y and | px | + | py | + | xy | < R. We consider the function u : W p → R + ∪ { } ,(4.18) u ( z ) = sup ξ ∈ Σ z (cid:12)(cid:12)(cid:12) Hess z f ( ξ, ξ ) − cot K ( | pz | ) · sin ( | ξ, ↑ pz | ) (cid:12)(cid:12)(cid:12) . For any point z ∈ Reg f ∩ B p ( R ), Hess z f is a bilinear form on T z and Hess z f ( ↑ pz , ↑ pz ) = 0.Let Λ z = { ξ ∈ Σ z : ∠ ( ξ, ↑ pz ) = π/ } . By Lemma 4.1, we have
Hess z f ( ξ, ξ ) = H δz f = cot K ( | pz | ) on Λ z for some subsequence δ of θ , and hence u ( z ) = 0 . Since
Reg f has full measure in B p ( R ) , we conclude that u ≡ B p ( R ).Given any positive number ǫ > ǫ ≪ min (cid:8) | px | , | py | , | xy | , R − ( | px | + | py | + | xy | ) (cid:9) . Let x ∈ B x ( ǫ ) and y ∈ B y ( ǫ ), and let γ x ,y ( s ) be a geodesic from x to y . By triangleinequality, it is easy to see | px | + | x y | + | py | < R as ǫ is sufficiently small. Thus γ x ,y ∈ B p ( R ) . Set u x ,y ( s ) = u ( γ x ,y ( s )) . By applying Proposition 4.2 to A = B x ( ǫ ) , A = B y ( ǫ ), W = B p ( R ) and function u , we know that there exist two points x ∈ B x ( ǫ ) and y ∈ B y ( ǫ )such that u x ,y ( s ) = 0 almost everywhere on (0 , | x y | ).Consider a s ∈ (0 , | x , y | ) such that u x ,y ( s ) = 0. Set z = γ x ,y ( s ), ζ + = γ + x ,y ( s )and ζ − = γ − x ,y ( s ). Then we have Hess z f ( ζ + , ζ + ) = Hess z f ( ζ − , ζ − ) = cot K ( | pz | ) · sin ( | ζ + , ↑ pz | ) . Therefore, for function e f ( s ) = f ◦ γ x ,y ( s ), we get e f ( h + s ) e f ( s ) + h e f + ( s ) + F ( s ) · h / o ( h ) , e f ( − h + s ) e f ( s ) − h e f − ( s ) + F ( s ) · h / o ( h ) , (4.19)for any h >
0, where F ( s ) = cot K ( | pz | ) · sin ( | ζ + , ↑ pz | ) = cot K ( | pz | ) · (cid:0) − cos ( | ζ + , ↑ pz | ) (cid:1) . By the first variation formula of arc-length, we have e f + ( s ) = − cos( | ζ + , ↑ pz | ) and e f − ( s ) = − cos( | ζ − , ↑ pz | ) . Note that γ x ,y ∈ W p , | ζ + , ↑ pz | + | ζ − , ↑ pz | = π, which implies that e f ( s ) is continuously differential. Then by combining this with (4.19),we have e f ′′ ( s ) F ( s ) = cot K e f ( s ) · (cid:0) − e f ′ ( s ) (cid:1) for almost everywhere s ∈ (0 , | x y | ). Thus the function e h ( s ) = ̺ K ◦ e f ( s ) satisfies e h ′′ ( s ) − K e h ( s ) ICCI CURVATURE ON ALEXANDROV SPACES 17 for almost everywhere s ∈ (0 , | x y | ). On the other hand, the fact f is semi-concave impliesthat e h ′′ ( s ) < + ∞ for all s ∈ (0 , | x y | ). Thus, from 1.3(3) in [PP], we have e h ′′ − K e h. Letting ǫ → + , we can get point sequences { x i } and { y i } such that x i → x , y i → y and e h ′′ i − K e h i , where e h i = ̺ K ◦ f ◦ γ x i ,y i ( s ). Since the geodesic from x to y is unique, there exists asubsequence of geodesics γ x i ,y i , which converges to geodesic γ uniformly. Hence e h i convergesto h uniformly, and the desired result follows from 1.3(4) in [PP]. Therefore, we havecompleted the proof. (cid:3) Lemma 4.5.
Let σ ( t ) and ς ( t ) be two geodesics in B p ( R ) with σ (0) = ς (0) = p , and let ϕ ( τ, τ ′ ) = e ∠ K σ ( τ ) pς ( τ ′ ) be the comparison angle of ∠ σ ( τ ) pς ( τ ′ ) in the K − plane. Then, under the same assumptionsas Lemma 4.4, we have ϕ ( τ, τ ′ ) is non-increasing with respect to τ and τ ′ .(If K > , we add the assumption that τ + τ ′ + | σ ( τ ) ς ( τ ′ ) | < π/ √ K ).Proof. Firstly, we claim that for any triangle △ pxy , (if K >
0, we assume that | px | + | py | + | xy | < π/ √ K ), there exists a comparison triangle △ ¯ p ¯ x ¯ y in the K − plane M K such that(4.20) ∠ ¯ p ¯ x ¯ y pxy, ∠ ¯ p ¯ y ¯ x pyx. Indeed, for any triangle △ pxy ∈ B p ( R ), there exists a triangle △ b p b x b y in M K such that | b p b x | = | px | , | b x b y | = | xy | , ∠ b p b x b y = ∠ pxy, and by Lemma 4.4, we have | b p b y | > | py | . So by an obvious reason, we get the required triangle △ ¯ p ¯ x ¯ y .Fix τ ′ > ς = ς ( τ ′ ). We only need to show ϕ ( τ ) := ϕ ( τ, τ ′ ) is non-increasingwith respect to τ. Let △ ¯ σ ( τ )¯ p ¯ ς be a comparison triangle of △ σ ( τ ) pς in the K − plane M K and extend thegeodesic ¯ p ¯ σ ( τ ) slightly longer to ¯ σ ( τ + s ) for small s > dist ς is λ − concave for some number λ ∈ R , we have(4.21) | ςσ ( τ + s ) | | ςσ ( τ ) | + s · (cid:0) − cos ∠ σ ( τ + s ) σ ( τ ) ς (cid:1) + s λ/ . On the other hand, we have(4.22) | ¯ ς ¯ σ ( τ + s ) | = | ¯ ς ¯ σ ( τ ) | + s · (cid:0) − cos ∠ ¯ σ ( τ + s )¯ σ ( τ )¯ ς (cid:1) + s ¯ λ/ o ( s )for some number ¯ λ ∈ R . Note from (4.20) that ∠ ¯ σ ( τ + s )¯ σ ( τ )¯ ς > ∠ σ ( τ + s ) σ ( τ ) ς. By combining this with (4.21), (4.22) and | ςσ ( τ ) | = | ¯ ς ¯ σ ( τ ) | , we have(4.23) | ςσ ( τ + s ) | | ¯ ς ¯ σ ( τ + s ) | + ( − λ + ¯ λ ) s + o ( s ) . Now, if
K >
0, by cosine law in M K , we havecos e ∠ K σ ( τ + s ) pς − cos e ∠ K σ ( τ ) pς = cos( √ K | ςσ ( τ + s ) | ) − cos( √ K | ¯ ς ¯ σ ( τ + s ) | )sin( √ K | pσ ( τ + s ) | ) · sin( √ K | pς | ) > − ( λ + ¯ λ )sin( √ K | pσ ( τ + s ) | ) · sin( √ K | pς | ) · s . Hence, we get d + dτ cos e ∠ K σ ( τ ) pς > . If K
0, using a similar argument, we can get d + dτ e ∠ K σ ( τ ) pς . Therefore we havecompleted the proof of the lemma. (cid:3) Maximal diameter theorem
The main purpose of this section is to prove Theorem 1.8.Bonnet–Myers’ theorem asserts that if an n -dimensional Riemannian manifold has Ric > n −
1, then its diameter π. Furthermore, its fundamental group is finite.The first assertion, the diameter estimate, has been extend to metric measure spacewith CD ( n, n −
1) (see [S2]) or
M CP ( n, n −
1) (see [O1]). Since our condition
Ric > n − CD ( n, n − n -dimensional Alexandrov space M with Ric ( M ) > n − ∂M = ∅ .Now we consider the second assertion: finiteness of the fundamental group. Proposition 5.1.
Let M be an n -dimensional Alexandrov space without boundary and Ric ( M ) > n − . The order of fundamental group of M , ord π ( M ) , satisfies ord π ( M ) ω n vol ( M ) where ω n is the volume of n -dimensional standard sphere S n . In particular, if add assump-tion vol ( M ) > ω n / , M is simply connected.Proof. Let f M be the universal covering of M . We have Ric ( f M ) > n −
1. Therefore, byBishop–Gromov volume comparison theorem (see Corollary A.3 in Appendix), we getord π ( M ) · vol ( M ) = vol ( f M ) ω n . This completes the proof. (cid:3)
Now, we are in position to prove Theorem 1.8. We rewrite it as following
Theorem 5.2.
Let M be an n -dimensional Alexandrov space with Ric ( M ) > n − and ∂M = ∅ . If diam ( M ) = π , then M is isometric to suspension [0 , π ] × sin N, where N is anAlexandrov space with curvature > . Proof.
Takes two points p, q ∈ M such that | pq | = π. Exactly as in Riemannian manifold case, by using Bishop–Gromov volume comparisontheorem, we have the following assertions:
Fact: (i) For any point x ∈ M , there holds | px | + | qx | = π. This implies W p = W q = M \{ p, q } . (ii) For any x ∈ M , we can extend the geodesic px to a geodesic from p to q . We willdenote it by pxq. ICCI CURVATURE ON ALEXANDROV SPACES 19 (iii) For any non-degenerate triangle △ pxy , we have | px | + | py | + | xy | < π. (iv) For any direction ξ ∈ Σ p , there exists a geodesic γ ξ such that γ ξ (0) = p, γ + ξ (0) = ξ and its length is equal to π. Indeed, the first assertion (i) is an immediate consequence of Bishop–Gromov volumecomparison theorem (see, for example, page 271 in [P]). Gluing geodesics px and qx , theresult curve has length = π = | pq | . Thus it is a geodesic. This proves the second assertion(ii). The third assertion (iii) follows directly from triangle inequality | px | + | py | + | xy | < | px | + | py | + | qx | + | qy | π. To show (iv), we consider a sequence of direction ξ i ∈ Σ p such that ξ i → ξ and there existsgeodesics α i with α i (0) = p and α + i (0) = ξ i . From (ii), we can extend each α i to a newgeodesic with length = π , denoted by α i again. By Arzela–Ascoli Theorem, we can take alimit from some subsequence of α i . Clearly, the limit is the desired geodesic. This provesthe last assertion (iv).Let f = dist p and ¯ f = dist q . For any point x = p, q , we set Λ x ⊂ Σ x all of directionswhich are vertical with the geodesic pxq .Fix a sequence θ = { θ j } ∞ j =1 ∈ S . By Theorem 3.3, we can find a subsequence δ ⊂ θ suchthat(5.1) ∆ δ f ( x ) ( n − · cot( | px | ) and ∆ δ ¯ f ( x ) ( n − · cot( | qx | ) . The above fact (i) implies f + ¯ f = π . Thus(5.2) H δx ¯ f ( ξ ) = − lim inf s → s ∈ δ f ◦ exp x ( s · ξ ) − f ( x ) s / . By Definition 3.1, we have H δ ′ x f > − H δx ¯ f for any subsequence δ ′ ⊂ δ . Hence, by combiningthis with (5.1) and the definition of ∆ δ f , we get∆ δ ′ f ( x ) > − ∆ δ ¯ f ( x ) > − ( n − · cot( | qx | ) = ( n − · cot( | px | ) . Note also that ∆ δ ′ f ( x ) ∆ δ f ( x ) . By combining this with (5.1), this implies that∆ δ ′ f ( x ) = ( n − · cot( | px | )for any subsequence δ ′ ⊂ δ .¿From Lemma 4.4, − cos f is cos f − concave in B p ( π ) \{ p } = W p . Given any geodesic σ ( s ) : [0 , L ] → W p with L < π , we have(5.3) ( − cos f ◦ σ ) ′′ ( s ) cos f ◦ σ ( s ) , ∀ s ∈ (0 , L ) . Similarly, − cos ¯ f is cos ¯ f − concave in W q = W p and(5.4) ( − cos ¯ f ◦ σ ) ′′ ( s ) cos ¯ f ◦ σ ( s ) , ∀ s ∈ (0 , L ) . Since f + ¯ f = π , cos f = − cos ¯ f , by combining this with (5.3) and (5.4), we get(5.5) ( − cos f ◦ σ ) ′′ ( s ) = cos f ◦ σ ( s ) , ∀ s ∈ (0 , L ) . Denote by M + = (cid:8) x ∈ M : f ( x ) π/ (cid:9) , M − = (cid:8) x ∈ M : f ( x ) > π/ (cid:9) and N = M + ∩ M − = { x ∈ M : f ( x ) = π/ } . Set v x = (geodesic pxq ) ∩ N, which is consisting of a single point. We claim that N is totally geodesic in M .Indeed, take any two points v , v ∈ N with | v v | < π . Let σ ( s ) be a geodesic connected v and v . By (5.5) and noting thatcos f ( v ) = cos f ( v ) = 0 , we have cos f ◦ σ ( s ) ≡ . This tells us σ ⊂ N and N is totally geodesic.Now we are ready to prove that M is isometric to suspension [0 , π ] × sin N . Consider anytwo points x, y ∈ M \{ p, q } .If x, y ∈ M + , we know from Lemma 4.5 that(5.6) e ∠ xpy > e ∠ v x pv y and e ∠ xqy e ∠ v x qv y . Note from Fact (i) that e ∠ xpy = e ∠ xqy. Thus we obtain(5.7) e ∠ xpy = e ∠ v x pv y . Clearly, if x, y ∈ M − , the same argument also deduces the equality (5.7).While if x ∈ M + and y ∈ M − , by Lemma 4.5 again, we have e ∠ xpy > e ∠ v x py = e ∠ v x pv y and e ∠ xpy e ∠ xpv y = e ∠ v x pv y , which implies the equality (5.7).Then by applying the cosine law to the comparison triangle, we getcos( | xy | ) = cos( | px | ) · cos( | py | ) + sin( | px | ) · sin( | py | ) cos e ∠ v x pv y . This proves that M is isometric to suspension [0 , π ] × sin N. It remains to show that N has curvature > . We define a map Φ : N → Σ p byΦ( v ) = ↑ vp , ∀ v ∈ N. Since N ⊂ W p and | pv | = π/ v ∈ N , Φ is well defined.Given two points v , v ∈ N , for any x ∈ M lies in geodesic pv q and any x ∈ M liesin geodesic pv q , the equality (5.7) implies e ∠ x py = e ∠ v pv = | v v | . Since ∠ v pv = lim x → p,x → p e ∠ x py , we have | ↑ v p ↑ v p | Σ p = | v v | . This shows that Φ is an isometrical embedding. On the other hand, by Fact (iv), Φ issurjective. Therefore, Φ is an isometry. Thus N has curvature >
1. Therefore, we havecompleted the proof of the theorem. (cid:3)
Corollary 5.3.
Let M be an n -dimensional Alexandrov space with Ric ( M ) > n − and ∂M = ∅ . If rad ( M ) = π , then M is isometric to the sphere S n with standard metric.Proof. For any point p ∈ M , there exists a point q such that | pq | = π . From the proofof theorem 5.2, we have that − cos dist p is cos dist p − concave in B p ( π ) \{ p } . Thus M hascurvature >
1. It is well-known (see,for example, Lemma 10.9.10 in [BBI]) that an n -dimensional Alexandrov space with curvature > rad = π must be isometric to thesphere S n with standard metric. (cid:3) ICCI CURVATURE ON ALEXANDROV SPACES 21
Remark . Colding in [C] had proved the corollary for limit spaces of Riemannian mani-folds. That is, if M i is a sequence of m − dimensional Riemannian manifolds with Ric M i ≥ m − X with rad X = π , then X is isometric to the sphere S m ′ with standard metric for some integer m ′ m. Splitting theorem
In this section, M will always denote an n -dimensional Alexandrov space with curvaturebounded below locally, Ric ( M ) > ∂M = ∅ . The main purpose of this section is toprove Theorem 1.7.A curve γ : [0 , + ∞ ) → M is called a ray if | γ ( s ) γ ( t ) | = s − t for any 0 t < s < + ∞ . Acurve γ : ( −∞ , + ∞ ) → M is called a line if | γ ( s ) γ ( t ) | = s − t for any −∞ < t < s < + ∞ . For a line γ , obviously, γ | [0 , + ∞ ) and γ | ( −∞ , form two rays.Given a ray γ ( t ), we define the Busemann function b γ for γ on M by b γ ( x ) = lim t → + ∞ (cid:0) t − | xγ ( t ) | (cid:1) . Clearly, it is well-defined and is a 1-Lipschitz function.¿From now on, in this section, we fix a line γ ( t ) in M and set γ + = γ | [0 , + ∞ ) , γ − = γ | ( −∞ , .Let b + and b − be the Busemann functions for rays γ + and γ − , respectively.Let us recall what is the proof of the splitting theorem in the smooth case. When M is a smooth Riemannian manifold, Cheeger–Gromoll in [CG] used the standard Laplaciancomparison and the maximum principle to conclude that b + and b − are harmonic on M .Then the elliptic regularity theory implies that they are smooth. The important stepis to use Bochner formula to show that both ∇ b + and ∇ b − are parallel. Consequently,the splitting theorem follows directly from de Rham decomposition theorem. In [EH],Eschenburg–Heintze gave a proof avoiding the elliptic regularity; while the Bochner formulais essentially used. But for the general Alexandrov spaces case, the main difficulty is thelack of Bochner formula.We begin with a lemma which was proved by Kuwae and Shioya for Alexandrov spaceswith M CP ( n,
0) and hence for Alexandrov spaces with nonnegative Ricci curvature. (Seelemma 6.5 and the proof of theorem 1.3 in [KS1]).
Lemma 6.1. b + ( x ) + b − ( x ) ≡ , on M . Lemma 6.2.
For any point x ∈ M , there exists a unique line γ x such that x = γ x (0) and b + ◦ γ x is a linear function with ( b + ◦ γ x ) ′ = 1 .Proof. Existence. If x ∈ γ , then we can write x = γ ( t ). Hence we set γ x ( t ) = γ ( t + t ) , which is a desired line.We then consider the case x γ . Let σ t, + ( s ) be a geodesic from x to γ + ( t ). By usingArzela–Ascoli Theorem, we can take a sequence t j → + ∞ such that σ t j , + converges to alimit curve σ ∞ , + ( s ) : [0 , + ∞ ) → M . It is easy to check ( see, for example, page 286 in [P])that σ ∞ , + is 1-Lipschitz and b + ◦ σ ∞ , + ( s ) = s + b + ◦ σ ∞ , + (0) = s + b + ( x ) , for all s > . By a similar construction, we can obtain a 1-Lipschitz curve σ ∞ , − ( s ′ ) : ( −∞ , → M suchthat σ ∞ , − (0) = x and b − ◦ σ ∞ , − ( s ′ ) = − s ′ + b − ( x ) , for all s ′ . Let σ ∞ = σ ∞ , + ∪ σ ∞ , − : ( −∞ , + ∞ ) → M . This is a 1-Lipschitz curve. By Lemma 6.1, wehave(6.1) b + ◦ σ ∞ ( s ) = s + b + ( x ) , for all s ∈ ( −∞ , + ∞ ) . Then for any −∞ < t < s < ∞ , by (6.1), we get s − t = b + ◦ σ ∞ ( s ) − b + ◦ σ ∞ ( t ) | σ ∞ ( s ) σ ∞ ( t ) | s − t. Thus σ ∞ is a line. The equation (6.1) shows that it is a desired line. Uniqueness.
Argue by contradiction. Suppose that there exist two such lines γ , γ .The equations ( b + ◦ γ ) ′ = ( b + ◦ γ ) ′ = 1 implies b + ◦ γ ( −
1) = b + ( x ) − b + ◦ γ (1) = b + ( x ) + 1Hence b + ◦ γ (1) − b + ◦ γ ( −
1) = 2 . Since b + is 1-Lipschitz, we get(6.2) | γ ( − γ (1) | > b + ◦ γ (1) − b + ◦ γ ( −
1) = 2 . On the other hand,
Length (cid:0) γ ([ − , ∪ γ ([0 , (cid:1) = 2 . Thus γ ([ − , ∪ γ ([0 , M is non-branching. Theproof of the lemma is completed. (cid:3) For any point x ∈ M , we take the line γ x in Lemma 6.2. Let L x = { ξ ∈ T x | ∠ ( ξ, γ + x (0)) = ∠ ( ξ, γ − x (0)) = π/ } , Λ x = { ξ ∈ Σ x | ∠ ( ξ, γ + x (0)) = ∠ ( ξ, γ − x (0)) = π/ } . Given a sequence θ := { θ j } ∈ S , we define a function H θx b + : Λ x → R by H θx b + ( ξ ) def = lim sup s → , s ∈ θ b + ◦ exp x ( s · ξ ) − b + ( x ) s / θ b + ( x ) def = ( n − · I Λ x H θx b + ( ξ ) . In the following Lemma 6.3, we will prove that both b + and b − are semi-concave. Thus,by lemma 6.1, H θx b + is well defined and is locally bounded. It is easy to see that H θx b + ismeasurable, so ∆ θ b + ( x ) is also well defined. Lemma 6.3. b + ( x ) is a semi-concave function in M . Moreover, for any point x ∈ M andany sequence θ = { θ j } ∈ S , there exists a subsequence δ ⊂ θ such that ∆ δ b + ( x ) . Proof.
Fix a point x ∈ M , we will construct a semi-concave support function for b + near x .We take the line γ x in Lemma 6.2 and choose a point p ∈ γ x such that b + ( p ) ≪ b + ( x ) . The equation ( b + ◦ γ x ) ′ = 1 implies(6.3) b + ( x ) − b + ( p ) = | px | . On the other hand, since b + is 1-Lipschitz, we have(6.4) b + ( y ) − b + ( p ) | py | for any y ∈ M . By combining (6.3) and (6.4), we know that function dist p ( · ) + b + ( p )supports b + near x .This tells us b + is a semi-concave function. Furthermore, from Theorem 3.3, we can finda subsequence e δ ⊂ θ such that ∆ e δ b + ( x ) ( n − / | px | . By letting | px | → ∞ and a diagonalargument, we can choose a subsequence δ ⊂ e δ such that ∆ δ b + ( x ) . Therefore the proofof the lemma is completed. (cid:3)
ICCI CURVATURE ON ALEXANDROV SPACES 23
The following lemma is similar to Lemma 4.4.
Lemma 6.4.
Assume that for each point x ∈ M , there exists a sequence θ := { θ j } ∈ S such that ∆ θ ′ b + ( x ) = 0 for any subsequence θ ′ ⊂ θ . Then b + is a concave function in M .Proof. It suffices to show that b + is concave on an arbitrarily given bounded open setΩ ⊂ M . Clearly, we may assume M has curvature > k Ω in Ω for some constant k Ω .In following, we divide the proof into three steps.Step 1. Let γ x be the line in Lemma 6.2. Replacing equation (3.6) and (3.7) by thefacts that | b + ( y ) − b + ( z ) | = | yz | for any y, z ∈ γ x and b + is 1-Lipschitz, the same proof inLemma 3.2 shows that the lemma also holds when we replace f = dist p by b + .Step 2. Similar as Lemma 4.1, we want to show H δx b + = 0 almost everywhere in Λ x ,for some subsequence δ = { δ j } ⊂ θ .We now follow the proof of Lemma 4.1. Firstly, from Lemma 6.3, we know that both b + and b − are semi-concave. In turn, Lemma 6.1 gives a bound for H θx b + . Secondly, we useLemma 3.2 for b + (i.e., the above Step 1) and replace Theorem 3.3 by the above Lemma6.3 in the proof of Lemma 4.1. We repeat the same proof of Lemma 4.1 to get H δx b + = 0almost everywhere in Λ x , for some subsequence δ ⊂ θ .Step 3. Following the proof of Lemma 4.4, we then deduce that b + ( x ) is concave in Ω . Therefore b + ( x ) is concave in M and the proof of the lemma is completed. (cid:3) Now, we are in a position to prove Theorem 1.7.
Proof of Theorem 1.7.
Given a sequence θ = { θ j } ∈ S , from Lemma 6.3, we can find asubsequence δ ⊂ θ such that(6.5) ∆ δ b + ( x ) δ b − ( x ) . By the definition of ∆ δ b + ( x ) and ∆ δ b − ( x ), we have∆ δ ′ b + ( x ) ∆ δ b + ( x ) and ∆ δ ′ b − ( x ) ∆ δ b − ( x )for any subsequence δ ′ ⊂ δ. So (6.5) holds for any subsequence δ ′ ⊂ δ .On the other hand, by Lemma 6.1 and the definition of ∆ θ b + ( x ), we have∆ ϑ b + ( x ) + ∆ ϑ b − ( x ) > ϑ = { ϑ j } ∈ S . Therefore, by combining with (6.5), we get∆ δ ′ b + ( x ) = 0 and ∆ δ ′ b − ( x ) = 0for any subsequence δ ′ ⊂ δ. Then we can apply Lemma 6.4 to conclude that both b + and b − are concave. By usingLemma 6.1 again, we deduce that b + ◦ ς ( s ) is a linear function on any geodesic ς ( s ) in M .In particular, the level surfaces L ( a ) := b − ( a ) are totally geodesic for all a ∈ R .Set N = L (0) = b − (0). It is an Alexandrov space with curvature bounded below locally.When M is an Alexandrov space with curvature > − κ for some κ >
0. Mashiko, in[Ma], proved that if there exists a function u such that u ◦ γ is a linear function for anygeodesic γ ⊂ M and u ∈ D , (see [Ma] for the definition of the class of D , ), then M is isometric to a direct product R × Y over an Alexandrov space Y has curvature > − κ . Later in [AB], Alexander and Bishop removed the condition u ∈ D , .Since we do not assume that M has a uniform lower curvature bound, we adapt Mashiko’sargument as follows.For any x ∈ N and any a ∈ R , let γ x be the line obtained in Lemma 6.2. Note that ( b + ◦ γ x )( s ) ′ = 1 which implies ∇ b + ( γ x ( s )) = γ + x ( s ). Thus γ x is a gradientcurve of b + .It is easy to check that γ x ∩ L ( a ) is a set of single point. We define Φ a : N → L ( a ) byΦ a ( x ) = γ x ∩ L ( a ) . Φ a and Φ − a are the gradient flows of b + and b − , respectively. Since agradient flow of a concave function is non-expanding, we have that Φ a is an isometry.Now we are ready to show that M is isometric to the direct product R × N . Considerany two points x, y ∈ M .Without loss of generality, we may assume that x ∈ N and y ∈ L ( a ) with a >
0. Let z = γ y ∩ N , where γ y comes from Lemma 6.2.We take a C curve σ ( s ) ⊂ N with σ (0) = x and σ ( Length ( σ )) = z, | σ ′ ( s ) | = 1 . Definea new curve ¯ σ ( s ) by ¯ σ ( s ) = γ σ ( s ) (cid:16) alength ( σ ) · s (cid:17) . Clearly, we have ¯ σ (0) = x , ¯ σ ( length ( σ )) = γ z ( a ) = y and(6.6) b + (¯ σ ( s )) = alength ( σ ) · s. Fixed any s ∈ (0 , Length ( σ )), we set u = σ ( s ) and v = ¯ σ ( s ).We claim that(6.7) ∠ ( ∇ u b + , σ + ( s )) = ∠ ( ↑ vu , σ + ( s )) = π/ . Indeed, | vσ ( s ′ ) | > b + ( v ) − b + ( σ ( s ′ )) = b + ( v ) = | vu | for any s ′ ∈ (0 , Length ( σ )). Then by the first variation formula of arc-length, we have(6.8) ∠ ( ↑ vu , σ + ( s )) > π/ ∠ ( ↑ vu , σ − ( s )) > π/ . On the other hand,(6.9) ∠ ( ↑ vu , σ + ( s )) + ∠ ( ↑ vu , σ − ( s )) = π. Thus the desired (6.7) follows from (6.8) and (6.9).Now let us calculate the length of the curve ¯ σ. Clearly, we may assume that a neighborhood of ¯ σ has curvature > k (for some k < s ∈ (0 , length ( σ )). Let h > w = ¯ σ ( s + h ) and w = γ σ ( s + h ) (cid:0) alength ( σ ) · s (cid:1) (see figure 1). xz y )( su σ = )( sv σ = w w (cid:51)(cid:39)(cid:41) !" (cid:3)(cid:5)(cid:83)(cid:71)(cid:73)(cid:41)(cid:68)(cid:70)(cid:87)(cid:82)(cid:85)(cid:92)(cid:3)(cid:51)(cid:85)(cid:82)(cid:5)(cid:3) $ (cid:90)(cid:90)(cid:90)(cid:17)(cid:73)(cid:76)(cid:81)(cid:72)(cid:83)(cid:85)(cid:76)(cid:81)(cid:87)(cid:17)(cid:70)(cid:81) Figure 1.
By cosine law in 0 − plane R , we have(6.10) | ¯ σ ( s + h )¯ σ ( s ) | = | v ¯ w | = | vw | + | w ¯ w | − | w ¯ w | · | vw | · cos e ∠ vw ¯ w. Note that(6.11) | v ¯ w | = | σ ( s ) σ ( s + h ) | = | σ + ( s ) · h + o ( h ) | = h + o ( h ) , ICCI CURVATURE ON ALEXANDROV SPACES 25 (6.12) | w ¯ w | = ( b + ( ¯ w ) − b + ( w )) = aLength ( σ ) · h. By using Lemma 11.2 in [BGP], we have(6.13) e ∠ k vw ¯ w → ∠ vw ¯ w = π/ h → . On the other hand, note that(6.14) e ∠ vw ¯ w − e ∠ k vw ¯ w → h → . We have cos e ∠ vw ¯ w → h → . Combining this and (6.10)–(6.12), we have(6.15) | ¯ σ ( s + h )¯ σ ( s ) | = (cid:16) (cid:0) aLength ( σ ) (cid:1) (cid:17) · h + o ( h ) . Hence, | ¯ σ ( s ) | + = (cid:16) (cid:0) aLength ( σ ) (cid:1) (cid:17) / . Similarly, we can get | ¯ σ ( s ) | − = (cid:16) (cid:0) aLength ( σ ) (cid:1) (cid:17) / . So(6.16)
Length (¯ σ ) = Z length ( σ )0 | ¯ σ | ′ ds = (cid:16) a + (cid:0) Length ( σ ) (cid:1) (cid:17) / . If we take σ to be a geodesic xz , we get, from (6.16), that(6.17) | xy | ( Length (¯ σ )) = | xz | + a = | xz | + | yz | . While if we take σ to be the projection of a geodesic xy to N , we get, from (6.16), that(6.18) | xy | = ( Length ( σ )) + a > | xz | + | yz | . The combination of (6.17) and (6.18) implies that(6.19) | xy | = | xz | + | yz | . This says that M is isometric to the direct product N × R .Lastly, we need prove that N has nonnegative Ricci curvature.Let γ ( t ) : ( − ℓ, ℓ ) → N be a geodesic in N . Assume that N has curvature > K in aneighborhood of γ and for some K <
0. Otherwise, there is nothing to prove. Hence M has curvature > K in a neighborhood of γ in M .Let p and q be two interior points in γ . We denote the tangent spaces, exponential mapin N (or M , resp.) by T p N , exp Np (or T p M = T ( p, M , exp Mp = exp M ( p, , resp.) andΛ Np = { ξ ∈ T Np : (cid:10) ξ, γ ′ (cid:11) = 0 } . Let Λ Mp := Λ M ( p, = { ξ ∈ T Mp : h ξ, γ ′ i = 0 } . Then Λ Mp = S (Λ Np ) with vertex ζ ± , where ζ ± are the directions along factor R in M = N × R . For any ξ ∈ T p N , we have(6.20) exp Mp ( ξ, t ) = (exp Np ( ξ ) , t ) . Suppose that a family of continuous functions { g ( γ ( t ) , ( ξ, η ) } − ℓ 0) in M and Z Λ Mp g ( γ ( t ) , ( ξ, η ) d vol Λ Mp > − ǫ for a given small number ǫ > Given a sequence { e s j } ∈ S , the isometry T : L M ( p, → L M ( q, and subsequence { s j } ⊂ { e s j } come from the definition of Condition ( RC ). Recall Petrunin’s construction for T , we canassume that T ( ζ + ) = ζ + , hence T : L Np ⊂ T Np → L Nq ⊂ T Nq . Given a quasi-geodesic σ ( s ) in N , setting ¯ σ ( s ) = ( σ ( as ) , bs ) for any two number a, b ∈ R with a + b = 1, we will prove that ¯ σ ( s ) is a quasi-geodesic in M .Let u ( z, r ) be a λ − concave function, defined in a neighborhood of γ in M = N × R . Sofunction u ( · , r ) is λ − concave in N and u ( z, · ) is λ − concave in R for all r ∈ R and z ∈ N .Since σ is quasi-geodesic in N , we have u ′′ (cid:0) σ ( as ) , r (cid:1) a · λ for all r ∈ R . Now u ′′ (cid:0) σ ( as ) , bs (cid:1) ( a + b ) · λ = λ. By definition of quasi-geodesic [Pet3], we get that ¯ σ ( s ) is a quasi-geodesic in M .Fix any nonnegative number l and l . Let ξ ∈ Λ Np . For any constant A ∈ R , we have(see figure 2) γ + ζ β p M p LA )0,( ),( ∈ + ζξ Np L ∈ ξ RNM ×= N (cid:51)(cid:39)(cid:41) !" (cid:3)(cid:5)(cid:83)(cid:71)(cid:73)(cid:41)(cid:68)(cid:70)(cid:87)(cid:82)(cid:85)(cid:92)(cid:3)(cid:51)(cid:85)(cid:82)(cid:5)(cid:3) $ (cid:90)(cid:90)(cid:90)(cid:17)(cid:73)(cid:76)(cid:81)(cid:72)(cid:83)(cid:85)(cid:76)(cid:81)(cid:87)(cid:17)(cid:70)(cid:81) Figure 2. | exp Np ( s j · l ξ ) , exp Nq ( s j · l T ξ ) | = (cid:12)(cid:12)(cid:0) exp Np ( s j · l ξ ) , s j l Aζ + (cid:1) , (cid:0) exp Nq ( s j · l T ξ ) , s j l Aζ + (cid:1)(cid:12)(cid:12) − A ( l − l ) · s j = (cid:12)(cid:12) exp Mp (cid:0) s j · l ( ξ, Aζ + ) (cid:1) , (cid:0) exp Mq (cid:0) s j · l ( T ξ, Aζ + ) (cid:1)(cid:12)(cid:12) − A ( l − l ) · s j | pq | + s j · (cid:16) ( l − l ) − g p ( ξ, Aζ + ) · (1 + A )3 | pq | ( l + l · l + l ) (cid:17) + o ( s j ) . (6.21)We set β = ∠ (cid:0) ( ξ, , ( ξ, Aζ + ) (cid:1) and then A = tan β , β ∈ ( − π/ , π/ t ∈ ( − ℓ, ℓ ) and A ∈ R , we define a function g A,γ ( t ) : Λ Nγ ( t ) → R by g A,γ ( t ) ( ξ ) : = g ( γ ( t ) , ( ξ, Aζ + ) · (1 + A )= g ( γ ( t ) , ( ξ, Aζ + ) / cos β. (6.22)¿From (6.21), for any A ∈ R , the family of continuous functions { g A,γ ( t ) ( ξ ) } − ℓ On the other hand, we have − ǫ Z Λ Mp g ( γ ( t ) , ( ξ, η ) d vol Λ Mp = Z Λ Np Z π/ − π/ g ( γ ( t ) , ( ξ, η ) cos n − βdβd vol Λ Np = Z Λ Np Z π/ − π/ g A,γ ( t ) ( ξ ) cos n βdβd vol Λ Np = Z π/ − π/ Z Λ Np g A,γ ( t ) ( ξ ) cos n βd vol Λ Np dβ. (6.23)Thus, we can choose some A ∈ R such that Z Λ Np (cid:0) g A,γ ( t ) ( ξ ) (cid:1) d vol Λ Np > − c n · ǫ, for some constant c n . This completes the proof that N has nonnegative Ricci curvature.Therefore the proof of Theorem 1.7 is completed. (cid:3) Appendix A.In the Appendix, we will recall the definition of curvature-dimension condition CD ( n, k )which is given by Sturm [S2] and Lott–Villani [LV1] (see also book [V]). After that we willpresent a proof, due to Petrunin [Pet2], for the statement that an n -dimensional Alexandrovspace with Ricci curvature > ( n − K and with ∂M = ∅ must satisfy CD ( n, ( n − K ) . Let ( X, d, m ) be a metric measure space, where ( X, d ) is a complete separable metricspace.Given two measures µ and ν on X , a measure q on X × X is called a coupling (or transference plan ) of µ and ν if q ( A × X ) = µ ( A ) and q ( X × A ) = ν ( A )for all measurable A ⊂ X .The L − W asserstein distance between two measures µ, ν is defined by d W ( µ, ν ) = inf q Z X × X d ( x, y ) dq ( x, y )where infimum runs over all coupling q of µ and ν . (If µ ( X ) = ν ( X ), we set d W ( µ, ν ) = + ∞ .)Let P ( X ) be the space of all probability measures ν on X with finite second moments: Z X d ( o, x ) dν ( x ) < ∞ for some (hence all) point o ∈ X . L − Wasserstein space is a complete metric space ( P ( X ) , d W ). (see [S1] for the geome-try of L − Wasserstein space.) Fix a Borel measure m on X . We denote L − Wassersteinspace by P ( X, d ) and its subspace of m − absolutely continuous measures is denoted by P ( X, d, m ). Given k ∈ R , n ∈ (1 , ∞ ], t ∈ [0 , 1] and two points x, y ∈ X , we define β ( k,n ) t as follows:(1) If 0 < t 1, then β ( k,n ) t ( x, y ) := exp (cid:0) k (1 − t ) · d ( x , x ) (cid:1) if n = ∞ , ∞ if n < ∞ , k > α > π, (cid:0) sin( tα ) t sin α (cid:1) n − if n < ∞ , k > α ∈ [0 , π ) , n < ∞ , k = 0 , (cid:16) sinh( tα ) t sinh α (cid:17) n − if n < ∞ , k < , where α = d ( x, y ) · p | k | / ( n − β ( k,n )0 ( x, y ) = 1 . The curvature-dimension condition CD ( n, k ) is defined as follows (see 29.8 and 30.32 in[V]): Definition A.1 . Let ( X, d, m ) be a non-branching locally compact complete separable geo-desic space equipped with a locally finite measure m .Given two real numbers k and n with n > 1, The metric measure space ( X, d, m ) is saidto satisfy the curvature-dimension condition CD ( n, k ) if and only if for each pair compactlysupported µ , µ ∈ P ( X, d, m ) there exist an optimal coupling q of µ = ̺ m and µ = ̺ m ,and a geodesic path µ t : [0 , → P ( X, d ) connecting µ and µ , with H n ( µ t | m ) − (1 − t ) Z X × X (cid:16) ̺ ( x ) β ( k,n )1 − t ( x, y ) (cid:17) − /n dq ( x, y ) − t Z X × X (cid:16) ̺ ( y ) β ( k,n ) t ( x, y ) (cid:17) − /n dq ( x, y )(A.1)for all t ∈ [0 , H n ( ·| m ) : P ( X, d ) → R is R´enyi entropy functional with respect to m , H n ( µ | m ) := − Z X ̺ − /n dµ and ̺ denotes the density of the absolutely continuous part in the Lebesgue decomposition µ = ̺m + µ c of µ. ¿From now on, in the appendix, M will always denote an n -dimensional Alexandrov spacewith Ric ( M ) > ( n − K and ∂M = ∅ . Our purpose of this appendix is to prove the following proposition, which is essentiallydue to Petrunin [Pet2]. Proposition A.2. Let M be an n -dimensional Alexandrov space without boundary and Ric ( M ) > ( n − K . Let vol denote the n -dimensional Hausdorff measure on M . Then themetric measure space ( M, | · ·| , vol ) satisfies CD ( n, ( n − K ) . ¿From [S2], we know that the curvature-dimension condition CD ( n, ( n − K ) impliesBishop–Gromov volume comparison theorem. Consequently, we get the following Corollary A.3. Let M be as in above proposition. Then the function, for any p ∈ M , vol B p ( r ) vol B nK ( r ) Lott–Villani and Sturm defined curvature dimension condition on general metric measure spaces. constant-speed shortest curve defined on [0 , ICCI CURVATURE ON ALEXANDROV SPACES 29 is non-increasing in r > , where B nK ( r ) is a geodesic ball of radius r in the n -dimensionalsimply connected Riemannian manifold with constant sectional curvature K . Before beginning the proof of Proposition A.2, let us review some indispensable materials.For a continuous function f , we define its Hamilton–Jacobi shift H t f for time t > H t f def = inf y ∈ M (cid:8) f ( y ) + 12 t | xy | (cid:9) . Denote by f t = H t f . A solution of α + ( t ) = ∇ α ( t ) f t is called a f t − gradient curve .Refer to [Pet2] for the existence and uniqueness of f t − gradient curve and basic proposi-tions of Hamilton–Jacobi shifts. Now we list only facts that is necessary for us to prove theabove Proposition A.2. Fact A: Let f : M → R be bounded and continuous function and f t = H t f. Assume γ : (0 , → M is a f t − gradient curve which is also a constant-speed shortest curve. Wehave :( i ) f t ( x ) f t ( y ) + | xy | t − t ) for any t > t > x, y ∈ M ;( ii ) f t ( γ ( t )) = f t ( γ ( t )) + | γ ( t ) γ ( t ) | t − t ) ;( iii ) ∇ f t = γ + and |∇ f t | = | γ ( t ) γ ( t ) | t − t = | γ (0) γ (1) | .The following result is a modification of the proposition 2.2 in [Pet2], where we replacethe condition curvature > K by the condition Ric ( M ) > ( n − K. Proposition A.4. Let M be an n -dimensional Alexandrov space with Ricci curvature > ( n − K . f : M → R be bounded and continuous function and f t = H t f. Assume γ :(0 , → M is a f t − gradient curve which is also a constant-speed shortest curve. Supposethat the bilinear form Hess γ ( t ) f t is defined for almost all t ∈ (0 , . Then h ′ T − h T ,h ′ V − ( n − K | γ (0) γ (1) | − h V n − in the sense of distributions, where h T ( t ) def = Hess γ ( t ) f t (cid:0) γ + | γ + | , γ + | γ + | (cid:1) and h V is the trace of Hess γ ( t ) f t in the vertical space L γ ( t ) , i.e., h V ( t ) def = T race L Hess γ ( t ) f t . Proof. Since the bilinear form Hess γ ( t ) f t is defined for almost all t ∈ (0 , T γ ( t ) , t ∈ (0 , n -dimensional Euclidean space. In particular,all L γ ( t ) , t ∈ (0 , R n − .Take two points 0 < t < t < , we may assume that Hess γ ( t ) f t is defined at t and t .Denote by the direction ξ t = γ + ( t ) / | γ + ( t ) | , t ∈ (0 , f t (cid:0) γ ( t + s ) (cid:1) = f t (cid:0) γ ( t ) (cid:1) + s · h∇ f t , γ + ( t ) i + s · Hess γ ( t ) f t ( ξ t , ξ t ) · | γ + ( t ) | + o ( s ) and f t (cid:0) γ ( t + ls ) (cid:1) = f t (cid:0) γ ( t ) (cid:1) + ls · h∇ f t , γ + ( t ) i + ( ls ) · Hess γ ( t ) f t ( ξ t , ξ t ) · | γ + ( t ) | + o ( s )for any l > 0. Combining these and the Fact A, we get l · h T ( t ) − h T ( t ) ( l − t − t for any l > l = (cid:0) − ( t − t ) h T ( t ) (cid:1) − (when t − t suffices small, 1 − ( t − t ) h T ( t )is positive), we get h T ( t ) − h T ( t ) t − t − h T ( t ) · h T ( t ) . That is, h ′ T − h T . Fix arbitrary ǫ > 0. By our definition of Ricci curvature > ( n − K along γ , there existsa continuous function family { g γ ( t ) } 0. By combining (A.3), (A.4) and the Fact A, we get l · Hess γ ( t ) f t ( η, η ) − Hess γ ( t ) f t ( T η, T η ) ( l − t − t − ( t − t ) | γ + | · (cid:0) g γ ( t ) ( η ) − ǫ (cid:1) · l + l + 13(A.5)for any l > . Set τ = t − t and G = | γ + ( t ) | · ( g γ ( t ) ( η ) − ǫ ). By choosing l = (cid:0) /τ + τ G/ (cid:1) · (cid:0) /τ − τ G/ − Hess γ ( t ) f t ( η, η ) (cid:1) − (when τ suffices small, 1 /τ − τ G/ − Hess γ ( t ) f t ( η, η ) and l are positive), we get (cid:0) τ − τ G/ (cid:1) · (cid:0) Hess γ ( t ) f t ( η, η ) − Hess γ ( t ) f t ( T η, T η ) (cid:1) − Hess γ ( t ) f t ( η, η ) · Hess γ ( t ) f t ( T η, T η ) − G + τ G / . (A.6) ICCI CURVATURE ON ALEXANDROV SPACES 31 Note the simple fact that for an bilinear form β ( a, a ) on a m − dimensional inner productspace V m , trace V m β = mvol ( S ) Z S β ( a, a ) da, where S is the unit sphere of V m with canonical measure. By taking trace for Hess γ ( t ) f t ( and Hess γ ( t ) f t ) in L γ ( t ) (and L γ ( t ) , respectively), we get, from (A.2) and (A.6), that h V ( t ) − h V ( t ) τ − n − (cid:0) h V ( t ) + h V ( t ) (cid:1) − ( n − K − ǫ ) | γ + ( t ) | + o (1)(A.7)when we fix t and let t → t . On the other hand, by setting l = 1 in (A.5) and taking trace, we have h V ( t ) − h V ( t ) τ − ( n − K − ǫ ) | γ + ( t ) | . This and (A.7) tell us that h V is locally Lipschitz almost everywhere in (0,1).By using (A.7), the arbitrariness of ǫ and Fact A (iii), we get h ′ V − ( n − K |∇ f t | − h V n − . Therefore, we have completed the proof of this proposition. (cid:3) Now we can follow Petrunin’s argument in [Pet2] to prove the above Proposition A.2. Proof of Proposition A.2. Let µ , µ ∈ P ( M, d, m ) with compactly supported sets spt ( µ ) , spt ( µ )and µ t ∈ P ( M, d ) be a geodesic path. We have spt ( µ t ) ⊂ [ x ∈ spt ( µ ) , y ∈ spt ( µ ) γ x,y ∀ t ∈ [0 , , where γ x,y is any one geodesic path between x and y . Thus we can choose a big enoughball B such that spt ( µ t ) ⊂ B for all t ∈ [0 , k such that M has curvature > k in B .As shown in [V, 7.22], there is a probability measure Π on the space of all geodesicpaths in M such that if Γ = spt (Π) and e t : Γ → M is evaluation map e t ( γ ) = γ ( t ) then µ t = ( e t ) Π. Let Γ be equipped a metric | γ γ ′ | Γ := max t ∈ [0 , | γ ( t ) γ ′ ( t ) | . According to [V, 5.10], there are a pair of optimal price functions φ and ψ on M suchthat φ ( y ) − ψ ( x ) | xy | for any x, y ∈ M and equality holds for any ( x, y ) ∈ spt (cid:0) ( e , e ) Π (cid:1) . By considering the Hamilton–Jacobi shifts ψ t = H t ψ and φ t = H − t ( − ψ ) , Petrunin in [Pet2] proved that, for any t ∈ (0 , µ t is absolutely continuous and theevaluation map e t is bi-Lipschitz (where the bi-Lipschitz constant depends on k ). Hencefor any measure χ on M , there is uniquely determined one-parameter family of pull-backmeasures χ ∗ t on Γ such that χ ∗ t ( E ) = χ ( e t E ) for any Borel subset E ⊂ Γ . (Refer to [Pet2]for details), Fix the measure e ν = vol ∗ t =1 / on Γ. We write vol ∗ t = e w t · e ν for some Borel function w t : Γ → R , since e t is bi-Lipschitz and vol ∗ t is absolutely continuous with respect to e ν forany t ∈ (0 , − a.e. γ ∈ Γ,(A.8) w t = Z tt ∂w s ∂s ds a . e . t ∈ (0 , ∂w t ∂t = h t a . e . t ∈ (0 , h t ( γ ) = T raceHess γ ( t ) φ t . Noting that h t = h T ( t ) + h V ( t ), we set w (1) t = Z tt h T ( s ) ds, B ( t ) = exp( w (1) t )and w (2) t = Z tt h V ( s ) ds, B ( t ) = exp (cid:0) w (2) t n − (cid:1) . By applying Proposition A.4, we get B ( t ) > (1 − t ) B (0) + tB (1) ,B ( t ) > (1 − t ) β / ( n − − t B (0) + tβ / ( n − t B (1) , (A.11)where β t = β (cid:0) ( n − K,n (cid:1) t ( γ (0) , γ (1)) . Setting D ( t ) = exp( w t /n ) and using H¨older inequality (cid:0) a + b (cid:1) /n · ( c + d ) ( n − /n > a /n · c ( n − /n + b /n · d ( n − /n ∀ a, b, c, d > , we have D ( t ) = B /n · B ( n − /n > (cid:16) (1 − t ) B (0) + tB (1) (cid:17) n · (cid:16) (1 − t ) β n − − t B (0) + tβ n − t B (1) (cid:17) n − n > (1 − t ) β /n − t B (0) B (0) ( n − /n + tβ /nt B (1) B (1) ( n − /n = (1 − t ) β /n − t D (0) + tβ /nt D (1) . (A.12)Note that Petrunin in [Pet2] had represented H n ( µ t | m ) in terms of w t ( γ ) as following, H n ( µ t | m ) = − Z Γ exp( w t ( γ ) /n ) · ad Πfor some non-negative Borel function a : Γ → R . The combination of this with (A.12)implies the desired inequality (A.1) in the definition of CD ( n, ( n − K ). Therefore we havecompleted the proof of Proposition A.2. (cid:3) ICCI CURVATURE ON ALEXANDROV SPACES 33 References [A] M. Anderson, On the popology of complete manifolds of non-negative Ricci curvature , Topology, 29(1),41-55 (1990).[AB] S. Alexander, R. 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