Eulerian side: Bakry-Émery cd ( k , n ) theory The BE approach to the theory of Ricci lower bounds is functional-analytic and requires (particularly for the study of higher order operators, as Hessians) the existence of a nice algebra of “smooth” functions and a self-adjoint, measure-preserving semigroup P t in an abstract measure space ( X , F , m ) . Then we define the operator ∆ as the infinitesimal generator of P t , so that dt P t f = ∆ P t f . Considering also the bilinear form E canonically associated to d P t , we introduce a “metric” structure, namely a carré du champ Γ( f ) , inspired by the calculus identity |∇ f | 2 w = �∇ f , ∇ ( fw ) � − 1 2 �∇ f 2 , ∇ w � i.e. (by integration on X ) Γ( f ) w dm = E ( f , fw ) − 1 � 2 E ( f 2 , w ) . X Then, the induced blinear form Γ( f , f ′ ) plays the role of �∇ f , ∇ f ′ � . Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 4 / 26
Eulerian side: Bakry-Émery cd ( k , n ) theory The BE approach to the theory of Ricci lower bounds is functional-analytic and requires (particularly for the study of higher order operators, as Hessians) the existence of a nice algebra of “smooth” functions and a self-adjoint, measure-preserving semigroup P t in an abstract measure space ( X , F , m ) . Then we define the operator ∆ as the infinitesimal generator of P t , so that dt P t f = ∆ P t f . Considering also the bilinear form E canonically associated to d P t , we introduce a “metric” structure, namely a carré du champ Γ( f ) , inspired by the calculus identity |∇ f | 2 w = �∇ f , ∇ ( fw ) � − 1 2 �∇ f 2 , ∇ w � i.e. (by integration on X ) Γ( f ) w dm = E ( f , fw ) − 1 � 2 E ( f 2 , w ) . X Then, the induced blinear form Γ( f , f ′ ) plays the role of �∇ f , ∇ f ′ � . Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 4 / 26
Eulerian side: Bakry-Émery cd ( k , n ) theory The BE approach to the theory of Ricci lower bounds is functional-analytic and requires (particularly for the study of higher order operators, as Hessians) the existence of a nice algebra of “smooth” functions and a self-adjoint, measure-preserving semigroup P t in an abstract measure space ( X , F , m ) . Then we define the operator ∆ as the infinitesimal generator of P t , so that dt P t f = ∆ P t f . Considering also the bilinear form E canonically associated to d P t , we introduce a “metric” structure, namely a carré du champ Γ( f ) , inspired by the calculus identity |∇ f | 2 w = �∇ f , ∇ ( fw ) � − 1 2 �∇ f 2 , ∇ w � i.e. (by integration on X ) Γ( f ) w dm = E ( f , fw ) − 1 � 2 E ( f 2 , w ) . X Then, the induced blinear form Γ( f , f ′ ) plays the role of �∇ f , ∇ f ′ � . Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 4 / 26
Eulerian side: Bakry-Émery cd ( k , n ) theory The BE approach to the theory of Ricci lower bounds is functional-analytic and requires (particularly for the study of higher order operators, as Hessians) the existence of a nice algebra of “smooth” functions and a self-adjoint, measure-preserving semigroup P t in an abstract measure space ( X , F , m ) . Then we define the operator ∆ as the infinitesimal generator of P t , so that dt P t f = ∆ P t f . Considering also the bilinear form E canonically associated to d P t , we introduce a “metric” structure, namely a carré du champ Γ( f ) , inspired by the calculus identity |∇ f | 2 w = �∇ f , ∇ ( fw ) � − 1 2 �∇ f 2 , ∇ w � i.e. (by integration on X ) Γ( f ) w dm = E ( f , fw ) − 1 � 2 E ( f 2 , w ) . X Then, the induced blinear form Γ( f , f ′ ) plays the role of �∇ f , ∇ f ′ � . Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 4 / 26
Eulerian side: Bakry-Émery cd ( k , n ) theory Having ∆ and Γ at our disposal, we can define cd ( k , n ) , for the structure ( X , F , m , P t ) (equivalently ( X , F , m , E ) ). Definition. We say that ( X , F , m , P t ) satisfies cd ( k , n ) if 1 2 ∆Γ( f ) ≥ 1 n (∆ f ) 2 + Γ( f , ∆ f ) + k Γ( f ) . This definition is inspired/motivated by the classical Bochner identity 1 2 ∆ |∇ f | 2 = | Hess ( f ) | 2 + �∇ f , ∇ ∆ f � + Ric ( ∇ f , ∇ f ) , where the inequalities are due to dim ≤ n , Ric ≥ kI . This definition is strongly consistent with the smooth case. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 5 / 26
Eulerian side: Bakry-Émery cd ( k , n ) theory Having ∆ and Γ at our disposal, we can define cd ( k , n ) , for the structure ( X , F , m , P t ) (equivalently ( X , F , m , E ) ). Definition. We say that ( X , F , m , P t ) satisfies cd ( k , n ) if 1 2 ∆Γ( f ) ≥ 1 n (∆ f ) 2 + Γ( f , ∆ f ) + k Γ( f ) . This definition is inspired/motivated by the classical Bochner identity 1 2 ∆ |∇ f | 2 = | Hess ( f ) | 2 + �∇ f , ∇ ∆ f � + Ric ( ∇ f , ∇ f ) , where the inequalities are due to dim ≤ n , Ric ≥ kI . This definition is strongly consistent with the smooth case. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 5 / 26
Eulerian side: Bakry-Émery cd ( k , n ) theory Having ∆ and Γ at our disposal, we can define cd ( k , n ) , for the structure ( X , F , m , P t ) (equivalently ( X , F , m , E ) ). Definition. We say that ( X , F , m , P t ) satisfies cd ( k , n ) if 1 2 ∆Γ( f ) ≥ 1 n (∆ f ) 2 + Γ( f , ∆ f ) + k Γ( f ) . This definition is inspired/motivated by the classical Bochner identity 1 2 ∆ |∇ f | 2 = | Hess ( f ) | 2 + �∇ f , ∇ ∆ f � + Ric ( ∇ f , ∇ f ) , where the inequalities are due to dim ≤ n , Ric ≥ kI . This definition is strongly consistent with the smooth case. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 5 / 26
Eulerian side: Bakry-Émery cd ( k , n ) theory Having ∆ and Γ at our disposal, we can define cd ( k , n ) , for the structure ( X , F , m , P t ) (equivalently ( X , F , m , E ) ). Definition. We say that ( X , F , m , P t ) satisfies cd ( k , n ) if 1 2 ∆Γ( f ) ≥ 1 n (∆ f ) 2 + Γ( f , ∆ f ) + k Γ( f ) . This definition is inspired/motivated by the classical Bochner identity 1 2 ∆ |∇ f | 2 = | Hess ( f ) | 2 + �∇ f , ∇ ∆ f � + Ric ( ∇ f , ∇ f ) , where the inequalities are due to dim ≤ n , Ric ≥ kI . This definition is strongly consistent with the smooth case. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 5 / 26
Lagrangian side: Lott-Villani and Sturm CD ( K , N ) theory This side of the theory involves the theory of Optimal Transport and the induced metric structure on ( P 2 ( X ) , W 2 ) , where W 2 is the Wasserstein distance with cost=distance 2 , namely �� � W 2 d 2 ( x , y ) d Σ( x , y ) : ( π 1 ) # Σ = µ, ( π 2 ) # Σ = ν 2 ( µ, ν ) := inf . X × X I will not enter into the details, but just mention that, in geodesic metric spaces, constant speed geodesics µ t , 0 ≤ t ≤ 1, beween two probability measures µ 0 , µ 1 ∈ P 2 ( X ) are “geodesic plans”, i.e. there exists π ∈ P ( Geo ( X )) with � � for all t ∈ [ 0 , 1 ] , φ bounded Borel. φ d µ t = φ ( γ ( t )) d π ( γ ) Geo ( X ) “Mass moves with constant speed along geodesics” Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 6 / 26
Lagrangian side: Lott-Villani and Sturm CD ( K , N ) theory This side of the theory involves the theory of Optimal Transport and the induced metric structure on ( P 2 ( X ) , W 2 ) , where W 2 is the Wasserstein distance with cost=distance 2 , namely �� � W 2 d 2 ( x , y ) d Σ( x , y ) : ( π 1 ) # Σ = µ, ( π 2 ) # Σ = ν 2 ( µ, ν ) := inf . X × X I will not enter into the details, but just mention that, in geodesic metric spaces, constant speed geodesics µ t , 0 ≤ t ≤ 1, beween two probability measures µ 0 , µ 1 ∈ P 2 ( X ) are “geodesic plans”, i.e. there exists π ∈ P ( Geo ( X )) with � � for all t ∈ [ 0 , 1 ] , φ bounded Borel. φ d µ t = φ ( γ ( t )) d π ( γ ) Geo ( X ) “Mass moves with constant speed along geodesics” Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 6 / 26
Lagrangian side: Lott-Villani and Sturm CD ( K , N ) theory This side of the theory involves the theory of Optimal Transport and the induced metric structure on ( P 2 ( X ) , W 2 ) , where W 2 is the Wasserstein distance with cost=distance 2 , namely �� � W 2 d 2 ( x , y ) d Σ( x , y ) : ( π 1 ) # Σ = µ, ( π 2 ) # Σ = ν 2 ( µ, ν ) := inf . X × X I will not enter into the details, but just mention that, in geodesic metric spaces, constant speed geodesics µ t , 0 ≤ t ≤ 1, beween two probability measures µ 0 , µ 1 ∈ P 2 ( X ) are “geodesic plans”, i.e. there exists π ∈ P ( Geo ( X )) with � � for all t ∈ [ 0 , 1 ] , φ bounded Borel. φ d µ t = φ ( γ ( t )) d π ( γ ) Geo ( X ) “Mass moves with constant speed along geodesics” Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 6 / 26
Lagrangian side: Lott-Villani and Sturm CD ( K , N ) theory In the case N = ∞ (no upper bound on dimension), the LSV definition of CD ( K , ∞ ) requires K -convexity of the Boltzmann-Shannon entropy �� X ρ log ρ dm if µ = ρ m ; Ent m ( µ ) := otherwise + ∞ along W 2 -geodesics µ t , namely Ent m ( µ t ) ≤ ( 1 − t ) Ent m ( µ 0 ) + t Ent m ( µ 1 ) − K 2 t ( 1 − t ) W 2 2 ( µ 0 , µ 1 ) . The roles of d and of m are nicely decoupled. Definition motivated by the classical inequality ≥ |∇ φ ( x ) | 2 � � ′′ − log J t ( x ) � + Ric x � � � ∇ φ ( x ) , ∇ φ ( x ) , � N � t = 0 with T t ( x ) = exp x ( − t ∇ φ ( x )) , J t ( x ) = det ∇ T t ( x ) . As for cd ( k , n ) , this is strongly consistent (Cordero-McCann-Schmuckenschläger, Sturm-Von Renesse). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 7 / 26
Lagrangian side: Lott-Villani and Sturm CD ( K , N ) theory In the case N = ∞ (no upper bound on dimension), the LSV definition of CD ( K , ∞ ) requires K -convexity of the Boltzmann-Shannon entropy �� X ρ log ρ dm if µ = ρ m ; Ent m ( µ ) := otherwise + ∞ along W 2 -geodesics µ t , namely Ent m ( µ t ) ≤ ( 1 − t ) Ent m ( µ 0 ) + t Ent m ( µ 1 ) − K 2 t ( 1 − t ) W 2 2 ( µ 0 , µ 1 ) . The roles of d and of m are nicely decoupled. Definition motivated by the classical inequality ≥ |∇ φ ( x ) | 2 � � ′′ − log J t ( x ) � + Ric x � � � ∇ φ ( x ) , ∇ φ ( x ) , � N � t = 0 with T t ( x ) = exp x ( − t ∇ φ ( x )) , J t ( x ) = det ∇ T t ( x ) . As for cd ( k , n ) , this is strongly consistent (Cordero-McCann-Schmuckenschläger, Sturm-Von Renesse). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 7 / 26
Lagrangian side: Lott-Villani and Sturm CD ( K , N ) theory In the case N = ∞ (no upper bound on dimension), the LSV definition of CD ( K , ∞ ) requires K -convexity of the Boltzmann-Shannon entropy �� X ρ log ρ dm if µ = ρ m ; Ent m ( µ ) := otherwise + ∞ along W 2 -geodesics µ t , namely Ent m ( µ t ) ≤ ( 1 − t ) Ent m ( µ 0 ) + t Ent m ( µ 1 ) − K 2 t ( 1 − t ) W 2 2 ( µ 0 , µ 1 ) . The roles of d and of m are nicely decoupled. Definition motivated by the classical inequality ≥ |∇ φ ( x ) | 2 � � ′′ − log J t ( x ) � + Ric x � � � ∇ φ ( x ) , ∇ φ ( x ) , � N � t = 0 with T t ( x ) = exp x ( − t ∇ φ ( x )) , J t ( x ) = det ∇ T t ( x ) . As for cd ( k , n ) , this is strongly consistent (Cordero-McCann-Schmuckenschläger, Sturm-Von Renesse). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 7 / 26
Lagrangian side: Lott-Villani and Sturm CD ( K , N ) theory In the case N = ∞ (no upper bound on dimension), the LSV definition of CD ( K , ∞ ) requires K -convexity of the Boltzmann-Shannon entropy �� X ρ log ρ dm if µ = ρ m ; Ent m ( µ ) := otherwise + ∞ along W 2 -geodesics µ t , namely Ent m ( µ t ) ≤ ( 1 − t ) Ent m ( µ 0 ) + t Ent m ( µ 1 ) − K 2 t ( 1 − t ) W 2 2 ( µ 0 , µ 1 ) . The roles of d and of m are nicely decoupled. Definition motivated by the classical inequality ≥ |∇ φ ( x ) | 2 � � ′′ − log J t ( x ) � + Ric x � � � ∇ φ ( x ) , ∇ φ ( x ) , � N � t = 0 with T t ( x ) = exp x ( − t ∇ φ ( x )) , J t ( x ) = det ∇ T t ( x ) . As for cd ( k , n ) , this is strongly consistent (Cordero-McCann-Schmuckenschläger, Sturm-Von Renesse). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 7 / 26
Lagrangian side: Lott-Villani and Sturm CD ( K , N ) theory In the case N < ∞ the definition is more involved, since Ent m has to be replaced by the Reny’s N -dimensional entropies � ρ − ρ 1 − 1 / N � if µ = ρ m + µ ⊥ � R N ( µ ) := N dm X and the coefficients ( 1 − t ) , t in the convexity inequality have to be replaced by suitable distorsion coefficients τ s κ ( θ ) (0 ≤ s ≤ 1, θ ≥ 0) with κ = K / N : � � � ( d ( x 1 , x 0 )) ρ − 1 / N κ ( d ( x 1 , x 0 )) ρ − 1 / N τ 1 − t ( x 0 )+ τ t R N ( µ t ) ≤ − ( x 1 ) d Σ( x 0 , x 1 ) . κ 0 1 In the simpler case K = 0, since τ s 0 ( θ ) = s , this is simply convexity of R N . A more recent variant in the choice of coefficients, considered by Bacher- Sturm, leads to the so-called CD ∗ ( K , N ) condition. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 8 / 26
Lagrangian side: Lott-Villani and Sturm CD ( K , N ) theory In the case N < ∞ the definition is more involved, since Ent m has to be replaced by the Reny’s N -dimensional entropies � ρ − ρ 1 − 1 / N � if µ = ρ m + µ ⊥ � R N ( µ ) := N dm X and the coefficients ( 1 − t ) , t in the convexity inequality have to be replaced by suitable distorsion coefficients τ s κ ( θ ) (0 ≤ s ≤ 1, θ ≥ 0) with κ = K / N : � � � ( d ( x 1 , x 0 )) ρ − 1 / N κ ( d ( x 1 , x 0 )) ρ − 1 / N τ 1 − t ( x 0 )+ τ t R N ( µ t ) ≤ − ( x 1 ) d Σ( x 0 , x 1 ) . κ 0 1 In the simpler case K = 0, since τ s 0 ( θ ) = s , this is simply convexity of R N . A more recent variant in the choice of coefficients, considered by Bacher- Sturm, leads to the so-called CD ∗ ( K , N ) condition. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 8 / 26
Lagrangian side: Lott-Villani and Sturm CD ( K , N ) theory In the case N < ∞ the definition is more involved, since Ent m has to be replaced by the Reny’s N -dimensional entropies � ρ − ρ 1 − 1 / N � if µ = ρ m + µ ⊥ � R N ( µ ) := N dm X and the coefficients ( 1 − t ) , t in the convexity inequality have to be replaced by suitable distorsion coefficients τ s κ ( θ ) (0 ≤ s ≤ 1, θ ≥ 0) with κ = K / N : � � � ( d ( x 1 , x 0 )) ρ − 1 / N κ ( d ( x 1 , x 0 )) ρ − 1 / N τ 1 − t ( x 0 )+ τ t R N ( µ t ) ≤ − ( x 1 ) d Σ( x 0 , x 1 ) . κ 0 1 In the simpler case K = 0, since τ s 0 ( θ ) = s , this is simply convexity of R N . A more recent variant in the choice of coefficients, considered by Bacher- Sturm, leads to the so-called CD ∗ ( K , N ) condition. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 8 / 26
Adding the “Riemannian” assumption The BE theory is “Riemannian” in nature and it leads to a very efficient calculus and to powerful and sinthetic proofs of geometric and functional inequalities, very often with sharp constants (isoperimetric, Poincaré, Logarithmic Sobolev, etc.) with contributions by many authors Bakry, Gentil, Ledoux, Hino, Wang,...... The LSV theory not only is built on different structures (m.m.s. instead of semigroups/Dirichlet forms), but also covers more classes of spaces, as for instance Finsler spaces. Its great merit, also in connection with the analysis of Ricci limit spaces, is the stability w.r.t. measured Gromov-Hausdorff convergence, we recall here one of the many equivalent definitions: We say thay ( X i , d i , m i ) MGH-converge to ( X , d , m ) if we can find isometric embeddings j i : X i → Z, j : X → Z with j i ( X i ) → j ( X ) in the Hausdorff sense and ( j i ) # m i → j # m weakly in Z. Given these substantial differences, can we find a closer link between the two theories? Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 9 / 26
Adding the “Riemannian” assumption The BE theory is “Riemannian” in nature and it leads to a very efficient calculus and to powerful and sinthetic proofs of geometric and functional inequalities, very often with sharp constants (isoperimetric, Poincaré, Logarithmic Sobolev, etc.) with contributions by many authors Bakry, Gentil, Ledoux, Hino, Wang,...... The LSV theory not only is built on different structures (m.m.s. instead of semigroups/Dirichlet forms), but also covers more classes of spaces, as for instance Finsler spaces. Its great merit, also in connection with the analysis of Ricci limit spaces, is the stability w.r.t. measured Gromov-Hausdorff convergence, we recall here one of the many equivalent definitions: We say thay ( X i , d i , m i ) MGH-converge to ( X , d , m ) if we can find isometric embeddings j i : X i → Z, j : X → Z with j i ( X i ) → j ( X ) in the Hausdorff sense and ( j i ) # m i → j # m weakly in Z. Given these substantial differences, can we find a closer link between the two theories? Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 9 / 26
Adding the “Riemannian” assumption The BE theory is “Riemannian” in nature and it leads to a very efficient calculus and to powerful and sinthetic proofs of geometric and functional inequalities, very often with sharp constants (isoperimetric, Poincaré, Logarithmic Sobolev, etc.) with contributions by many authors Bakry, Gentil, Ledoux, Hino, Wang,...... The LSV theory not only is built on different structures (m.m.s. instead of semigroups/Dirichlet forms), but also covers more classes of spaces, as for instance Finsler spaces. Its great merit, also in connection with the analysis of Ricci limit spaces, is the stability w.r.t. measured Gromov-Hausdorff convergence, we recall here one of the many equivalent definitions: We say thay ( X i , d i , m i ) MGH-converge to ( X , d , m ) if we can find isometric embeddings j i : X i → Z, j : X → Z with j i ( X i ) → j ( X ) in the Hausdorff sense and ( j i ) # m i → j # m weakly in Z. Given these substantial differences, can we find a closer link between the two theories? Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 9 / 26
Adding the “Riemannian” assumption The BE theory is “Riemannian” in nature and it leads to a very efficient calculus and to powerful and sinthetic proofs of geometric and functional inequalities, very often with sharp constants (isoperimetric, Poincaré, Logarithmic Sobolev, etc.) with contributions by many authors Bakry, Gentil, Ledoux, Hino, Wang,...... The LSV theory not only is built on different structures (m.m.s. instead of semigroups/Dirichlet forms), but also covers more classes of spaces, as for instance Finsler spaces. Its great merit, also in connection with the analysis of Ricci limit spaces, is the stability w.r.t. measured Gromov-Hausdorff convergence, we recall here one of the many equivalent definitions: We say thay ( X i , d i , m i ) MGH-converge to ( X , d , m ) if we can find isometric embeddings j i : X i → Z, j : X → Z with j i ( X i ) → j ( X ) in the Hausdorff sense and ( j i ) # m i → j # m weakly in Z. Given these substantial differences, can we find a closer link between the two theories? Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 9 / 26
Adding the “Riemannian” assumption The BE theory is “Riemannian” in nature and it leads to a very efficient calculus and to powerful and sinthetic proofs of geometric and functional inequalities, very often with sharp constants (isoperimetric, Poincaré, Logarithmic Sobolev, etc.) with contributions by many authors Bakry, Gentil, Ledoux, Hino, Wang,...... The LSV theory not only is built on different structures (m.m.s. instead of semigroups/Dirichlet forms), but also covers more classes of spaces, as for instance Finsler spaces. Its great merit, also in connection with the analysis of Ricci limit spaces, is the stability w.r.t. measured Gromov-Hausdorff convergence, we recall here one of the many equivalent definitions: We say thay ( X i , d i , m i ) MGH-converge to ( X , d , m ) if we can find isometric embeddings j i : X i → Z, j : X → Z with j i ( X i ) → j ( X ) in the Hausdorff sense and ( j i ) # m i → j # m weakly in Z. Given these substantial differences, can we find a closer link between the two theories? Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 9 / 26
Adding the “Riemannian” assumption to the CD theory Given a metric measure structure ( X , d , m ) , we define the slope (or local Lipschitz constant) of f : X → R by | f ( y ) − f ( x ) | |∇ f | ( x ) := lim sup . d ( y , x ) y → x Then, following Cheeger, we can define a kind of Dirichlet energy, by the L 2 ( X , m ) relaxation of |∇ f | 2 : Ch ( f ) := 1 � � � � |∇ f n | 2 dm : f n ∈ Lip ( X ) , | f n − f | 2 dm → 0 2 inf lim inf . n →∞ X X The functional Ch is convex and lower semicontinuous, and encodes in a subtle way properties of the distance and of the measure. By localizing this construction, one builds a pseudo “gradient” |∇ f | ∗ , called minimal relaxed slope , for which Ch ( f ) = 1 X |∇ f | 2 � ∗ dm and standard calculus properties 2 hold. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 10 / 26
Adding the “Riemannian” assumption to the CD theory Given a metric measure structure ( X , d , m ) , we define the slope (or local Lipschitz constant) of f : X → R by | f ( y ) − f ( x ) | |∇ f | ( x ) := lim sup . d ( y , x ) y → x Then, following Cheeger, we can define a kind of Dirichlet energy, by the L 2 ( X , m ) relaxation of |∇ f | 2 : Ch ( f ) := 1 � � � � |∇ f n | 2 dm : f n ∈ Lip ( X ) , | f n − f | 2 dm → 0 2 inf lim inf . n →∞ X X The functional Ch is convex and lower semicontinuous, and encodes in a subtle way properties of the distance and of the measure. By localizing this construction, one builds a pseudo “gradient” |∇ f | ∗ , called minimal relaxed slope , for which Ch ( f ) = 1 X |∇ f | 2 � ∗ dm and standard calculus properties 2 hold. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 10 / 26
Adding the “Riemannian” assumption to the CD theory Given a metric measure structure ( X , d , m ) , we define the slope (or local Lipschitz constant) of f : X → R by | f ( y ) − f ( x ) | |∇ f | ( x ) := lim sup . d ( y , x ) y → x Then, following Cheeger, we can define a kind of Dirichlet energy, by the L 2 ( X , m ) relaxation of |∇ f | 2 : Ch ( f ) := 1 � � � � |∇ f n | 2 dm : f n ∈ Lip ( X ) , | f n − f | 2 dm → 0 2 inf lim inf . n →∞ X X The functional Ch is convex and lower semicontinuous, and encodes in a subtle way properties of the distance and of the measure. By localizing this construction, one builds a pseudo “gradient” |∇ f | ∗ , called minimal relaxed slope , for which Ch ( f ) = 1 X |∇ f | 2 � ∗ dm and standard calculus properties 2 hold. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 10 / 26
Adding the “Riemannian” assumption to the CD theory Given a metric measure structure ( X , d , m ) , we define the slope (or local Lipschitz constant) of f : X → R by | f ( y ) − f ( x ) | |∇ f | ( x ) := lim sup . d ( y , x ) y → x Then, following Cheeger, we can define a kind of Dirichlet energy, by the L 2 ( X , m ) relaxation of |∇ f | 2 : Ch ( f ) := 1 � � � � |∇ f n | 2 dm : f n ∈ Lip ( X ) , | f n − f | 2 dm → 0 2 inf lim inf . n →∞ X X The functional Ch is convex and lower semicontinuous, and encodes in a subtle way properties of the distance and of the measure. By localizing this construction, one builds a pseudo “gradient” |∇ f | ∗ , called minimal relaxed slope , for which Ch ( f ) = 1 X |∇ f | 2 � ∗ dm and standard calculus properties 2 hold. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 10 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces The class of m.m.s. RCD ( K , ∞ ) , introduced in [AGS], can be defined by one of the following equivalent conditions: (i) ( X , d , m ) is a CD ( K , ∞ ) space and the L 2 heat flow P t is linear; (ii) ( X , d , m ) is a CD ( K , ∞ ) space and Ch is a quadratic form; (iii) the heat flow t �→ P t f , when seen as a curve of measures t �→ µ t = P t f m , satisfies the EVI (evolution variational inequality) � 1 � 1 � � d 2 W 2 2 W 2 + Ent m ( µ t ) ≤ Ent m ( µ ) 2 ( µ t , µ ) + K 2 ( µ t , µ ) dt for all µ ∈ P 2 ( X ) . Notice that (iii) combines in a nice way the two basic ingredients of the LSV theory, namely Ent m and W 2 . Not only, (iii) encodes both the curvature condition and the quadraticity of Ch , and much more. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 11 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces The class of m.m.s. RCD ( K , ∞ ) , introduced in [AGS], can be defined by one of the following equivalent conditions: (i) ( X , d , m ) is a CD ( K , ∞ ) space and the L 2 heat flow P t is linear; (ii) ( X , d , m ) is a CD ( K , ∞ ) space and Ch is a quadratic form; (iii) the heat flow t �→ P t f , when seen as a curve of measures t �→ µ t = P t f m , satisfies the EVI (evolution variational inequality) � 1 � 1 � � d 2 W 2 2 W 2 + Ent m ( µ t ) ≤ Ent m ( µ ) 2 ( µ t , µ ) + K 2 ( µ t , µ ) dt for all µ ∈ P 2 ( X ) . Notice that (iii) combines in a nice way the two basic ingredients of the LSV theory, namely Ent m and W 2 . Not only, (iii) encodes both the curvature condition and the quadraticity of Ch , and much more. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 11 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces The class of m.m.s. RCD ( K , ∞ ) , introduced in [AGS], can be defined by one of the following equivalent conditions: (i) ( X , d , m ) is a CD ( K , ∞ ) space and the L 2 heat flow P t is linear; (ii) ( X , d , m ) is a CD ( K , ∞ ) space and Ch is a quadratic form; (iii) the heat flow t �→ P t f , when seen as a curve of measures t �→ µ t = P t f m , satisfies the EVI (evolution variational inequality) � 1 � 1 � � d 2 W 2 2 W 2 + Ent m ( µ t ) ≤ Ent m ( µ ) 2 ( µ t , µ ) + K 2 ( µ t , µ ) dt for all µ ∈ P 2 ( X ) . Notice that (iii) combines in a nice way the two basic ingredients of the LSV theory, namely Ent m and W 2 . Not only, (iii) encodes both the curvature condition and the quadraticity of Ch , and much more. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 11 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces The class of m.m.s. RCD ( K , ∞ ) , introduced in [AGS], can be defined by one of the following equivalent conditions: (i) ( X , d , m ) is a CD ( K , ∞ ) space and the L 2 heat flow P t is linear; (ii) ( X , d , m ) is a CD ( K , ∞ ) space and Ch is a quadratic form; (iii) the heat flow t �→ P t f , when seen as a curve of measures t �→ µ t = P t f m , satisfies the EVI (evolution variational inequality) � 1 � 1 � � d 2 W 2 2 W 2 + Ent m ( µ t ) ≤ Ent m ( µ ) 2 ( µ t , µ ) + K 2 ( µ t , µ ) dt for all µ ∈ P 2 ( X ) . Notice that (iii) combines in a nice way the two basic ingredients of the LSV theory, namely Ent m and W 2 . Not only, (iii) encodes both the curvature condition and the quadraticity of Ch , and much more. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 11 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces Later on, this equivalence has been extended to the dimensional case by Erbar-Kuwada-Sturm, replacing (1) by � 1 � 1 � � � 1 − U N ( µ ) � d ≤ N dt σ 2 + K σ 2 2 W 2 ( µ t , µ ) 2 W 2 ( µ t , µ ) K / N K / N 2 U N ( µ t ) for all µ ∈ P 2 ( X ) , with U N a dimensional modification of Ent m : − 1 � � U N ( µ ) := exp N Ent m ( µ ) . More recently, A-Mondino-Savaré found characterizations of RCD ( K , N ) spaces involving EVI properties of Reny’s entropies, thus closing the circle. However, in this case one has to replace the curve µ t = P t f m with µ t = S t f m , where S t is the nonlinear diffusion semigroup given by dtS t f = 1 N ∆( S t f ) 1 − 1 d N . Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 12 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces Later on, this equivalence has been extended to the dimensional case by Erbar-Kuwada-Sturm, replacing (1) by � 1 � 1 � � � 1 − U N ( µ ) � d ≤ N dt σ 2 + K σ 2 2 W 2 ( µ t , µ ) 2 W 2 ( µ t , µ ) K / N K / N 2 U N ( µ t ) for all µ ∈ P 2 ( X ) , with U N a dimensional modification of Ent m : − 1 � � U N ( µ ) := exp N Ent m ( µ ) . More recently, A-Mondino-Savaré found characterizations of RCD ( K , N ) spaces involving EVI properties of Reny’s entropies, thus closing the circle. However, in this case one has to replace the curve µ t = P t f m with µ t = S t f m , where S t is the nonlinear diffusion semigroup given by dtS t f = 1 N ∆( S t f ) 1 − 1 d N . Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 12 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces Later on, this equivalence has been extended to the dimensional case by Erbar-Kuwada-Sturm, replacing (1) by � 1 � 1 � � � 1 − U N ( µ ) � d ≤ N dt σ 2 + K σ 2 2 W 2 ( µ t , µ ) 2 W 2 ( µ t , µ ) K / N K / N 2 U N ( µ t ) for all µ ∈ P 2 ( X ) , with U N a dimensional modification of Ent m : − 1 � � U N ( µ ) := exp N Ent m ( µ ) . More recently, A-Mondino-Savaré found characterizations of RCD ( K , N ) spaces involving EVI properties of Reny’s entropies, thus closing the circle. However, in this case one has to replace the curve µ t = P t f m with µ t = S t f m , where S t is the nonlinear diffusion semigroup given by dtS t f = 1 N ∆( S t f ) 1 − 1 d N . Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 12 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces Adding assumptions to the CD ( K , N ) theory might be dangerous, since one might lose stability w.r.t. MGH convergence. For instance the geodesic assumption works, while the non-branching assumption (which still plays a role in some proofs) does not. However, it can proved (for instance using EVI) that the combination of the Riemannian and curvature conditions is stable! Among the nice properties of RCD ( K , ∞ ) we mention in particular: • Strong Feller property, namely that P t , t > 0, maps L ∞ ( X , m ) to C b ( X ) ; • Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm): optimal geodesic plans between measures ≪ m are concentrated on a set of nonbranching geodesics. Because of the Riemannian condition, now it makes sense to compare the BE and LSV theories. This fundamental equivalence result has been proved first by [AGS] in the adimensional case N = ∞ , and then in general by Erbar-Kuwada-Sturm. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 13 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces Adding assumptions to the CD ( K , N ) theory might be dangerous, since one might lose stability w.r.t. MGH convergence. For instance the geodesic assumption works, while the non-branching assumption (which still plays a role in some proofs) does not. However, it can proved (for instance using EVI) that the combination of the Riemannian and curvature conditions is stable! Among the nice properties of RCD ( K , ∞ ) we mention in particular: • Strong Feller property, namely that P t , t > 0, maps L ∞ ( X , m ) to C b ( X ) ; • Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm): optimal geodesic plans between measures ≪ m are concentrated on a set of nonbranching geodesics. Because of the Riemannian condition, now it makes sense to compare the BE and LSV theories. This fundamental equivalence result has been proved first by [AGS] in the adimensional case N = ∞ , and then in general by Erbar-Kuwada-Sturm. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 13 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces Adding assumptions to the CD ( K , N ) theory might be dangerous, since one might lose stability w.r.t. MGH convergence. For instance the geodesic assumption works, while the non-branching assumption (which still plays a role in some proofs) does not. However, it can proved (for instance using EVI) that the combination of the Riemannian and curvature conditions is stable! Among the nice properties of RCD ( K , ∞ ) we mention in particular: • Strong Feller property, namely that P t , t > 0, maps L ∞ ( X , m ) to C b ( X ) ; • Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm): optimal geodesic plans between measures ≪ m are concentrated on a set of nonbranching geodesics. Because of the Riemannian condition, now it makes sense to compare the BE and LSV theories. This fundamental equivalence result has been proved first by [AGS] in the adimensional case N = ∞ , and then in general by Erbar-Kuwada-Sturm. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 13 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces Adding assumptions to the CD ( K , N ) theory might be dangerous, since one might lose stability w.r.t. MGH convergence. For instance the geodesic assumption works, while the non-branching assumption (which still plays a role in some proofs) does not. However, it can proved (for instance using EVI) that the combination of the Riemannian and curvature conditions is stable! Among the nice properties of RCD ( K , ∞ ) we mention in particular: • Strong Feller property, namely that P t , t > 0, maps L ∞ ( X , m ) to C b ( X ) ; • Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm): optimal geodesic plans between measures ≪ m are concentrated on a set of nonbranching geodesics. Because of the Riemannian condition, now it makes sense to compare the BE and LSV theories. This fundamental equivalence result has been proved first by [AGS] in the adimensional case N = ∞ , and then in general by Erbar-Kuwada-Sturm. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 13 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces Adding assumptions to the CD ( K , N ) theory might be dangerous, since one might lose stability w.r.t. MGH convergence. For instance the geodesic assumption works, while the non-branching assumption (which still plays a role in some proofs) does not. However, it can proved (for instance using EVI) that the combination of the Riemannian and curvature conditions is stable! Among the nice properties of RCD ( K , ∞ ) we mention in particular: • Strong Feller property, namely that P t , t > 0, maps L ∞ ( X , m ) to C b ( X ) ; • Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm): optimal geodesic plans between measures ≪ m are concentrated on a set of nonbranching geodesics. Because of the Riemannian condition, now it makes sense to compare the BE and LSV theories. This fundamental equivalence result has been proved first by [AGS] in the adimensional case N = ∞ , and then in general by Erbar-Kuwada-Sturm. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 13 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces Adding assumptions to the CD ( K , N ) theory might be dangerous, since one might lose stability w.r.t. MGH convergence. For instance the geodesic assumption works, while the non-branching assumption (which still plays a role in some proofs) does not. However, it can proved (for instance using EVI) that the combination of the Riemannian and curvature conditions is stable! Among the nice properties of RCD ( K , ∞ ) we mention in particular: • Strong Feller property, namely that P t , t > 0, maps L ∞ ( X , m ) to C b ( X ) ; • Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm): optimal geodesic plans between measures ≪ m are concentrated on a set of nonbranching geodesics. Because of the Riemannian condition, now it makes sense to compare the BE and LSV theories. This fundamental equivalence result has been proved first by [AGS] in the adimensional case N = ∞ , and then in general by Erbar-Kuwada-Sturm. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 13 / 26
RCD ( K , ∞ ) and RCD ( K , N ) spaces Adding assumptions to the CD ( K , N ) theory might be dangerous, since one might lose stability w.r.t. MGH convergence. For instance the geodesic assumption works, while the non-branching assumption (which still plays a role in some proofs) does not. However, it can proved (for instance using EVI) that the combination of the Riemannian and curvature conditions is stable! Among the nice properties of RCD ( K , ∞ ) we mention in particular: • Strong Feller property, namely that P t , t > 0, maps L ∞ ( X , m ) to C b ( X ) ; • Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm): optimal geodesic plans between measures ≪ m are concentrated on a set of nonbranching geodesics. Because of the Riemannian condition, now it makes sense to compare the BE and LSV theories. This fundamental equivalence result has been proved first by [AGS] in the adimensional case N = ∞ , and then in general by Erbar-Kuwada-Sturm. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 13 / 26
Equivalence of RCD and BE Theorem. RCD ( K , N ) m.m.s. satisfy the BE ( k , n ) condition with F = B ( d ) , E = Ch , k = K , n = N . Conversely, assume that E is strongly local, and that for some Hausdorff topology τ in X generating F one has: (a) P t , t > 0 , maps L ∞ ( X , m ) to C b ( X , τ ) ; (b) the Biroli-Mosco intrinsic distance d E ( x , y ) := sup {| f ( x ) − f ( y ) | : f ∈ C b ( X ) , Γ( f ) ≤ 1 } x , y ∈ X induces the topology τ ; (c) the BE ( k , n ) condition 1 2 ∆Γ( f ) ≥ 1 n (∆ f ) 2 + Γ( f , ∆ f ) + k Γ( f ) . holds. Then ( X , d E , m ) is a RCD ( K , N ) space with K = k , N = n . Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 14 / 26
Equivalence of RCD and BE Theorem. RCD ( K , N ) m.m.s. satisfy the BE ( k , n ) condition with F = B ( d ) , E = Ch , k = K , n = N . Conversely, assume that E is strongly local, and that for some Hausdorff topology τ in X generating F one has: (a) P t , t > 0 , maps L ∞ ( X , m ) to C b ( X , τ ) ; (b) the Biroli-Mosco intrinsic distance d E ( x , y ) := sup {| f ( x ) − f ( y ) | : f ∈ C b ( X ) , Γ( f ) ≤ 1 } x , y ∈ X induces the topology τ ; (c) the BE ( k , n ) condition 1 2 ∆Γ( f ) ≥ 1 n (∆ f ) 2 + Γ( f , ∆ f ) + k Γ( f ) . holds. Then ( X , d E , m ) is a RCD ( K , N ) space with K = k , N = n . Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 14 / 26
Equivalence of RCD and BE Theorem. RCD ( K , N ) m.m.s. satisfy the BE ( k , n ) condition with F = B ( d ) , E = Ch , k = K , n = N . Conversely, assume that E is strongly local, and that for some Hausdorff topology τ in X generating F one has: (a) P t , t > 0 , maps L ∞ ( X , m ) to C b ( X , τ ) ; (b) the Biroli-Mosco intrinsic distance d E ( x , y ) := sup {| f ( x ) − f ( y ) | : f ∈ C b ( X ) , Γ( f ) ≤ 1 } x , y ∈ X induces the topology τ ; (c) the BE ( k , n ) condition 1 2 ∆Γ( f ) ≥ 1 n (∆ f ) 2 + Γ( f , ∆ f ) + k Γ( f ) . holds. Then ( X , d E , m ) is a RCD ( K , N ) space with K = k , N = n . Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 14 / 26
Characterizations based on contractivity of P t This also lead to other characterizations of RCD based on gradient contractivity: |∇ P t f | 2 ≤ e − 2 Kt P t |∇ f | 2 ( N = ∞ ) When N < ∞ , setting κ = K / N we need to consider different times s , t (Bolley, Gentil, Guillin, Erbar, Kuwada, Sturm) � 1 � 1 � � s 2 e − K ( t + s ) s 2 2 W 2 ( P t f m , P s g m ) ≤ 2 W 2 ( f m , g m ) κ κ 1 − e − K ( t + s ) � ( √ t − √ s ) 2 1 � + . 2 ( t + s ) κ Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 15 / 26
Characterizations based on contractivity of P t This also lead to other characterizations of RCD based on gradient contractivity: |∇ P t f | 2 ≤ e − 2 Kt P t |∇ f | 2 ( N = ∞ ) When N < ∞ , setting κ = K / N we need to consider different times s , t (Bolley, Gentil, Guillin, Erbar, Kuwada, Sturm) � 1 � 1 � � s 2 e − K ( t + s ) s 2 2 W 2 ( P t f m , P s g m ) ≤ 2 W 2 ( f m , g m ) κ κ 1 − e − K ( t + s ) � ( √ t − √ s ) 2 1 � + . 2 ( t + s ) κ Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 15 / 26
Local structure of RCD spaces Splitting. (Gigli) If a RCD ( 0 , N ) space ( X , d , m ) contains a line (namely an isometric embedding of R ), then m ) × ( R , d eu , L 1 ) ( X , d , m ) ∼ ( Y , ˜ ( Y , ˜ m ) ∈ RCD ( 0 , N − 1 ) . d , ˜ d , ˜ with It extends to a nonsmooth setting the classical result of Toponogov, Cheeger-Gromoll, already proved for Ricci limit spaces by Cheeger-Colding. Euclidean tangents and rectifiability. (Mondino-Naber, after Gigli- Mondino-Rajala) A RCD ( K , N ) space ( X , d , m ) can be covered m-almost all by countably many X i , each bi-Lipschitz to a subset of R k ( i ) , k ( i ) ≤ N. The constancy of dimension is an open problem, and there is still a (big?) gap with the structure of Ricci limit spaces. On the other hand, the development of a good calculus in RCD spaces does not depend on these “local” regularity results. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 16 / 26
Local structure of RCD spaces Splitting. (Gigli) If a RCD ( 0 , N ) space ( X , d , m ) contains a line (namely an isometric embedding of R ), then m ) × ( R , d eu , L 1 ) ( X , d , m ) ∼ ( Y , ˜ ( Y , ˜ m ) ∈ RCD ( 0 , N − 1 ) . d , ˜ d , ˜ with It extends to a nonsmooth setting the classical result of Toponogov, Cheeger-Gromoll, already proved for Ricci limit spaces by Cheeger-Colding. Euclidean tangents and rectifiability. (Mondino-Naber, after Gigli- Mondino-Rajala) A RCD ( K , N ) space ( X , d , m ) can be covered m-almost all by countably many X i , each bi-Lipschitz to a subset of R k ( i ) , k ( i ) ≤ N. The constancy of dimension is an open problem, and there is still a (big?) gap with the structure of Ricci limit spaces. On the other hand, the development of a good calculus in RCD spaces does not depend on these “local” regularity results. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 16 / 26
Local structure of RCD spaces Splitting. (Gigli) If a RCD ( 0 , N ) space ( X , d , m ) contains a line (namely an isometric embedding of R ), then m ) × ( R , d eu , L 1 ) ( X , d , m ) ∼ ( Y , ˜ ( Y , ˜ m ) ∈ RCD ( 0 , N − 1 ) . d , ˜ d , ˜ with It extends to a nonsmooth setting the classical result of Toponogov, Cheeger-Gromoll, already proved for Ricci limit spaces by Cheeger-Colding. Euclidean tangents and rectifiability. (Mondino-Naber, after Gigli- Mondino-Rajala) A RCD ( K , N ) space ( X , d , m ) can be covered m-almost all by countably many X i , each bi-Lipschitz to a subset of R k ( i ) , k ( i ) ≤ N. The constancy of dimension is an open problem, and there is still a (big?) gap with the structure of Ricci limit spaces. On the other hand, the development of a good calculus in RCD spaces does not depend on these “local” regularity results. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 16 / 26
Local structure of RCD spaces Splitting. (Gigli) If a RCD ( 0 , N ) space ( X , d , m ) contains a line (namely an isometric embedding of R ), then m ) × ( R , d eu , L 1 ) ( X , d , m ) ∼ ( Y , ˜ ( Y , ˜ m ) ∈ RCD ( 0 , N − 1 ) . d , ˜ d , ˜ with It extends to a nonsmooth setting the classical result of Toponogov, Cheeger-Gromoll, already proved for Ricci limit spaces by Cheeger-Colding. Euclidean tangents and rectifiability. (Mondino-Naber, after Gigli- Mondino-Rajala) A RCD ( K , N ) space ( X , d , m ) can be covered m-almost all by countably many X i , each bi-Lipschitz to a subset of R k ( i ) , k ( i ) ≤ N. The constancy of dimension is an open problem, and there is still a (big?) gap with the structure of Ricci limit spaces. On the other hand, the development of a good calculus in RCD spaces does not depend on these “local” regularity results. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 16 / 26
Local structure of RCD spaces Splitting. (Gigli) If a RCD ( 0 , N ) space ( X , d , m ) contains a line (namely an isometric embedding of R ), then m ) × ( R , d eu , L 1 ) ( X , d , m ) ∼ ( Y , ˜ ( Y , ˜ m ) ∈ RCD ( 0 , N − 1 ) . d , ˜ d , ˜ with It extends to a nonsmooth setting the classical result of Toponogov, Cheeger-Gromoll, already proved for Ricci limit spaces by Cheeger-Colding. Euclidean tangents and rectifiability. (Mondino-Naber, after Gigli- Mondino-Rajala) A RCD ( K , N ) space ( X , d , m ) can be covered m-almost all by countably many X i , each bi-Lipschitz to a subset of R k ( i ) , k ( i ) ≤ N. The constancy of dimension is an open problem, and there is still a (big?) gap with the structure of Ricci limit spaces. On the other hand, the development of a good calculus in RCD spaces does not depend on these “local” regularity results. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 16 / 26
Some geometric/functional inequalities Local Poincaré. (Buser, Lott-Villani, Rajala) In CD ( K , ∞ ) spaces � | f − f x , r | dm ≤ re K − r 2 � f ∈ Lip ( X ) . |∇ f | dm B r ( x ) B 2 r ( x ) Global Poincaré (spectral gap). (Lott-Villani, Sturm) In CD ( K , N ) spaces with N > 1, K > 0, | f | 2 dm ≤ N − 1 � � |∇ f | 2 dm f ∈ Lip ( X ) . KN X X Li-Yau. (Garofalo-Mondino, Jiang, after Baudoin-Garofalo, Bakry-Ledoux) |∇ log P t f | 2 − d dt log P t f ≤ N ( f > 0, in RCD ( 0 , N ) spaces) 2 t (and many more, transport, Logarithmic Sobolev, isoperimetric, log- Harnack,...) Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 17 / 26
Some geometric/functional inequalities Local Poincaré. (Buser, Lott-Villani, Rajala) In CD ( K , ∞ ) spaces � | f − f x , r | dm ≤ re K − r 2 � f ∈ Lip ( X ) . |∇ f | dm B r ( x ) B 2 r ( x ) Global Poincaré (spectral gap). (Lott-Villani, Sturm) In CD ( K , N ) spaces with N > 1, K > 0, | f | 2 dm ≤ N − 1 � � |∇ f | 2 dm f ∈ Lip ( X ) . KN X X Li-Yau. (Garofalo-Mondino, Jiang, after Baudoin-Garofalo, Bakry-Ledoux) |∇ log P t f | 2 − d dt log P t f ≤ N ( f > 0, in RCD ( 0 , N ) spaces) 2 t (and many more, transport, Logarithmic Sobolev, isoperimetric, log- Harnack,...) Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 17 / 26
Some geometric/functional inequalities Local Poincaré. (Buser, Lott-Villani, Rajala) In CD ( K , ∞ ) spaces � | f − f x , r | dm ≤ re K − r 2 � f ∈ Lip ( X ) . |∇ f | dm B r ( x ) B 2 r ( x ) Global Poincaré (spectral gap). (Lott-Villani, Sturm) In CD ( K , N ) spaces with N > 1, K > 0, | f | 2 dm ≤ N − 1 � � |∇ f | 2 dm f ∈ Lip ( X ) . KN X X Li-Yau. (Garofalo-Mondino, Jiang, after Baudoin-Garofalo, Bakry-Ledoux) |∇ log P t f | 2 − d dt log P t f ≤ N ( f > 0, in RCD ( 0 , N ) spaces) 2 t (and many more, transport, Logarithmic Sobolev, isoperimetric, log- Harnack,...) Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 17 / 26
Some geometric/functional inequalities Local Poincaré. (Buser, Lott-Villani, Rajala) In CD ( K , ∞ ) spaces � | f − f x , r | dm ≤ re K − r 2 � f ∈ Lip ( X ) . |∇ f | dm B r ( x ) B 2 r ( x ) Global Poincaré (spectral gap). (Lott-Villani, Sturm) In CD ( K , N ) spaces with N > 1, K > 0, | f | 2 dm ≤ N − 1 � � |∇ f | 2 dm f ∈ Lip ( X ) . KN X X Li-Yau. (Garofalo-Mondino, Jiang, after Baudoin-Garofalo, Bakry-Ledoux) |∇ log P t f | 2 − d dt log P t f ≤ N ( f > 0, in RCD ( 0 , N ) spaces) 2 t (and many more, transport, Logarithmic Sobolev, isoperimetric, log- Harnack,...) Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 17 / 26
Some geometric/functional inequalities Local Poincaré. (Buser, Lott-Villani, Rajala) In CD ( K , ∞ ) spaces � | f − f x , r | dm ≤ re K − r 2 � f ∈ Lip ( X ) . |∇ f | dm B r ( x ) B 2 r ( x ) Global Poincaré (spectral gap). (Lott-Villani, Sturm) In CD ( K , N ) spaces with N > 1, K > 0, | f | 2 dm ≤ N − 1 � � |∇ f | 2 dm f ∈ Lip ( X ) . KN X X Li-Yau. (Garofalo-Mondino, Jiang, after Baudoin-Garofalo, Bakry-Ledoux) |∇ log P t f | 2 − d dt log P t f ≤ N ( f > 0, in RCD ( 0 , N ) spaces) 2 t (and many more, transport, Logarithmic Sobolev, isoperimetric, log- Harnack,...) Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 17 / 26
Some comparison results Bishop-Gromov. In CD ( K , N ) spaces m ( B r ( x )) m ( B s ( x )) ≤ V K , N ( r ) 0 < s ≤ r . V K , N ( s ) Laplacian comparison. (Gigli) In CD ( K , N ) spaces (with ˜ τ 0 , N ≡ N ) 1 2 ∆ d 2 ( · , z ) ≤ ˜ in the sense of distributions. � � τ K , N d ( · , z ) ∀ z ∈ X , Lévy-Gromov. (Cavalletti-Mondino, after Klartag) In essentially nonbranching CD ( K , N ) spaces ( X , d , m ) , for E ⊂ X one has m ( X ) ≥ | ∂ B | | ∂ E | | M | where M is the model space and B is the isoperimetric region in M with the same volume fraction, i.e. m ( E ) / m ( X ) = | B | / | M | . It covers also K < 0, via the model spaces of E.Milman. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 18 / 26
Some comparison results Bishop-Gromov. In CD ( K , N ) spaces m ( B r ( x )) m ( B s ( x )) ≤ V K , N ( r ) 0 < s ≤ r . V K , N ( s ) Laplacian comparison. (Gigli) In CD ( K , N ) spaces (with ˜ τ 0 , N ≡ N ) 1 2 ∆ d 2 ( · , z ) ≤ ˜ in the sense of distributions. � � τ K , N d ( · , z ) ∀ z ∈ X , Lévy-Gromov. (Cavalletti-Mondino, after Klartag) In essentially nonbranching CD ( K , N ) spaces ( X , d , m ) , for E ⊂ X one has m ( X ) ≥ | ∂ B | | ∂ E | | M | where M is the model space and B is the isoperimetric region in M with the same volume fraction, i.e. m ( E ) / m ( X ) = | B | / | M | . It covers also K < 0, via the model spaces of E.Milman. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 18 / 26
Some comparison results Bishop-Gromov. In CD ( K , N ) spaces m ( B r ( x )) m ( B s ( x )) ≤ V K , N ( r ) 0 < s ≤ r . V K , N ( s ) Laplacian comparison. (Gigli) In CD ( K , N ) spaces (with ˜ τ 0 , N ≡ N ) 1 2 ∆ d 2 ( · , z ) ≤ ˜ in the sense of distributions. � � τ K , N d ( · , z ) ∀ z ∈ X , Lévy-Gromov. (Cavalletti-Mondino, after Klartag) In essentially nonbranching CD ( K , N ) spaces ( X , d , m ) , for E ⊂ X one has m ( X ) ≥ | ∂ B | | ∂ E | | M | where M is the model space and B is the isoperimetric region in M with the same volume fraction, i.e. m ( E ) / m ( X ) = | B | / | M | . It covers also K < 0, via the model spaces of E.Milman. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 18 / 26
Some comparison results Bishop-Gromov. In CD ( K , N ) spaces m ( B r ( x )) m ( B s ( x )) ≤ V K , N ( r ) 0 < s ≤ r . V K , N ( s ) Laplacian comparison. (Gigli) In CD ( K , N ) spaces (with ˜ τ 0 , N ≡ N ) 1 2 ∆ d 2 ( · , z ) ≤ ˜ in the sense of distributions. � � τ K , N d ( · , z ) ∀ z ∈ X , Lévy-Gromov. (Cavalletti-Mondino, after Klartag) In essentially nonbranching CD ( K , N ) spaces ( X , d , m ) , for E ⊂ X one has m ( X ) ≥ | ∂ B | | ∂ E | | M | where M is the model space and B is the isoperimetric region in M with the same volume fraction, i.e. m ( E ) / m ( X ) = | B | / | M | . It covers also K < 0, via the model spaces of E.Milman. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 18 / 26
Some comparison results Bishop-Gromov. In CD ( K , N ) spaces m ( B r ( x )) m ( B s ( x )) ≤ V K , N ( r ) 0 < s ≤ r . V K , N ( s ) Laplacian comparison. (Gigli) In CD ( K , N ) spaces (with ˜ τ 0 , N ≡ N ) 1 2 ∆ d 2 ( · , z ) ≤ ˜ in the sense of distributions. � � τ K , N d ( · , z ) ∀ z ∈ X , Lévy-Gromov. (Cavalletti-Mondino, after Klartag) In essentially nonbranching CD ( K , N ) spaces ( X , d , m ) , for E ⊂ X one has m ( X ) ≥ | ∂ B | | ∂ E | | M | where M is the model space and B is the isoperimetric region in M with the same volume fraction, i.e. m ( E ) / m ( X ) = | B | / | M | . It covers also K < 0, via the model spaces of E.Milman. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 18 / 26
Heat flow/1 The standard analytic interpretation of the heat flow is the L 2 gradient flow of the Dirichlet energy. In our m.m.s. context, the role of the Dirichlet energy is played by Cheeger’s energy Ch , so that according to the Komura-Brezis theory, we look for solutions to d dt f t = ∆ f t where, by definition, − ∆ f = − ∆ d , m f is the element with minimal norm in the subgradient ∂ Ch ( f ) of Ch at f : � � � ξ ∈ L 2 ( X , m ) : Ch ( g ) ≥ Ch ( f ) + ∂ Ch ( f ) := ξ ( g − f ) dm ∀ g . X The Komura-Brezis theory is very robust, with existence, uniqueness and regularization results which work even when Ch is not quadratic. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 19 / 26
Heat flow/1 The standard analytic interpretation of the heat flow is the L 2 gradient flow of the Dirichlet energy. In our m.m.s. context, the role of the Dirichlet energy is played by Cheeger’s energy Ch , so that according to the Komura-Brezis theory, we look for solutions to d dt f t = ∆ f t where, by definition, − ∆ f = − ∆ d , m f is the element with minimal norm in the subgradient ∂ Ch ( f ) of Ch at f : � � � ξ ∈ L 2 ( X , m ) : Ch ( g ) ≥ Ch ( f ) + ∂ Ch ( f ) := ξ ( g − f ) dm ∀ g . X The Komura-Brezis theory is very robust, with existence, uniqueness and regularization results which work even when Ch is not quadratic. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 19 / 26
Heat flow/1 The standard analytic interpretation of the heat flow is the L 2 gradient flow of the Dirichlet energy. In our m.m.s. context, the role of the Dirichlet energy is played by Cheeger’s energy Ch , so that according to the Komura-Brezis theory, we look for solutions to d dt f t = ∆ f t where, by definition, − ∆ f = − ∆ d , m f is the element with minimal norm in the subgradient ∂ Ch ( f ) of Ch at f : � � � ξ ∈ L 2 ( X , m ) : Ch ( g ) ≥ Ch ( f ) + ∂ Ch ( f ) := ξ ( g − f ) dm ∀ g . X The Komura-Brezis theory is very robust, with existence, uniqueness and regularization results which work even when Ch is not quadratic. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 19 / 26
Heat flow/1 The standard analytic interpretation of the heat flow is the L 2 gradient flow of the Dirichlet energy. In our m.m.s. context, the role of the Dirichlet energy is played by Cheeger’s energy Ch , so that according to the Komura-Brezis theory, we look for solutions to d dt f t = ∆ f t where, by definition, − ∆ f = − ∆ d , m f is the element with minimal norm in the subgradient ∂ Ch ( f ) of Ch at f : � � � ξ ∈ L 2 ( X , m ) : Ch ( g ) ≥ Ch ( f ) + ∂ Ch ( f ) := ξ ( g − f ) dm ∀ g . X The Komura-Brezis theory is very robust, with existence, uniqueness and regularization results which work even when Ch is not quadratic. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 19 / 26
Heat flow/2 Another interpretation of the heat flow in Euclidean spaces originated within the theory of optimal transport, in the 90’s, with the work of Jordan- Kinderlehrer-Otto and with the development of the “Otto-calculus” in the space of probability measures, initially motivated by the analysis of the long time behaviour of nonlinear diffusion equations. When we want to identify a specific PDE with a gradient flow x ′ = −∇ F we can play both with the energy F and with the metric structure, since ∇ F (unlike dF ) does depend on the metric structure. JKO realized that, by replacing the Dirichlet energy with the Boltzmann- Shannon entropy, and the L 2 distance with the quadratic Wasserstein distance in P 2 ( R n ) , we can recover again the heat equation! In m.m.s. we can give a meaning to the gradient flow (following De Giorgi) by looking at the maximal rate of energy dissipation: � t Ent m ( µ t ) + 1 µ s | 2 + |∇ − Ent m | 2 ( µ s ) ds ≤ Ent m ( µ 0 ) ∀ t ≥ 0 . | ˙ 2 0 Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 20 / 26
Heat flow/2 Another interpretation of the heat flow in Euclidean spaces originated within the theory of optimal transport, in the 90’s, with the work of Jordan- Kinderlehrer-Otto and with the development of the “Otto-calculus” in the space of probability measures, initially motivated by the analysis of the long time behaviour of nonlinear diffusion equations. When we want to identify a specific PDE with a gradient flow x ′ = −∇ F we can play both with the energy F and with the metric structure, since ∇ F (unlike dF ) does depend on the metric structure. JKO realized that, by replacing the Dirichlet energy with the Boltzmann- Shannon entropy, and the L 2 distance with the quadratic Wasserstein distance in P 2 ( R n ) , we can recover again the heat equation! In m.m.s. we can give a meaning to the gradient flow (following De Giorgi) by looking at the maximal rate of energy dissipation: � t Ent m ( µ t ) + 1 µ s | 2 + |∇ − Ent m | 2 ( µ s ) ds ≤ Ent m ( µ 0 ) ∀ t ≥ 0 . | ˙ 2 0 Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 20 / 26
Heat flow/2 Another interpretation of the heat flow in Euclidean spaces originated within the theory of optimal transport, in the 90’s, with the work of Jordan- Kinderlehrer-Otto and with the development of the “Otto-calculus” in the space of probability measures, initially motivated by the analysis of the long time behaviour of nonlinear diffusion equations. When we want to identify a specific PDE with a gradient flow x ′ = −∇ F we can play both with the energy F and with the metric structure, since ∇ F (unlike dF ) does depend on the metric structure. JKO realized that, by replacing the Dirichlet energy with the Boltzmann- Shannon entropy, and the L 2 distance with the quadratic Wasserstein distance in P 2 ( R n ) , we can recover again the heat equation! In m.m.s. we can give a meaning to the gradient flow (following De Giorgi) by looking at the maximal rate of energy dissipation: � t Ent m ( µ t ) + 1 µ s | 2 + |∇ − Ent m | 2 ( µ s ) ds ≤ Ent m ( µ 0 ) ∀ t ≥ 0 . | ˙ 2 0 Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 20 / 26
Heat flow/2 Another interpretation of the heat flow in Euclidean spaces originated within the theory of optimal transport, in the 90’s, with the work of Jordan- Kinderlehrer-Otto and with the development of the “Otto-calculus” in the space of probability measures, initially motivated by the analysis of the long time behaviour of nonlinear diffusion equations. When we want to identify a specific PDE with a gradient flow x ′ = −∇ F we can play both with the energy F and with the metric structure, since ∇ F (unlike dF ) does depend on the metric structure. JKO realized that, by replacing the Dirichlet energy with the Boltzmann- Shannon entropy, and the L 2 distance with the quadratic Wasserstein distance in P 2 ( R n ) , we can recover again the heat equation! In m.m.s. we can give a meaning to the gradient flow (following De Giorgi) by looking at the maximal rate of energy dissipation: � t Ent m ( µ t ) + 1 µ s | 2 + |∇ − Ent m | 2 ( µ s ) ds ≤ Ent m ( µ 0 ) ∀ t ≥ 0 . | ˙ 2 0 Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 20 / 26
Heat flow/3 Theorem. [AGS] In a general class of m.m.s., that includes all CD ( K , ∞ ) spaces, the two gradient flows coincide in the (invariant) class � � � f ∈ L 2 ( X , m ) : f ≥ 0 , f dm = 1 . X This identification is fundamental for many reasons. The main one is maybe related to the proof of stability of heat flows w.r.t. measured Gromov- Hausdorff convergence: since this notion is Lagrangian in nature, we need a corresponding notion of heat flow to deal with it. As a matter of fact, it would be hard to obtain a similar result using only the BE theory. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 21 / 26
Heat flow/3 Theorem. [AGS] In a general class of m.m.s., that includes all CD ( K , ∞ ) spaces, the two gradient flows coincide in the (invariant) class � � � f ∈ L 2 ( X , m ) : f ≥ 0 , f dm = 1 . X This identification is fundamental for many reasons. The main one is maybe related to the proof of stability of heat flows w.r.t. measured Gromov- Hausdorff convergence: since this notion is Lagrangian in nature, we need a corresponding notion of heat flow to deal with it. As a matter of fact, it would be hard to obtain a similar result using only the BE theory. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 21 / 26
Heat flow/3 Theorem. [AGS] In a general class of m.m.s., that includes all CD ( K , ∞ ) spaces, the two gradient flows coincide in the (invariant) class � � � f ∈ L 2 ( X , m ) : f ≥ 0 , f dm = 1 . X This identification is fundamental for many reasons. The main one is maybe related to the proof of stability of heat flows w.r.t. measured Gromov- Hausdorff convergence: since this notion is Lagrangian in nature, we need a corresponding notion of heat flow to deal with it. As a matter of fact, it would be hard to obtain a similar result using only the BE theory. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 21 / 26
Heat flow/3 Theorem. [AGS] In a general class of m.m.s., that includes all CD ( K , ∞ ) spaces, the two gradient flows coincide in the (invariant) class � � � f ∈ L 2 ( X , m ) : f ≥ 0 , f dm = 1 . X This identification is fundamental for many reasons. The main one is maybe related to the proof of stability of heat flows w.r.t. measured Gromov- Hausdorff convergence: since this notion is Lagrangian in nature, we need a corresponding notion of heat flow to deal with it. As a matter of fact, it would be hard to obtain a similar result using only the BE theory. Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 21 / 26
Key tools: superposition principle and Kuwada lemma The superposition principle represents currents as superposition of elementary currents associated to curves (L.C.Young, Smirnov, Paolini-Stepanov); in the optimal transport context (AGS) it represents nonnegative solutions to the continuity equation d dt µ t + div ( v t µ t ) = 0 t ∈ ( 0 , 1 ) as marginals µ t of positive finite measures π in C ([ 0 , 1 ]; X ) concentrated on solutions to the ODE γ ′ = v t ( γ ) . A metric version of this result is possible (Lisini). Kuwada lemma. If µ t = P t f m with f ≥ 0 , � X f dm = 1 , then µ t ∈ AC 2 � [ 0 , 1 ]; ( P 2 ( X ) , W 2 ) � with |∇ P t f | 2 � µ t | 2 ≤ for L 1 -a.e. t > 0 . ∗ | ˙ dm P t f { P t f > 0 } Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 22 / 26
Key tools: superposition principle and Kuwada lemma The superposition principle represents currents as superposition of elementary currents associated to curves (L.C.Young, Smirnov, Paolini-Stepanov); in the optimal transport context (AGS) it represents nonnegative solutions to the continuity equation d dt µ t + div ( v t µ t ) = 0 t ∈ ( 0 , 1 ) as marginals µ t of positive finite measures π in C ([ 0 , 1 ]; X ) concentrated on solutions to the ODE γ ′ = v t ( γ ) . A metric version of this result is possible (Lisini). Kuwada lemma. If µ t = P t f m with f ≥ 0 , � X f dm = 1 , then µ t ∈ AC 2 � [ 0 , 1 ]; ( P 2 ( X ) , W 2 ) � with |∇ P t f | 2 � µ t | 2 ≤ for L 1 -a.e. t > 0 . ∗ | ˙ dm P t f { P t f > 0 } Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 22 / 26
Key tools: superposition principle and Kuwada lemma The superposition principle represents currents as superposition of elementary currents associated to curves (L.C.Young, Smirnov, Paolini-Stepanov); in the optimal transport context (AGS) it represents nonnegative solutions to the continuity equation d dt µ t + div ( v t µ t ) = 0 t ∈ ( 0 , 1 ) as marginals µ t of positive finite measures π in C ([ 0 , 1 ]; X ) concentrated on solutions to the ODE γ ′ = v t ( γ ) . A metric version of this result is possible (Lisini). Kuwada lemma. If µ t = P t f m with f ≥ 0 , � X f dm = 1 , then µ t ∈ AC 2 � [ 0 , 1 ]; ( P 2 ( X ) , W 2 ) � with |∇ P t f | 2 � µ t | 2 ≤ for L 1 -a.e. t > 0 . ∗ | ˙ dm P t f { P t f > 0 } Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 22 / 26
Key tools: superposition principle and Kuwada lemma The superposition principle represents currents as superposition of elementary currents associated to curves (L.C.Young, Smirnov, Paolini-Stepanov); in the optimal transport context (AGS) it represents nonnegative solutions to the continuity equation d dt µ t + div ( v t µ t ) = 0 t ∈ ( 0 , 1 ) as marginals µ t of positive finite measures π in C ([ 0 , 1 ]; X ) concentrated on solutions to the ODE γ ′ = v t ( γ ) . A metric version of this result is possible (Lisini). Kuwada lemma. If µ t = P t f m with f ≥ 0 , � X f dm = 1 , then µ t ∈ AC 2 � [ 0 , 1 ]; ( P 2 ( X ) , W 2 ) � with |∇ P t f | 2 � µ t | 2 ≤ for L 1 -a.e. t > 0 . ∗ | ˙ dm P t f { P t f > 0 } Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 22 / 26
Metric Sobolev spaces and weak gradients/1 A closely related identification has to do with definitions of Sobolev spaces and weak notions of gradient in m.m.s. Recall that the minimal relaxed slope |∇ f | ∗ , is the local object that provides integral representation to Ch : Ch ( f ) = 1 � |∇ f | 2 ∗ dm ∀ f ∈ D ( Ch ) . 2 X It has all the natural properties a weak gradient should have, for instance locality f = g on B |∇ f | ∗ = |∇ g | ∗ m -a.e. in B = ⇒ and chain rule |∇ ( φ ◦ f ) | ∗ = | φ ′ ( f ) ||∇ f | ∗ m -a.e. in X . This weak gradient is useful when doing “vertical” (Eulerian) variations ǫ �→ f + ǫ g (i.e. in the dependent variable). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 23 / 26
Metric Sobolev spaces and weak gradients/1 A closely related identification has to do with definitions of Sobolev spaces and weak notions of gradient in m.m.s. Recall that the minimal relaxed slope |∇ f | ∗ , is the local object that provides integral representation to Ch : Ch ( f ) = 1 � |∇ f | 2 ∗ dm ∀ f ∈ D ( Ch ) . 2 X It has all the natural properties a weak gradient should have, for instance locality f = g on B |∇ f | ∗ = |∇ g | ∗ m -a.e. in B = ⇒ and chain rule |∇ ( φ ◦ f ) | ∗ = | φ ′ ( f ) ||∇ f | ∗ m -a.e. in X . This weak gradient is useful when doing “vertical” (Eulerian) variations ǫ �→ f + ǫ g (i.e. in the dependent variable). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 23 / 26
Metric Sobolev spaces and weak gradients/1 A closely related identification has to do with definitions of Sobolev spaces and weak notions of gradient in m.m.s. Recall that the minimal relaxed slope |∇ f | ∗ , is the local object that provides integral representation to Ch : Ch ( f ) = 1 � |∇ f | 2 ∗ dm ∀ f ∈ D ( Ch ) . 2 X It has all the natural properties a weak gradient should have, for instance locality f = g on B |∇ f | ∗ = |∇ g | ∗ m -a.e. in B = ⇒ and chain rule |∇ ( φ ◦ f ) | ∗ = | φ ′ ( f ) ||∇ f | ∗ m -a.e. in X . This weak gradient is useful when doing “vertical” (Eulerian) variations ǫ �→ f + ǫ g (i.e. in the dependent variable). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 23 / 26
Metric Sobolev spaces and weak gradients/1 A closely related identification has to do with definitions of Sobolev spaces and weak notions of gradient in m.m.s. Recall that the minimal relaxed slope |∇ f | ∗ , is the local object that provides integral representation to Ch : Ch ( f ) = 1 � |∇ f | 2 ∗ dm ∀ f ∈ D ( Ch ) . 2 X It has all the natural properties a weak gradient should have, for instance locality f = g on B |∇ f | ∗ = |∇ g | ∗ m -a.e. in B = ⇒ and chain rule |∇ ( φ ◦ f ) | ∗ = | φ ′ ( f ) ||∇ f | ∗ m -a.e. in X . This weak gradient is useful when doing “vertical” (Eulerian) variations ǫ �→ f + ǫ g (i.e. in the dependent variable). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 23 / 26
Metric Sobolev spaces and weak gradients/1 A closely related identification has to do with definitions of Sobolev spaces and weak notions of gradient in m.m.s. Recall that the minimal relaxed slope |∇ f | ∗ , is the local object that provides integral representation to Ch : Ch ( f ) = 1 � |∇ f | 2 ∗ dm ∀ f ∈ D ( Ch ) . 2 X It has all the natural properties a weak gradient should have, for instance locality f = g on B |∇ f | ∗ = |∇ g | ∗ m -a.e. in B = ⇒ and chain rule |∇ ( φ ◦ f ) | ∗ = | φ ′ ( f ) ||∇ f | ∗ m -a.e. in X . This weak gradient is useful when doing “vertical” (Eulerian) variations ǫ �→ f + ǫ g (i.e. in the dependent variable). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 23 / 26
Metric Sobolev spaces and weak gradients/1 A closely related identification has to do with definitions of Sobolev spaces and weak notions of gradient in m.m.s. Recall that the minimal relaxed slope |∇ f | ∗ , is the local object that provides integral representation to Ch : Ch ( f ) = 1 � |∇ f | 2 ∗ dm ∀ f ∈ D ( Ch ) . 2 X It has all the natural properties a weak gradient should have, for instance locality f = g on B |∇ f | ∗ = |∇ g | ∗ m -a.e. in B = ⇒ and chain rule |∇ ( φ ◦ f ) | ∗ = | φ ′ ( f ) ||∇ f | ∗ m -a.e. in X . This weak gradient is useful when doing “vertical” (Eulerian) variations ǫ �→ f + ǫ g (i.e. in the dependent variable). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 23 / 26
Metric Sobolev spaces and weak gradients/2 But, when computing variations of the entropy in the LSV theory, the “horizontal” (Lagrangian) variations ǫ → f ( γ ǫ ) (i.e. in the independent variable) are necessary. These are related to another notion of weak gradient, denoted |∇ f | w , and defined as follows. Let us recall, first, the notion of upper gradient (Heinonen-Koskela): it is a function G satisfying � ( ∗ ) | f ( γ 1 ) − f ( γ 0 ) | ≤ G γ on all absolutely continuous curves γ . Obviously G ≥ |∇ f | in a “smooth” setting and the smallest upper gradient is precisely |∇ f | . In AGS we considered the so-called weak upper gradient property by requiring (*) along “almost all” curves γ in AC 2 ([ 0 , 1 ]; X ) . Then, we define |∇ f | w as the weak upper gradient G with smallest L 2 ( X , m ) norm. This is related to a notion introduced by Koskela-MacManus, Shanmugalingham, but with a different notion of null set of curves, based on p -Modulus (Beurling-Ahlfors). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 24 / 26
Metric Sobolev spaces and weak gradients/2 But, when computing variations of the entropy in the LSV theory, the “horizontal” (Lagrangian) variations ǫ → f ( γ ǫ ) (i.e. in the independent variable) are necessary. These are related to another notion of weak gradient, denoted |∇ f | w , and defined as follows. Let us recall, first, the notion of upper gradient (Heinonen-Koskela): it is a function G satisfying � ( ∗ ) | f ( γ 1 ) − f ( γ 0 ) | ≤ G γ on all absolutely continuous curves γ . Obviously G ≥ |∇ f | in a “smooth” setting and the smallest upper gradient is precisely |∇ f | . In AGS we considered the so-called weak upper gradient property by requiring (*) along “almost all” curves γ in AC 2 ([ 0 , 1 ]; X ) . Then, we define |∇ f | w as the weak upper gradient G with smallest L 2 ( X , m ) norm. This is related to a notion introduced by Koskela-MacManus, Shanmugalingham, but with a different notion of null set of curves, based on p -Modulus (Beurling-Ahlfors). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 24 / 26
Metric Sobolev spaces and weak gradients/2 But, when computing variations of the entropy in the LSV theory, the “horizontal” (Lagrangian) variations ǫ → f ( γ ǫ ) (i.e. in the independent variable) are necessary. These are related to another notion of weak gradient, denoted |∇ f | w , and defined as follows. Let us recall, first, the notion of upper gradient (Heinonen-Koskela): it is a function G satisfying � ( ∗ ) | f ( γ 1 ) − f ( γ 0 ) | ≤ G γ on all absolutely continuous curves γ . Obviously G ≥ |∇ f | in a “smooth” setting and the smallest upper gradient is precisely |∇ f | . In AGS we considered the so-called weak upper gradient property by requiring (*) along “almost all” curves γ in AC 2 ([ 0 , 1 ]; X ) . Then, we define |∇ f | w as the weak upper gradient G with smallest L 2 ( X , m ) norm. This is related to a notion introduced by Koskela-MacManus, Shanmugalingham, but with a different notion of null set of curves, based on p -Modulus (Beurling-Ahlfors). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 24 / 26
Metric Sobolev spaces and weak gradients/2 But, when computing variations of the entropy in the LSV theory, the “horizontal” (Lagrangian) variations ǫ → f ( γ ǫ ) (i.e. in the independent variable) are necessary. These are related to another notion of weak gradient, denoted |∇ f | w , and defined as follows. Let us recall, first, the notion of upper gradient (Heinonen-Koskela): it is a function G satisfying � ( ∗ ) | f ( γ 1 ) − f ( γ 0 ) | ≤ G γ on all absolutely continuous curves γ . Obviously G ≥ |∇ f | in a “smooth” setting and the smallest upper gradient is precisely |∇ f | . In AGS we considered the so-called weak upper gradient property by requiring (*) along “almost all” curves γ in AC 2 ([ 0 , 1 ]; X ) . Then, we define |∇ f | w as the weak upper gradient G with smallest L 2 ( X , m ) norm. This is related to a notion introduced by Koskela-MacManus, Shanmugalingham, but with a different notion of null set of curves, based on p -Modulus (Beurling-Ahlfors). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 24 / 26
Metric Sobolev spaces and weak gradients/2 But, when computing variations of the entropy in the LSV theory, the “horizontal” (Lagrangian) variations ǫ → f ( γ ǫ ) (i.e. in the independent variable) are necessary. These are related to another notion of weak gradient, denoted |∇ f | w , and defined as follows. Let us recall, first, the notion of upper gradient (Heinonen-Koskela): it is a function G satisfying � ( ∗ ) | f ( γ 1 ) − f ( γ 0 ) | ≤ G γ on all absolutely continuous curves γ . Obviously G ≥ |∇ f | in a “smooth” setting and the smallest upper gradient is precisely |∇ f | . In AGS we considered the so-called weak upper gradient property by requiring (*) along “almost all” curves γ in AC 2 ([ 0 , 1 ]; X ) . Then, we define |∇ f | w as the weak upper gradient G with smallest L 2 ( X , m ) norm. This is related to a notion introduced by Koskela-MacManus, Shanmugalingham, but with a different notion of null set of curves, based on p -Modulus (Beurling-Ahlfors). Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 24 / 26
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