One dimensional weighted Ricci curvature and displacement convexity of entropies
OONE DIMENSIONAL WEIGHTED RICCI CURVATUREAND DISPLACEMENT CONVEXITY OF ENTROPIES
YOHEI SAKURAI
Abstract.
In the present paper, we prove that a lower 1-weightedRicci curvature bound is equivalent to a convexity of entropies onthe Wasserstein space. Based on such characterization, we provideinequalities of Brunn-Minkowski type, Pr´ekopa-Leindler type andseveral functional inequalities under the curvature bound. Introduction
The aim of this article is to characterize a lower 1-weighted Riccicurvature bound in terms of a convexity of entropies on the Wasser-stein space. Such characterization enables us to produce inequalitiesof Brunn-Minkowski type, Pr´ekopa-Leindler type and some functionalinequalities under the lower 1-weighted Ricci curvature bound.For n ≥
2, let (
M, d, m ) be an n -dimensional weighted Riemannianmanifold, namely, M = ( M, g ) is an n -dimensional complete Riemann-ian manifold (without boundary), d is the Riemannian distance on M ,and m := e − f vol for some smooth f : M → R , where vol denotes theRiemannian volume measure on M . For N ∈ ( −∞ , ∞ ], the associated N -weighted Ricci curvature Ric Nf is defined as follows ([2], [8]):(1.1) Ric Nf := Ric g + Hess f − df ⊗ dfN − n , where Ric g is the Ricci curvature determined by g , and df and Hess f are the differential and the Hessian of f , respectively. For F : M → R ,we mean by Ric Nf,M ≥ F for every x ∈ M , and for every unit tangentvector v at x we have Ric Nf ( v ) ≥ F ( x ). Traditionally, the parameter N has been chosen from [ n, ∞ ], and in that case, we have already knownmany geometric and analytic properties (see e.g., [9], [19], [25]). On theother hand, very recently, in the complementary case of N ∈ ( −∞ , n ),various properties have begun to be studied (see e.g., [6], [7], [13], [15],[16], [17], [18], [26], [27]). Date : May 2, 2018.2010
Mathematics Subject Classification.
Primary 53C21; Secondary 49Q20.
Key words and phrases.
Weighted Ricci curvature; Optimal transport theory. a r X i v : . [ m a t h . DG ] M a y YOHEI SAKURAI
It is well-known that lower N -weighted Ricci curvature bounds canbe characterized by convexities of entropies on the Wasserstein spacevia optimal transport theory. Let us consider a curvature condition(1.2) Ric Nf,M ≥ K for K ∈ R . In the traditional case of N ∈ [ n, ∞ ], the characterization ofthe curvature condition (1.2) is due to von Renesse and Sturm [20], andSturm [21] for N = ∞ , and Sturm [22], [23], and Lott and Villani [10],[11] for N ∈ [ n, ∞ ). Based on such characterization results, for gen-eral metric measure spaces, Sturm [22], [23], and Lott and Villani [10],[11] have independently introduced the so-called curvature-dimension-condition CD(
K, N ) for K ∈ R and N ∈ [1 , ∞ ] that is equivalent to thecondition (1.2) when N ∈ [ n, ∞ ] on weighted Riemannian manifolds.Metric measure spaces satisfying the curvature-dimension condition ormore restricted version called Riemannian-curvature-dimension condi-tion introduced by Ambrosio, Gigli and Savar´e [1], and Erbar, Kuwadaand Sturm [5] have been widely studied from various perspectives.In the complementary case of N ∈ ( −∞ , n ), Ohta [15] has recentlycharacterized the condition (1.2) for N ∈ ( −∞ , K, N ) for K ∈ R and N ∈ ( −∞ , N = 0.Now, we are concerned with the characterization problem of lower N -weighted Ricci curvature bounds in the case of N ∈ (0 , n ). We focuson the case of N = 1, especially a curvature condition(1.3) Ric f,M ≥ ( n − κ e − fn − for κ ∈ R introduced by Wylie and Yeroshkin [27] from the view pointof the study of weighted affine connections. Wylie and Yeroshkin [27]have observed that the curvature condition (1.3) is equivalent to a lowerRicci curvature bound by ( n − κ with respect to some weighted affineconnection. They further established comparison geometry under thecondition (1.3) (more precisely, see Subsection 2.1). In this paper, wewill prove that the curvature condition (1.3) can be characterized by aconvexity of entropies on the Wasserstein space. Using the equivalence,we conclude inequalities of Brunn-Minkowski type, Pr´ekopa-Leindlertype and several functional inequalities under the condition (1.3).1.1. Main result.
To state our main theorem, we introduce a convex-ity property of entropies on the Wasserstein space. Let (
M, d, m ) be an n -dimensional weighted Riemannian manifold, where m = e − f vol for NE DIMENSIONAL WEIGHTED RICCI CURVATURE 3 some smooth function f : M → R . Let P acc ( M ) be the set of all com-pactly supported Borel probability measures on M that are absolutelycontinuous with respect to m .Let P ( M ) denote the set of all Borel probability measures µ on M satisfying (cid:82) M d ( x, x ) dµ ( x ) < ∞ for some x ∈ M . Let DC stand forthe set of all continuous convex functions U : [0 , ∞ ) → R with U (0) = 0such that a function ϕ U : (0 , ∞ ) → R defined by ϕ U ( r ) := r n U ( r − n ) isconvex. For U ∈ DC , a functional U m on P ( M ) is defined by(1.4) U m ( µ ) := (cid:90) M U ( ρ ) dm, where ρ is the density of the absolutely continuous part in the Lebesguedecomposition of µ with respect to m . For a function H ∈ DC definedby H ( r ) := n r (1 − r − n ), the functional H m on P ( M ) defined as (1.4)is called the R´enyi entropy .In order to introduce our convexity property of entropies, we need todefine a twisted coefficient. For t ∈ [0 , d f,t , d f : M × M → R by(1.5) d f,t ( x, y ) := inf γ (cid:90) t d ( x,y )0 e − f ( γ ( ξ )) n − dξ, d f := d f, , where the infimum is taken over all unit speed minimal geodesics γ :[0 , d ( x, y )] → M from x to y . The function d f has been called the re-parametrize distance in [27] (cf. Subsection 2.1). In the unweightedcase of f = 0, we have d f,t = t d . Notice that for t ∈ (0 , d f,t is not necessarily distance since the triangle inequality does nothold in general. We also remark that for t ∈ (0 , d f,t is not always symmetric. For κ ∈ R , let s κ ( t ) be a unique solution ofthe Jacobi equation ψ (cid:48)(cid:48) ( t ) + κ ψ ( t ) = 0 with ψ (0) = 0 , ψ (cid:48) (0) = 1. For t ∈ [0 , β κ,f,t : M × M → R ∪ {∞} by(1.6) β κ,f,t ( x, y ) := (cid:18) s κ ( d f,t ( x, y )) ts κ ( d f ( x, y )) (cid:19) n − if d f ( x, y ) ∈ [0 , C κ ); otherwise, β κ,f,t ( x, y ) := ∞ , where C κ denotes thediameter of the space form of constant curvature κ . Furthermore, let β κ,f,t denote a function on M × M defined as(1.7) β κ,f,t ( x, y ) := β κ,f,t ( y, x ) . We introduce the following notion:
Definition 1.1.
For κ ∈ R , we say that ( M, d, m ) has κ -twisted cur-vature bound if for every pair µ , µ ∈ P acc ( M ), there are an optimal YOHEI SAKURAI coupling π of ( µ , µ ), and a minimal geodesic ( µ t ) t ∈ [0 , in the L -Wasserstein space from µ to µ such that for all U ∈ DC and t ∈ (0 , U m ( µ t ) ≤ (1 − t ) (cid:90) M × M U (cid:32) ρ ( x ) β κ,f, − t ( x, y ) (cid:33) β κ,f, − t ( x, y ) ρ ( x ) dπ ( x, y )(1.8) + t (cid:90) M × M U (cid:18) ρ ( y ) β κ,f,t ( x, y ) (cid:19) β κ,f,t ( x, y ) ρ ( y ) dπ ( x, y ) , where ρ i is the density of µ i with respect to m for each i = 0 , Definition 1.2.
For κ ∈ R , we say that ( M, d, m ) has κ -weak twistedcurvature bound if for every pair µ , µ ∈ P acc ( M ), there exist an op-timal coupling π of ( µ , µ ), and a minimal geodesic ( µ t ) t ∈ [0 , in the L -Wasserstein space from µ to µ such that for H ∈ DC defined as H ( r ) := n r (1 − r − n ), and for every t ∈ (0 ,
1) the inequality (1.8) holds.
Remark . In the unweighted case where f = 0, the notion of the κ -twisted curvature bound coincides with that of the curvature-dimensioncondition CD(( n − κ, n ) in the sense of Lott and Villani [10], [11].Similarly, the notion of the κ -weak twisted curvature bound coincideswith that of the curvature-dimension condition CD(( n − κ, n ) in thesense of Sturm [22], [23].Our main result is the following characterization theorem: Theorem 1.1.
Let ( M, d, m ) be an n -dimensional weighted Riemann-ian manifold, where m := e − f vol for some smooth function f : M → R . Let κ ∈ R . Then the following statements are equivalent: (1) Ric f,M ≥ ( n − κ e − fn − ; (2) ( M, d, m ) has κ -twisted curvature bound; (3) ( M, d, m ) has κ -weak twisted curvature bound. For K ∈ R and N ∈ [ n, ∞ ], Lott and Villani [11] have characterizedthe curvature condition (1.2) by a convexity of entropies on the Wasser-stein space (see Theorem 4.22 in [11]). The Lott-Villani theorem in aspecial case where f = 0 , K = ( n − κ and N = n states that thestatements (1) and (2) in Theorem 1.1 are equivalent when f = 0.For K ∈ R and N ∈ [ n, ∞ ), Sturm [23] has characterized a conditionthat Ric M ≥ K and n ≤ N (see Theorem 1.7 in [23]), where Ric M ≥ K means that for every x ∈ M , and for every unit tangent vector v at x we have Ric g ( v ) ≥ K . The Sturm theorem in the special case where NE DIMENSIONAL WEIGHTED RICCI CURVATURE 5 K = ( n − κ and N = n tells us that the statements (1) and (3) inTheorem 1.1 are equivalent when f = 0.One of the key ingredients of the proof of Theorem 1.1 is to obtaininequalities for Jacobians of optimal transport maps that are associatedwith Ric f . We first show an inequality of Riccati type (see Lemma 3.3).From the inequality of Riccati type, we derive an inequality concerningthe concavity of the Jacobians under the curvature condition (1.3) (seeProposition 3.1). By using the concavity, we prove that the curvaturecondition (1.3) implies the convexity of entropies.1.2. Organization.
In Section 2, we review the works done by Wylieand Yeroshkin [27], and also recall basics of the optimal transport the-ory. In Section 3, we show key inequalities for the proof of Theorem 1.1.In Section 4, we prove Theorem 1.1. Furthermore, under the curvaturecondition (1.3), we conclude inequalities of Brunn-Minkowski type andPr´ekopa-Leindler type (see Corollaries 4.4 and 4.5). In Section 5, wepresent some applications of Theorem 1.1 (see Corollaries 5.4, 5.5, 5.6).2.
Preliminaries
Hereafter, for n ≥
2, let (
M, d, m ) denote an n -dimensional weightedRiemannian manifold, namely, M = ( M, g ) is an n -dimensional com-plete Riemannian manifold (without boundary), d is the Riemanniandistance on M , and m := e − f vol for some smooth function f : M → R ,where vol is the Riemannian volume measure on M .2.1. Geometric analysis on -weighted Ricci curvature. In thissubsection, we briefly recall the work done by Wylie and Yeroshkin [27]concerning the curvature condition (1.3).Wylie and Yeroshkin [27] have suggested a new approach to investi-gate geometric properties of weighted manifolds. Let α be a 1-form on M . The basic tool in [27] was a torsion free affine connection ∇ α U V := ∇ U V − α ( U ) V − α ( V ) U , where ∇ denotes the Levi-Civita connection induced from g . They havestudied weighted manifolds in view of this weighted affine connection.They have examined the relation between the 1-weighted Ricci cur-vature and the Ricci curvature induced from ∇ α . The ∇ α -curvaturetensor and the ∇ α -Ricci tensor are defined as R ∇ α ( U , V ) W := ∇ α U ∇ α V W − ∇ α V ∇ α U W − ∇ α [ U , V ] W , Ric ∇ α ( U , V ) := n (cid:88) i =1 g (cid:0) R ∇ α ( e i , U ) V , e i (cid:1) , (2.1) YOHEI SAKURAI where { e i } ni =1 is an orthonormal basis with respect to g . Let us considera closed 1-form α f on M defined by α f := dfn − . The first key observation in [27] is that Ric ∇ αf coincides with the 1-weighted Ricci tensor Ric f defined as (1.1) (see Proposition 3.3 in [27]).They also investigated geodesics for ∇ α . For x ∈ M , we denote by U x M the unit tangent sphere at x . For v ∈ U x M , let γ v : [0 , ∞ ) → M be the ( ∇ -)geodesic with initial conditions γ v (0) = x and γ (cid:48) v (0) = v .We now define a function s f,v : [0 , ∞ ] → [0 , S f,v, ∞ ] by s f,v ( t ) := (cid:90) t e − f ( γv ( ξ )) n − dξ, S f,v, ∞ := (cid:90) ∞ e − f ( γv ( ξ )) n − dξ. Let t f,v : [0 , S f,v, ∞ ] → [0 , ∞ ] be the inverse function of s f,v . The secondkey observation in [27] is that a curve (cid:98) γ f,v : [0 , S f,v, ∞ ) → M defined as (cid:98) γ f,v := γ v ◦ t f,v is a ∇ α f -geodesic (see Proposition 3.1 in [27]).Summarizing the above two key observations, Wylie and Yeroshkin[27] have concluded the following interpretation of the curvature condi-tion (1.3) in terms of the ∇ α f -Ricci curvature Ric ∇ αf defined as (2.1): Proposition 2.1 ([27]) . For κ ∈ R , the following are equivalent: (1) Ric f ( γ (cid:48) v ( t )) ≥ ( n − κ e − f ( γv ( t )) n − for all v ∈ U x M and t ∈ [0 , ∞ ) ; (2) Ric ∇ αf ( (cid:98) γ (cid:48) f,v ( s )) ≥ ( n − κ for all v ∈ U x M and s ∈ [0 , S f,v, ∞ ) . Keeping in mind Proposition 2.1, Wylie and Yeroshkin [27] has de-veloped comparison geometry under the curvature condition (1.3). Be-fore the work of them, Wylie [26] has obtained a splitting theorem ofCheeger-Gromoll type under the condition Ric
Nf,M ≥ N ∈ ( −∞ , d f definedas (1.5), a maximal diameter theorem of Cheng type for the deformedmetric e − fn − g , and a volume comparison of Bishop-Gromov type for theweighted volume measure e − n +1 n − f vol under the condition (1.3).For later convenience, we will review the diameter comparison. For x ∈ M , we denote by d x : M → R the distance function from x definedas d x ( y ) := d ( x, y ). For v ∈ U x M we set(2.2) τ x ( v ) := sup { t > | d x ( γ v ( t )) = t } , τ f,x ( v ) := s f,v ( τ x ( v )) . Wylie and Yeroshkin [27] have obtained the following comparison forthe re-parametrized distance d f (see Theorem 2.2 in [27]): NE DIMENSIONAL WEIGHTED RICCI CURVATURE 7
Theorem 2.2 ([27]) . For κ > , if Ric f,M ≥ ( n − κ e − fn − , then forall x ∈ M and v ∈ U x M we have τ f,x ( v ) ≤ π √ κ . Moreover, for the re-parametrized distance d f defined as (1 . , we have sup x,y ∈ M d f ( x, y ) ≤ π √ κ . Optimal transport.
We review some basic facts of the optimaltransport theory in our setting. We refer to [4], [12] (see also [14], [24]).Let P ( M ) be the set of all Borel probability measure on M . We de-note by c : M × M → R a cost function defined as c ( x, y ) := d ( x, y ) / µ, ν ∈ P ( M ) we consider a value inf F (cid:82) M c ( x, F ( x )) dµ ( x ), wherethe infimum is taken over all Borel measurable maps F : M → M suchthat the pushforward measure F µ of µ by F coincides with ν . A Borelmeasurable map F is said to be an optimal transport map from µ to ν if it attains the infimum.For µ, ν ∈ P ( M ) a Borel probability measure π on M × M is said tobe a coupling of ( µ, ν ) if π ( X × M ) = µ ( X ) and π ( M × X ) = ν ( X ) forall Borel subsets X ⊂ M . Let Π( µ, ν ) denote the set of all couplingsof ( µ, ν ). Let us consider a value inf π ∈ Π( µ,ν ) (cid:82) M × M c ( x, y ) dπ ( x, y ). Acoupling π ∈ Π( µ, ν ) is called optimal if it attains the infimum.Let X, Y ⊂ M be compact, and let φ : X → R ∪ {−∞} be a functionthat is not identically −∞ . The c -transformation φ c : Y → R ∪ {−∞} of φ relative to ( X, Y ) is defined as φ c ( y ) := inf x ∈ X { c ( x, y ) − φ ( x ) } .The function φ is said to be c -concave relative to ( X, Y ) if φ = ψ c forsome ψ : Y → R ∪ {−∞} with ψ (cid:54)≡ −∞ . If φ is c -concave relative to( X, Y ), then it is Lipschitz, and t φ is also c -concave for every t ∈ [0 , Theorem 2.3 ([3], [12]) . Let µ ∈ P acc ( M ) , and let ν ∈ P ( M ) be com-pactly supported. Let supp µ and supp ν denote their supports. Takecompact subsets X, Y ⊂ M with supp µ ⊂ X, supp ν ⊂ Y . Then thereis a c -concave function φ relative to ( X, Y ) such that a map F on X defined by F ( z ) := exp z ( −∇ φ ( z )) gives a unique optimal transport mapfrom µ to ν , where exp z is the exponential map at z , and ∇ φ is the gra-dient of φ . Moreover, for the identity map Id M of M , the pushforwardmeasure (Id M × F ) µ is a unique optimal coupling of ( µ, ν ) . For a relatively compact, open subset V ⊂ M , let X be its closure V . Let Y ⊂ M be compact, and let φ be a c -concave function relativeto ( X, Y ). It is well-known that the function φ is twice differentiable YOHEI SAKURAI m -almost everywhere on V due to the Alexandorov-Bangert theorem.For a map F on V defined as F ( z ) := exp z ( −∇ φ ( z )), if φ is twicedifferentiable at x ∈ V , then F ( x ) does not belong to the cut locusCut x of x , and the differential ( dF ) x of F at x is well-defined.We further recall the following: Theorem 2.4 ([4]) . Let µ, ν ∈ P acc ( M ) . Take relatively compact, opensubsets V, W ⊂ M with supp µ ⊂ V, supp ν ⊂ W . Assume that a c -concave function φ relative to ( V , W ) satisfies F µ = ν , where F is amap on V defined as F ( z ) := exp z ( −∇ φ ( z )) . Then for µ -almost every x ∈ V the following hold: (1) φ is twice differential at x ; (2) for every t ∈ [0 , the determinant det ( dF t ) x of dF t at x is posi-tive, where F t is a map on V defined by F t ( z ) := exp z ( − t ∇ φ ( z )) ; (3) ρ ( x ) = ρ ( F ( x )) e − f ( F ( x ))+ f ( x ) det( dF ) x , where ρ and ρ arethe densities of µ and of ν with respect to m , respectively. Let (
Z, d Z ) be a metric space. A curve γ : [0 , l ] → Z is said to be a minimal geodesic if there exists a ≥ t , t ∈ [0 , l ] wehave d Z ( γ ( t ) , γ ( t )) = a | t − t | . Moreover, if a = 1, then γ is calleda unit speed minimal geodesic .Let W : P ( M ) × P ( M ) → R denote a function defined as W ( µ, ν ) := inf π ∈ Π( µ,ν ) (cid:18)(cid:90) M × M d ( x, y ) dπ ( x, y ) (cid:19) . It is well-known that ( P ( M ) , W ) is a complete separable metric space,and the metric space is called the L -Wasserstein space over M .We summarize some well-known facts for interpolants: Proposition 2.5.
Let µ ∈ P acc ( M ) , and let ν ∈ P ( M ) be compactlysupported. Take relatively compact, open subsets V, W ⊂ M with supp µ ⊂ V, supp ν ⊂ W . Let φ be a c -concave function relative to ( V , W ) . For each t ∈ [0 , we put µ t := ( F t ) µ , where F t is a map on V defined as F t ( z ) := exp z ( − t ∇ φ ( z )) . Then the map F t gives a uniqueoptimal transport map from µ to µ t . Moreover, if ν ∈ P acc ( M ) and µ = ν , then ( µ t ) t ∈ [0 , is a unique minimal geodesic in ( P ( M ) , W ) from µ to ν , and it lies in P acc ( M ) . Key inequalities
In the present section, we will prove the following key inequality forthe proof of Theorem 1.1:
NE DIMENSIONAL WEIGHTED RICCI CURVATURE 9
Proposition 3.1.
Let
V, W ⊂ M be relatively compact, open subsets,and let φ be a c -concave function relative to ( V , W ) . Fix a point x ∈ V .Assume that φ is twice differentiable at x , and det ( dF t ) x > for every t ∈ [0 , , where F t is a map on V defined as F t ( z ) := exp z ( − t ∇ φ ( z )) .For each t ∈ [0 , we put (3.1) J t ( x ) := e − f ( F t ( x ))+ f ( x ) det( dF t ) x . For κ ∈ R , if Ric f,M ≥ ( n − κ e − fn − , then for every t ∈ (0 , J t ( x ) n ≥ (1 − t ) β κ,f, − t ( x, F ( x )) n J ( x ) n + t β κ,f,t ( x, F ( x )) n J ( x ) n , where β κ,f,t and β κ,f,t are defined as (1 . and (1 . , respectively. Throughout this section, as in Proposition 3.1, let
V, W ⊂ M denoterelatively compact, open subsets, and let φ denote a c -concave functionrelative to ( V , W ). Moreover, for a fixed point x ∈ V , we assume that φ is twice differentiable at x , and det ( dF t ) x > t ∈ [0 , Riccati inequalities.
Define a curve γ x : [0 , → M by γ x ( t ) := F t ( x ), and choose an orthonormal basis { e i } ni =1 of the tangent space at x with e n = γ (cid:48) x (0) / (cid:107) γ (cid:48) x (0) (cid:107) , where (cid:107) · (cid:107) is the canonical norm inducedfrom g . For each i = 1 , . . . , n , we define a Jacobi field E i along γ x by E i ( t ) := ( dF t ) x ( e i ). For each t ∈ [0 ,
1] let A ( t ) = ( a ij ( t )) be an n × n matrix determined by(3.2) E (cid:48) i ( t ) = n (cid:88) j =1 a ij ( t ) E j ( t ) . We define a function h x : [0 , → R by h x ( t ) := log det( dF t ) x − (cid:90) t a nn ( ξ ) dξ. It is well-known that the function h x satisfies the following inequalityof Riccati type (see e.g., [23], and Chapter 14 in [24]): Lemma 3.2.
For every t ∈ (0 , we have h (cid:48)(cid:48) x ( t ) ≤ − h (cid:48) x ( t ) n − − Ric g ( γ (cid:48) x ( t )) . We define a function l x : [0 , → R by(3.3) l x ( t ) := h x ( t ) − f ( γ x ( t )) + f ( x ) . For distance functions, Wylie and Yeroshkin [27] have obtained aninequality of Riccati type that is associated with Ric f (see Lemma 4.1in [27]). By using the same method, we have the following: Lemma 3.3.
For every t ∈ (0 , we have (cid:16) e f ( γx ( t )) n − l (cid:48) x ( t ) (cid:17) (cid:48) ≤ − e f ( γx ( t )) n − (cid:18) l (cid:48) x ( t ) n − f ( γ (cid:48) x ( t )) (cid:19) . Proof.
Put f x := f ◦ γ x . From Lemma 3.2 we deduce l (cid:48)(cid:48) x ( t ) = h (cid:48)(cid:48) x ( t ) − f (cid:48)(cid:48) x ( t ) ≤ − h (cid:48) x ( t ) n − − (Ric g ( γ (cid:48) x ( t )) + f (cid:48)(cid:48) x ( t ))= − l (cid:48) x ( t ) n − − l (cid:48) x ( t ) f (cid:48) x ( t ) n − − Ric f ( γ (cid:48) x ( t )) . Hence we have e − fx ( t ) n − (cid:16) e fx ( t ) n − l (cid:48) x ( t ) (cid:17) (cid:48) = l (cid:48)(cid:48) x ( t ) + 2 l (cid:48) x ( t ) f (cid:48) x ( t ) n − ≤ − l (cid:48) x ( t ) n − − Ric f ( γ (cid:48) x ( t )) . This proves the desired inequality. (cid:50)
Jacobian inequalities.
We recall the following elementary com-parison argument (see e.g., Theorem 14.28 in [24]):
Lemma 3.4.
For a > , let D : [0 , a ] → R be a non-negative contin-uous function that is C on (0 , a ) . Take κ ∈ R and d ≥ . Assume κ d ∈ ( −∞ , a − π ) . Then D (cid:48)(cid:48) + κ d D ≤ on (0 , a ) if and only if forall s , s ∈ [0 , a ] and λ ∈ [0 , , D ((1 − λ ) s + λs ) ≥ s κ ((1 − λ ) | s − s | d ) s κ ( | s − s | d ) D ( s )+ s κ ( λ | s − s | d ) s κ ( | s − s | d ) D ( s ) . We define a function D x : [0 , → R by D x ( t ) := exp (cid:18) l x ( t ) n − (cid:19) . Lemmas 3.3 and 3.4 yield the following concavity of the function D x : Lemma 3.5.
For κ ∈ R , if Ric f,M ≥ ( n − κ e − fn − , then for every t ∈ (0 , we have D x ( t ) ≥ s κ ( d f, − t ( F ( x ) , x )) s κ ( d f ( F ( x ) , x )) D x (0) + s κ ( d f,t ( x, F ( x ))) s κ ( d f ( x, F ( x ))) D x (1) , where d f,t and d f are defined as (1 . .Proof. We define a function s f,x : [0 , → R by s f,x ( t ) := (cid:90) t e − f ( γx ( ξ )) n − dξ. NE DIMENSIONAL WEIGHTED RICCI CURVATURE 11
Put a := s f,x (1), and let t f,x : [0 , a ] → [0 ,
1] be the inverse function of s f,x . We define functions (cid:98) l x , (cid:98) D x : [0 , a ] → R by (cid:98) l x := l x ◦ t f,x , (cid:98) D x := D x ◦ t f,x , where l x is defined as (3.3). For each s ∈ (0 , a ) we see(3.4) ( n − (cid:98) D (cid:48)(cid:48) x ( s ) (cid:98) D x ( s ) = (cid:98) l (cid:48)(cid:48) x ( s ) + (cid:98) l (cid:48) x ( s ) n − . We also define functions L x : [0 , → R and (cid:98) L x : [0 , a ] → R by L x ( t ) := e f ( γx ( t )) n − l (cid:48) x ( t ) , (cid:98) L x := L x ◦ t f,x . Note that (cid:98) l (cid:48) x ( s ) = (cid:98) L x ( s ). From Lemma 3.3, it follows that (cid:98) l (cid:48)(cid:48) x ( s ) = (cid:98) L (cid:48) x ( s ) = t (cid:48) f,x ( s ) L (cid:48) x ( t f,x ( s ))(3.5) ≤ − e f ( γx ( tf,x ( s ) )) n − (cid:18) l (cid:48) x ( t f,x ( s )) n − f ( γ (cid:48) x ( t f,x ( s ))) (cid:19) = − (cid:98) l (cid:48) x ( s ) n − − e f ( γx ( tf,x ( s ) )) n − Ric f ( γ (cid:48) x ( t f,x ( s ))) . Combining (3.4) and (3.5), we obtain( n − (cid:98) D (cid:48)(cid:48) x ( s ) (cid:98) D x ( s ) ≤ − e f ( γx ( tf,x ( s ) )) n − Ric f ( γ (cid:48) x ( t f,x ( s ))) ≤ − ( n − κ d ( x, y ) , where y := F ( x ). Therefore, (cid:98) D (cid:48)(cid:48) x + κ d ( x, y ) (cid:98) D x ≤ , a ).Since the c -concave function φ is twice differentiable at x , the curve γ x lies in the complement of Cut x . In particular, γ x is a unique minimalgeodesic from x to y , and hence a d ( x, y ) = d f ( x, y ) < τ f,x (cid:18) γ (cid:48) x (0) (cid:107) γ (cid:48) x (0) (cid:107) (cid:19) , where τ f,x is defined as (2.2). By Theorem 2.2, κ d ( x, y ) ∈ ( −∞ , a − π ).Lemma 3.4 implies that for all s , s ∈ [0 , a ] and λ ∈ [0 , (cid:98) D x ((1 − λ ) s + λs ) ≥ s κ ((1 − λ ) | s − s | d ( x, y )) s κ ( | s − s | d ( x, y )) (cid:98) D x ( s )(3.6) + s κ ( λ | s − s | d ( x, y )) s κ ( | s − s | d ( x, y )) (cid:98) D x ( s ) . For every s ∈ (0 , a ) we obtain (cid:98) D x ( s ) ≥ s κ (( a − s ) d ( x, y )) s κ ( a d ( x, y )) (cid:98) D x (0) + s κ ( s d ( x, y )) s κ ( a d ( x, y )) (cid:98) D x ( a ) by letting s → , s → a and λ → s/a in (3.6). For every t ∈ (0 , D x ( t ) ≥ s κ (( a − s f,x ( t )) d ( x, y )) s κ ( a d ( x, y )) D x (0) + s κ ( s f,x ( t ) d ( x, y )) s κ ( a d ( x, y )) D x (1) . From the uniqueness of the geodesic γ x , for every t ∈ [0 ,
1] we see( a − s f,x ( t )) d ( x, y ) = d f, − t ( y, x ) , s f,x ( t ) d ( x, y ) = d f,t ( x, y ) . This completes the proof. (cid:50)
We define a function D x : [0 , → R by D x ( t ) := exp (cid:18)(cid:90) t a nn ( ξ ) dξ (cid:19) , where a nn is determined by (3.2). Notice that for every t ∈ (0 , J t ( x ) = D x ( t ) n − D x ( t ) , where J t ( x ) is defined as (3.1).The following concavity of the function D x is well-known (see e.g.,[23], and Chapter 14 in [24]): Lemma 3.6.
For every t ∈ (0 , we have D x ( t ) ≥ (1 − t ) D x (0) + t D x (1) . Now, we prove Proposition 3.1.
Proof of Proposition 3.1.
For κ ∈ R , we assume Ric f,M ≥ ( n − κ e − fn − .From (3.7) we deduce that for every t ∈ (0 , J t ( x ) n = D x ( t ) − n D x ( t ) n . By Lemmas 3.5 and 3.6, and by the H¨older inequality, we obtain J t ( x ) n ≥ (1 − t ) n (cid:18) s κ (( d f, − t ( F ( x ) , x )) s κ ( d f ( F ( x ) , x )) (cid:19) − n J ( x ) n + t n (cid:18) s κ ( d f,t ( x, F ( x ))) s κ ( d f ( x, F ( x ))) (cid:19) − n J ( x ) n . The right hand side is equal to that of the desired one. Therefore, weconclude the proposition. (cid:50) Displacement convexity
In this section, we prove Theorem 1.1 by using Proposition 3.1, andthe same argument as that in the proof of Theorem 1.7 in [23].
NE DIMENSIONAL WEIGHTED RICCI CURVATURE 13
Curvature bounds imply displacement convexity.
First, weprove the following part of Theorem 1.1:
Proposition 4.1.
For κ ∈ R , if Ric f,M ≥ ( n − κ e − fn − , then ( M, d, m ) has κ -twisted curvature bound.Proof. Fix µ , µ ∈ P acc ( M ). Take relatively compact, open subsets V, W ⊂ M with supp µ ⊂ V, supp µ ⊂ W . Due to Theorem 2.3, thereexists a c -concave function φ relative to ( V , W ) such that a map F on V defined as F ( z ) := exp z ( −∇ φ ( z )) gives a unique optimal transportmap from µ to µ . Moreover, π := (Id M × F ) µ is a unique optimalcoupling of ( µ , µ ). For each t ∈ [0 ,
1] we define a map F t on V as F t ( z ) := exp z ( − t ∇ φ ( z )), and put µ t := ( F t ) µ . By Lemma 2.5, F t isa unique optimal transport map from µ to µ t , and µ t ∈ P acc ( M ). Let ρ t stand for the density of µ t with respect to m .We fix t ∈ (0 , µ -almost every x ∈ V thefollowing hold: (1) φ is twice differential at x ; (2) det ( dF u ) x > u ∈ [0 , ρ ( x ) = ρ ( F ( x )) J ( x ) = ρ t ( F t ( x )) J t ( x )hold, where J ( x ) and J t ( x ) are defined as (3.1). Proposition 3.1 impliesthat for µ -almost every x ∈ V we have(4.2) J t ( x ) n ≥ (1 − t ) β κ,f, − t ( x, F ( x )) n J ( x ) n + t β κ,f,t ( x, F ( x )) n J ( x ) n . Fix U ∈ DC . By using µ t = ( F t ) µ and (4.1), we see U m ( µ t ) = (cid:90) M U (cid:18) ρ ( x ) J t ( x ) (cid:19) J t ( x ) ρ ( x ) dµ ( x )= (cid:90) M ϕ U (cid:32)(cid:18) J t ( x ) ρ ( x ) (cid:19) n (cid:33) dµ ( x ) . where ϕ U denotes the function defined as ϕ U ( r ) := r n U ( r − n ). Noticethat ϕ U is non-increasing and convex. From (4.2) we derive U m ( µ t ) ≤ (1 − t ) (cid:90) M ϕ U (cid:32) β κ,f, − t ( x, F ( x )) n (cid:18) J ( x ) ρ ( x ) (cid:19) n (cid:33) dµ ( x )+ t (cid:90) M ϕ U (cid:32) β κ,f,t ( x, F ( x )) n (cid:18) J ( x ) ρ ( x ) (cid:19) n (cid:33) dµ ( x ) . Using the Jacobian equation (4.1) again, we obtain U m ( µ t ) ≤ (1 − t ) (cid:90) M ϕ U (cid:32) β κ,f, − t ( x, F ( x )) ρ ( x ) (cid:33) n dµ ( x )(4.3) + t (cid:90) M ϕ U (cid:32)(cid:18) β κ,f,t ( x, F ( x )) ρ ( F ( x )) (cid:19) n (cid:33) dµ ( x ) . Since π = (Id M × F ) µ , the right hand side of (4.3) is equal to thatof (1.8). We complete the proof. (cid:50) Displacement convexity implies curvature bounds.
For sub-sets
X, Y ⊂ M and t ∈ [0 , Z t ( X, Y ) be the set of all points γ ( t ),where γ : [0 , → M is a minimal geodesic with γ (0) ∈ X, γ (1) ∈ Y .Let us show that the weak twisted curvature bound implies the fol-lowing inequality of Brunn-Minkowski type: Lemma 4.2.
Let
X, Y ⊂ M denote two bounded Borel subsets with m ( X ) , m ( Y ) ∈ (0 , ∞ ) . For κ ∈ R , if ( M, d, m ) has κ -weak twistedcurvature bound, then for every t ∈ (0 , we have m ( Z t ( X, Y )) n ≥ (1 − t ) (cid:18) inf ( x,y ) ∈ X × Y β κ,f, − t ( x, y ) n (cid:19) m ( X ) n (4.4) + t (cid:18) inf ( x,y ) ∈ X × Y β κ,f,t ( x, y ) n (cid:19) m ( Y ) n . Proof.
Let 1 X and 1 Y be the characteristic functions of X and of Y ,respectively. We set ρ := 1 X m ( X ) , µ := ρ m, ρ := 1 Y m ( Y ) , µ := ρ m. By Proposition 2.5, there exists a unique minimal geodesic ( µ t ) t ∈ [0 , in( P ( M ) , W ) from µ to µ , and it lies in P acc ( M ). For each t ∈ (0 , ρ t stand for the density of µ t with respect to m . From the Jenseninequality one can derive m ( Z t ( X, Y )) n ≥ (cid:90) M ρ t ( x ) − n dm ( x ) . Since (
M, d, m ) has κ -weak twisted curvature bound, we have (cid:90) M ρ t ( x ) − n dm ( x ) ≥ (1 − t ) (cid:90) M × M ρ ( x ) − n β κ,f, − t ( x, y ) n dπ ( x, y )(4.5) + t (cid:90) M × M ρ ( y ) − n β κ,f,t ( x, y ) n dπ ( x, y ) , NE DIMENSIONAL WEIGHTED RICCI CURVATURE 15 where π is a unique optimal coupling of ( µ , µ ). The coupling π issupported on X × Y . Hence, the right hand side of (4.5) is boundedfrom below by(1 − t ) (cid:18) inf ( x,y ) ∈ X × Y β κ,f, − t ( x, y ) n (cid:19) (cid:90) M ρ ( x ) − n dm ( x )+ t (cid:18) inf ( x,y ) ∈ X × Y β κ,f,t ( x, y ) n (cid:19) (cid:90) M ρ ( y ) − n dm ( y )that is equal to the right hand side of (4.4). This proves the lemma. (cid:50) We next prove the following part of Theorem 1.1:
Proposition 4.3.
For κ ∈ R , if ( M, d, m ) has κ -weak twisted curvaturebound, then Ric f,M ≥ ( n − κ e − fn − .Proof. Fix x ∈ M and v ∈ U x M . For (cid:15) >
0, let γ : ( − (cid:15), (cid:15) ) → M be thegeodesic with γ (0) = x, γ (cid:48) (0) = v . Take δ ∈ (0 , (cid:15) ) and η ∈ (0 , δ ). For y ∈ M , we denote by B η ( y ) the open geodesic ball of radius η centeredat y . We set X := B η ( γ ( − δ )) and Y := B η ( γ ( δ )). From Lemma 4.2we deduce m (cid:16) Z ( X, Y ) (cid:17) n ≥ (cid:18) inf ( x,y ) ∈ X × Y β κ,f, ( x, y ) n (cid:19) m ( X ) n + 12 (cid:18) inf ( x,y ) ∈ X × Y β κ,f, ( x, y ) n (cid:19) m ( Y ) n , where the functions β κ,f, and β κ,f, are defined as (1.6) and (1.7),respectively. By letting η → η → m (cid:16) Z ( X, Y ) (cid:17) ω n η n n ≥ (cid:16) e − f ( γ ( − δ )) β κ,f, ( γ ( − δ ) , γ ( δ )) (cid:17) n (4.6) + 12 (cid:16) e − f ( γ ( δ )) β κ,f, ( γ ( − δ ) , γ ( δ )) (cid:17) n , where ω n denotes the volume of the unit ball in R n . Let us recall that the function d f,t is defined as (1.5). By the defini-tion of d f,t we see d f, ( γ ( δ ) , γ ( − δ )) = (cid:90) δ e − f ( γ ( ξ )) n − dξ,d f, ( γ ( − δ ) , γ ( δ )) = (cid:90) − δ e − f ( γ ( ξ )) n − dξ,d f ( γ ( − δ ) , γ ( δ )) = (cid:90) δ − δ e − f ( γ ( ξ )) n − dξ. Hence, the Taylor series of β κ,f, ( γ ( − δ ) , γ ( δ )) and β κ,f, ( γ ( − δ ) , γ ( δ ))with respect to δ at 0 are given as follows: β κ,f, ( γ ( − δ ) , γ ( δ )) = 1 − g (( ∇ f ) x , v ) δ + ( n − κ e − f ( x ) n − δ n − n − g (( ∇ f ) x , v ) δ O ( δ ) ,β κ,f, ( γ ( − δ ) , γ ( δ )) = 1 + g (( ∇ f ) x , v ) δ + ( n − κ e − f ( x ) n − δ n − n − g (( ∇ f ) x , v ) δ O ( δ ) . On the other hand, e − f ( γ ( − δ ))+ f ( x ) = 1 + g (( ∇ f ) x , v ) δ + (cid:0) g (( ∇ f ) x , v ) − Hess f ( v, v ) (cid:1) δ O ( δ ) ,e − f ( γ ( δ ))+ f ( x ) = 1 − g (( ∇ f ) x , v ) δ + (cid:0) g (( ∇ f ) x , v ) − Hess f ( v, v ) (cid:1) δ O ( δ ) . Substituting these series into (4.6), we havelim inf η → m (cid:16) Z ( X, Y ) (cid:17) ω n η n ≥ e − f ( x ) (cid:18) n − κ e − f ( x ) n − δ (cid:19) (4.7) + e − f ( x ) (cid:18) − Hess f ( v, v ) + g (( ∇ f ) x , v ) − n (cid:19) δ O ( δ ) . We recall the following fundamental inequality (see e.g., [23]):(4.8) lim sup η → m (cid:16) Z ( X, Y ) (cid:17) ω n η n ≤ e − f ( x ) (cid:18) g ( v ) δ (cid:19) + O ( δ ) . NE DIMENSIONAL WEIGHTED RICCI CURVATURE 17
Comparing (4.7) with (4.8), we obtainRic g ( v ) ≥ ( n − κ e − f ( x ) n − − Hess f ( v, v ) + g (( ∇ f ) x , v ) − n ;in particular, Ric f ( v ) ≥ ( n − κ e − f ( x ) n − . This completes the proof. (cid:50) We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
By Propositions 4.1 and 4.3, we complete theproof of Theorem 1.1. (cid:50)
Brunn-Minkowski and Pr´ekopa-Leindler inequalities.
Dueto Theorem 1.1 and Lemma 4.2, we obtain the following inequality ofBrunn-Minkowski type under the curvature condition (1.3):
Corollary 4.4.
Let
X, Y ⊂ M denote two bounded Borel subsets with m ( X ) , m ( Y ) ∈ (0 , ∞ ) . For κ ∈ R , if Ric f,M ≥ ( n − κ e − fn − , then forevery t ∈ (0 , we have m ( Z t ( X, Y )) n ≥ (1 − t ) (cid:18) inf ( x,y ) ∈ X × Y β κ,f, − t ( x, y ) n (cid:19) m ( X ) n + t (cid:18) inf ( x,y ) ∈ X × Y β κ,f,t ( x, y ) n (cid:19) m ( Y ) n . Let t ∈ (0 ,
1) and a, b ∈ [0 , ∞ ). For p ∈ R \ { } we define M pt ( a, b ) := ((1 − t ) a p + t b p ) p if ab (cid:54) = 0, and M pt ( a, b ) := 0 if ab = 0. As the limits, we further define M t ( a, b ) := a − t b t , M −∞ t ( a, b ) := min { a, b } . We also have the following inequality of Pr´ekopa-Leindler type (cf.Corollary 1.1 in [4], Corollary 9.4 in [14] and Theorem 19.18 in [24]):
Corollary 4.5.
For i = 0 , , let ψ i : M → R be non-negative, com-pactly supported, integrable functions. We take relatively compact, opensubsets V, W ⊂ M with supp ψ ⊂ V, supp ψ ⊂ W , and put X := V and Y := W . Let ψ : M → R be a non-negative function. For t ∈ (0 , and p ≥ − /n , we assume that for all ( x, y ) ∈ X × Y and z ∈ Z t ( X, Y ) , ψ ( z ) ≥ M pt (cid:32) ψ ( x ) β κ,f, − t ( x, y ) , ψ ( y ) β κ,f,t ( x, y ) (cid:33) . For κ ∈ R , if Ric f,M ≥ ( n − κ e − fn − , then we have (cid:90) M ψ dm ≥ M p np t (cid:18)(cid:90) M ψ dm, (cid:90) M ψ dm (cid:19) . Theorem 19.18 in [24] states that for K ∈ R and N ∈ [ n, ∞ ), the cur-vature condition (1.2) implies an inequality of Pr´ekopa-Leindler type.One can prove Corollary 4.5 only by replacing the role of Theorem 19.4in [24] with that of Proposition 3.1 in the proof. We omit the proof.5. Applications
In this last section, as applications of Theorem 1.1, we present severalfunctional inequalities under the curvature condition (1.3). Through-out this section, we always assume that M is compact, and the function f : M → R satisfies (cid:82) M e − f d vol = 1; in particular, m ∈ P ( M ).5.1. Derivatives of entropies.
For κ ∈ R , let us define a function (cid:98) β κ,f : M × M → R ∪ {∞} by(5.1) (cid:98) β κ,f ( x, y ) := (cid:32) e − f ( x ) n − d ( x, y ) s κ ( d f ( x, y )) (cid:33) n − if d f ( x, y ) ∈ [0 , C κ ); otherwise, (cid:98) β κ,f ( x, y ) := ∞ . Let c κ := s (cid:48) κ . We alsodefine a function (cid:101) β κ,f : M × M → R ∪ {∞} by(5.2) (cid:101) β κ,f ( x, y ) := n − n (cid:32) e − f ( x ) n − d ( x, y ) c κ ( d f ( x, y )) s κ ( d f ( x, y )) − (cid:33) if d f ( x, y ) ∈ [0 , C κ ); otherwise, (cid:101) β κ,f ( x, y ) := ∞ .We check the following basic properties of (cid:98) β κ,f and (cid:101) β κ,f : Lemma 5.1.
Let κ ∈ R . We take x, y ∈ M with d f ( x, y ) ∈ [0 , C κ ) .We assume y / ∈ Cut x . Then by letting t → we have β κ,f,t ( x, y ) → (cid:98) β κ,f ( x, y ) , (5.3) 1 − β κ,f, − t ( x, y ) n t → (cid:101) β κ,f ( x, y ) . (5.4) Proof.
First, we show (5.3). Since the point y does not belong to Cut x ,there exists a unique minimal geodesic γ : [0 , → M from x to y . Wedefine a function s f : [0 , → M by s f ( t ) := (cid:90) t e − f ( γ ( ξ )) n − dξ. The uniqueness of γ tells us that for every t ∈ [0 ,
1] we have d f,t ( x, y ) = s f ( t ) d ( x, y ). This implies s κ ( d f,t ( x, y )) t s κ ( d f ( x, y )) → s (cid:48) f (0) d ( x, y ) s κ ( d f ( x, y )) = e − f ( x ) n − d ( x, y ) s κ ( d f ( x, y )) NE DIMENSIONAL WEIGHTED RICCI CURVATURE 19 as t →
0. We obtain (5.3).We next show (5.4). For a unique minimal geodesic γ : [0 , → M from y to x , we define a function s f : [0 , → M as s f ( t ) := (cid:90) t e − f ( γ ( ξ )) n − dξ. The uniqueness of γ implies that d f,t ( y, x ) = s f ( t ) d ( y, x ) for every t ∈ [0 , G f : [0 , → R by G f ( t ) := β κ,f,t ( x, y ) n = (cid:18) s κ ( s f ( t ) d f,t ( y, x )) t s κ ( d f ( y, x )) (cid:19) − n . From direct computations we deduce G (cid:48) f (1) = n − n (cid:18) s (cid:48) f (1) d ( x, y ) c κ ( d f ( x, y )) s κ ( d f ( x, y )) − (cid:19) = (cid:101) β κ,f ( x, y ) . This proves (5.4). (cid:50)
For a positive Lipschitz function ρ on M with (cid:82) M ρ dm = 1, we put µ := ρ m . The generalized Fisher information I m ( µ ) of µ is defined as I m ( µ ) := (cid:90) M (cid:13)(cid:13) ∇ ρ − n (cid:13)(cid:13) ρ dm. Recall the following fact concerning the derivative of the R´enyi en-tropy H m defined as (1.4) for H ∈ DC (see e.g., Theorem 20.1 in [24]): Proposition 5.2.
For i = 0 , , let ρ i : M → R be positive Lipschitzfunctions with (cid:82) M ρ i dm = 1 . We put µ := ρ m and ν := ρ m . Thenfor a unique minimal geodesic ( µ t ) t ∈ [0 , in ( P ( M ) , W ) from µ to ν , (5.5) lim inf t → H m ( µ t ) − H m ( µ ) t ≥ − (cid:112) I m ( µ ) W ( µ, ν ) . Using Theorem 1.1 and Proposition 5.2, we prove the following:
Proposition 5.3.
For i = 0 , , let ρ i : M → R be positive Lipschitzfunctions with (cid:82) M ρ i dm = 1 . We put µ := ρ m and ν := ρ m . For κ ∈ R , if Ric f,M ≥ ( n − κ e − fn − , then we have H m ( µ ) ≤ (cid:112) I m ( µ ) W ( µ, ν ) + n (cid:90) M × M ρ ( x ) − n (cid:101) β κ,f ( x, y ) dπ ( x, y ) − n (cid:90) M × M ρ ( y ) − n (cid:16) (cid:98) β κ,f ( x, y ) n − (cid:17) dπ ( x, y ) − n (cid:90) M × M (cid:16) ρ ( y ) − n − (cid:17) dπ ( x, y ) , where π is a unique optimal coupling of ( µ, ν ) . Proof.
By Theorem 1.1, (
M, d, m ) has κ -weak twisted curvature bound.It follows that H m ( µ t ) ≤ n − (1 − t ) n (cid:90) M × M ρ ( x ) − n β κ,f, − t ( x, y ) n dπ ( x, y ) − t n (cid:90) M × M ρ ( y ) − n β κ,f,t ( x, y ) n dπ ( x, y ) , where ( µ t ) t ∈ [0 , is a unique minimal geodesic in ( P ( M ) , W ) from µ to ν . This leads to H m ( µ t ) − H m ( µ ) t ≤ n (cid:90) M × M ρ ( x ) − n − β κ,f, − t ( x, y ) n t dπ ( x, y )+ n (cid:90) M × M ρ ( x ) − n (cid:16) β κ,f, − t ( x, y ) n − (cid:17) dπ ( x, y ) − n (cid:90) M × M ρ ( y ) − n (cid:16) β κ,f,t ( x, y ) n − (cid:17) dπ ( x, y ) − n (cid:90) M × M (cid:16) ρ ( y ) − n − (cid:17) dπ ( x, y ) − H m ( µ ) . We remark that for a unique optimal transport map F from µ to ν ,and for µ -almost every x ∈ M we have F ( x ) / ∈ Cut x ; in particular,Theorem 2.2 implies d f ( x, F ( x )) ∈ [0 , C κ ). Therefore, by using π =(Id M × F ) µ and Lemma 5.1, we seelim sup t → H m ( µ t ) − H m ( µ ) t ≤ n (cid:90) M × M ρ ( x ) − n (cid:101) β κ,f ( x, y ) dπ ( x, y ) − n (cid:90) M × M ρ ( y ) − n (cid:16) (cid:98) β κ,f ( x, y ) n − (cid:17) dπ ( x, y ) − n (cid:90) M × M (cid:16) ρ ( y ) − n − (cid:17) dπ ( x, y ) − H m ( µ ) . Comparing this inequality with (5.5), we arrive at the desired one. (cid:50)
Functional inequalities.
We formulate three functional inequal-ities under the curvature condition (1.3).Let µ ∈ P ( M ) be absolutely continuous with respect to m . We saythat m is µ -constant if for a unique optimal transport map F from µ to m , it holds that d f ( x, F ( x )) = e − f ( x ) n − d ( x, F ( x )) on M .We first prove the following HWI inequality under the curvaturecondition (1.3) (cf. Theorem 20.10 in [24]): Corollary 5.4.
Let ρ : M → R denote a positive Lipschitz functionwith (cid:82) M ρ dm = 1 , and put µ := ρ m . We assume that f ≤ ( n − δ , NE DIMENSIONAL WEIGHTED RICCI CURVATURE 21 and that m is µ -constant. For κ > , if Ric f,M ≥ ( n − κ e − fn − , then H m ( µ ) ≤ (cid:112) I m ( µ ) W ( µ, m ) − ( n − κ e − δ { , sup ρ } − n W ( µ, m ) . Proof.
Let F be a unique optimal transport map from µ to m , and let π be a unique optimal coupling of ( µ, m ). Since π = (Id M × F ) µ , andsince m is µ -constant, on the support of π , (cid:98) β κ,f ( x, y ) = (cid:18) d f ( x, y ) s κ ( d f ( x, y )) (cid:19) n − , (cid:101) β κ,f ( x, y ) = n − n (cid:18) d f ( x, y ) c κ ( d f ( x, y )) s κ ( d f ( x, y )) − (cid:19) , where (cid:98) β κ,f and (cid:101) β κ,f are defined as (5.1) and as (5.2), respectively. Werecall the following elementary estimates (see e.g., Lemma 5.13 in [10]): (cid:16) α sin α (cid:17) − n − ≥ n − n α , − α cos α sin α ≥ α α ∈ [0 , π ]. By this elementary estimates and f ≤ ( n − δ , − (cid:16) (cid:98) β κ,f ( x, y ) n − (cid:17) ≤ − ( n − κ n d f ( x, y ) ≤ − ( n − κ e − δ n d ( x, y ) , (cid:101) β κ,f ( x, y ) ≤ − ( n − κ n d f ( x, y ) ≤ − ( n − κ e − δ n d ( x, y ) . Applying Proposition 5.3 to ρ = ρ and ρ = 1, and using the estimatesfor (cid:98) β κ,f and (cid:101) β κ,f , we see that H m ( µ ) is at most (cid:112) I m ( µ ) W ( µ, m ) − ( n − κ e − δ (cid:90) M × M ρ ( x ) − n d ( x, y ) dπ ( x, y ) − ( n − κ e − δ (cid:90) M × M d ( x, y ) dπ ( x, y ) , and hence H m ( µ ) ≤ (cid:112) I m ( µ ) W ( µ, m ) − ( n − κ e − δ { , sup ρ } − n (cid:90) M × M d ( x, y ) dπ ( x, y ) . By the optimality of π , the right hand side of the above inequality isequal to that of the desired one. We conclude Corollary 5.4. (cid:50) We further show the following Logarithmic Sobolev inequality underour curvature condition (cf. Theorem 21.7 in [24]):
Corollary 5.5.
Let ρ : M → R denote a positive Lipschitz functionwith (cid:82) M ρ dm = 1 , and put µ := ρ m . We assume that f ≤ ( n − δ ,and that m is µ -constant. For κ > , if Ric f,M ≥ ( n − κ e − fn − , then H m ( µ ) ≤ { , sup ρ } n n − κ e − δ I m ( µ ) . Proof.
For all a, b ∈ R and K > ab ≤ K a b K .
Using this elementary inequality, we have (cid:112) I m ( µ ) W ( µ, m ) ≤ { , sup ρ } n n − κ e − δ I m ( µ )+ ( n − κ e − δ { , sup ρ } − n W ( µ, m ) . From Corollary 5.4 one can derive Corollary 5.5. (cid:50)
Finally, we conclude the following finite dimensional transport energyinequality (cf. Theorem 22.37 in [24]):
Corollary 5.6.
Let ρ : M → R be positive, Lipschitz and (cid:82) M ρ dm = 1 .We put µ := ρ m . For κ > , if Ric f,M ≥ ( n − κ e − fn − , then we have n (cid:90) M × M (cid:98) β κ,f ( x, y ) n (cid:16) − ρ ( y ) − n (cid:17) dπ ( x, y ) ≥ (cid:90) M × M n (cid:32) e − f ( x ) n − d ( x, y ) s κ ( d f ( x, y )) (cid:33) − n dπ ( x, y ) − (cid:90) M × M (cid:32) ( n − e − f ( x ) n − d ( x, y ) c κ ( d f ( x, y )) s κ ( d f ( x, y )) + 1 (cid:33) dπ ( x, y ) , where π is a unique optimal coupling of ( m, µ ) .Proof. We apply Proposition 5.3 to ρ = 1 and ρ = ρ . From H m ( m ) =0 and I m ( m ) = 0 we deduce0 ≤ (cid:90) M × M (cid:16) (cid:101) β κ,f ( x, y ) − ρ ( y ) − n (cid:98) β κ,f ( x, y ) n (cid:17) dπ ( x, y ) + 1 . Hence, the left hand side of the desired inequality is at least n (cid:90) M × M (cid:16) (cid:98) β κ,f ( x, y ) n − (cid:101) β κ,f ( x, y ) − (cid:17) dπ ( x, y ) . NE DIMENSIONAL WEIGHTED RICCI CURVATURE 23
By substituting (5.1) and (5.2), we obtain the desired inequality. Thus,we complete the proof of Corollary 5.6. (cid:50)
Acknowledgements . The author is grateful to Professor Shin-ichi Ohtafor his useful comments. One of his comments leads the author to thestudy of the finite dimensional transport energy inequalities.
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