Curvature and Diffusion in the Heisenberg group Curvature and Diffusion in the Heisenberg group Nicolas JUILLET IRMA, Strasbourg Paris, IHP , October 2014 Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Outline 1 Notations and definitions Curvature 2 3 Diffusion Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Notations and definitions The subRiemannian H . A basis for left-invariant vector fields is X = E 1 − y Y = E 2 + x Z = [ X , Y ] = E 3 . 2 E 3 , 2 E 3 and A curve is horizontal if ˙ γ ∈ Vect ( X , Y ) for any t . Actually γ ( t ) = a ( t ) X ( γ ( t ))+ b ( t ) Y ( γ ( t )) . ˙ � | ˙ � It has norm | ˙ γ | = a 2 ( t )+ b 2 ( t ) and the length of γ is γ | . The diffusion operator is ∆ = X 2 + Y 2 . The horizontal gradient is ∇ f = X f X + Y f Y . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Notations and definitions The Riemannian H ε for ε > 0 Let ε > 0 and H ε := ( R 3 , d ε , L 3 ) . An orthonormal basis at point ( x , y , u ) is X = E 1 − y Y = E 2 + x ε Z = ε [ X , Y ] = ε E 3 . 2 E 3 , 2 E 3 and If ˙ γ ( t ) = ( a ( t ) X + b ( t ) Y + c ( t ) Z )( γ ( t )) � � � a 2 + b 2 + c 2 a 2 + b 2 + c 2 then | ˙ γ ( t ) | ε = ε 2 and length ε ( γ ) = ε 2 ( t ) dt . The gradient is ∇ ε f = ∇ f +( ε 2 Z f ) Z . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Notations and definitions Curves in P 2 ( X ) . For a metric measure space ( X , d , ν ) , two theories make use of curves in the space of probability measures P 2 ( X ) endowed with the transport distance W . Ricci bounds: geodesic curves. (Lott-Villani and Sturm) Heat diffusion: curves with maximal slope. (Ambrosio-Gigli-Savaré) Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Notations and definitions Comparaison of the Wasserstein distances Let W be the L 2 -minimal metric with respect to the Carnot-Carathéodory metric d CC and W ε with respect to d ε . W ε ≤ W , P 2 ( H ) = P 2 ( H ε ) as topological spaces. Lipschitz curves (resp. absolutely continuous) of P 2 ( H ) are Lipschitz (resp. absolutely continuous) in P 2 ( H ε ) . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Notations and definitions Geodesic curves Theorem (Ambrosio-Rigot, 2004) Let µ 0 , µ 1 ∈ P 2 ( H ) such that µ 0 is absolutely continuous. Then there is a unique optimal coupling π . It is π = ( Id ⊗ T ) # µ 0 for some map T. Moreover there is a unique geodesic between p and T ( p ) ( µ 0 -almost surely). In fact T ( p ) = p . exp H ( ∇ψ ( p ) , Z ψ ( p )) . Here ψ : H → R depends on µ 0 , µ 1 . The unique geodesic ( µ s ) s ∈ [ 0 , 1 ] between µ 0 and µ 1 is defined by µ s = ( T s ) # µ 0 where T s ( p ) = p . exp H ( s ∇ψ ( p ) , sZ ψ ( p )) . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Notations and definitions Absolutely continuous curves Theorem Let ( µ t ) t ∈ [ 0 , T ] be an absolutely continuous curve of P 2 ( H ) . Then for almost every t > 0 there exists a field of horizontal vectors v t ∈ Tan µ t P 2 ( H ) such that the continuity equation is satisfied d µ t / dt + div ( v t µ t ) = 0 . � � | v t | 2 d µ t = lim h → 0 + h − 1 W ( µ t , µ t + h ) =: Speed ( µ t ) . Moreover Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Notations and definitions The relative entropy The relative entropy of µ = ρ L is given by � H ( µ ) = H ( µ | L ) = ρ ln ( ρ )( x ) d L ( x ) . Big entropy: µ concentrated on a few space. Small entropy: µ take a lot of space. Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature Ricci curvature bounds on a Riemannian manifolds Theorem Let ( M , g ) be a Riemannian manifold of dimension N and K ∈ R . Then the following properties are equivalent. ∀ ( x , v ) ∈ TM , Ricci ( v , v ) ≥ Kg ( v , v ) 1 ∀ f ∈ C ∞ ( M ) , ∀ x ∈ M , Γ 2 ( f )( x ) ≥ K Γ( f )( x )+(∆ f ( x )) 2 / N 2 ( M , d g , vol g ) satisfies the Curvature Dimension Condition CD ( K , N ) , 3 ( M , d g , vol g ) satisfies the Measure Contraction Property MCP ( K , N ) 4 Under this condition, one can deduce a lot of analytico-geometric results. Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature MCP is satisfied Jacobian estimate For any e ∈ H . The contraction map M s e : f → M s ( e , f ) is differentiable with Jac ( M s e )( f ) ≥ s 5 . Equality case : e and f are on a horizontal line. As a consequence ( H , d c , L 3 ) satisfies MCP ( 0 , 5 ) : Measure Contraction Property MCP ( 0 , N ) for ( X , d , ν ) : for every point e ∈ X and for all s ∈ [ 0 , 1 ] , M s e # ν ≤ s N ν . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature K-convexity and Ricci curvature Theorem (Cordero–McCann–Schmuckenschläger and Sturm–von Renesse) Let M be a Riemannian manifold. It has Ricci curvature bounded from below by K ∈ R if and only if for every geodesic ( µ t ) t ∈ [ 0 , T ] in P 2 ( M ) of speed 1 the function t ∈ [ 0 , T ] �→ H ( µ t | Vol M ) − K . t 2 2 ∈ R is convex. This theorem has been turned into the definition of CD ( K , + ∞ ) for metric measure spaces: A space satisfies CD ( K , + ∞ ) if H ( . | ν ) is K -convex along every geodesic curve of ( P 2 ( X ) , W d ) . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature The curvature dimension condition is not satisfied. Theorem (J.) In P 2 ( H ) the entropy H with respect to L is not K-convex along geodesics (for any K ∈ R ). The curvature-dimension conditions introduced by Lott–Villani and Sturm are not satisfied Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature The general theorem is Theorem (J. 2009) In ( H n , L 2 n + 1 , d c ) , the Heisenberg group with the Lebesgue measure and the Carnot-Carathéodory distance, MCP ( K , N ) is true if and only if N ≥ 2 n + 3 and K ≤ 0 , CD ( K , N ) is false for every ( K , N ) . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature Sketch of proof a ′ a E F p ′ p 0 H b ′ b Let F be a small ball such that 0 H and the center of the ball are on a “bad" geodesic. For E we take the “geodesic inverse" of F . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature Sketch of proof a ′ a E F p ′ p 0 H b ′ b Let F be a small ball such that 0 H and the center of the ball are on a “bad" geodesic. For E we take the “geodesic inverse" of F . It turns out that L ( E ) = L ( F ) . � � M 1 / 2 ( E , F ) We want to prove L < L ( F ) because it is a contradiction to the Brunn-Minkowski inequality. Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature Sketch of proof a ′ a p ′ p 0 H b ′ b For each e ∈ E , the contracted set M 1 / 2 ( e , F ) is a sort of ellipsoid that contains 0 H . The volume of such an ellipsoid is 2 − 5 L ( F ) . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature Sketch of proof a ′ a p ′ p 0 H b ′ b For each e ∈ E , the contracted set M 1 / 2 ( e , F ) is a sort of ellipsoid that contains 0 H . The volume of such an ellipsoid is 2 − 5 L ( F ) . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature Sketch of proof a ′ a p ′ p 0 H b ′ b For each e ∈ E , the contracted set M 1 / 2 ( e , F ) is a sort of ellipsoid that contains 0 H . The volume of such an ellipsoid is 2 − 5 L ( F ) . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature Sketch of proof a ′ a p ′ p 0 H b ′ b The midset M 1 / 2 ( E , F ) is made of the reunion of these ellipsoids. Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Curvature Sketch of proof a ′ a p ′ p 0 H b ′ b All of them contains 0 H . Then M 1 / 2 ( E , F ) is an ellipsoid of size 2. Its volume is = L ( F ) 2 3 � · 2 − 5 L ( F ) � . 4 Then L ( M 1 / 2 ) < L ( F ) = L ( E ) . Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
Curvature and Diffusion in the Heisenberg group Diffusion After Jordan-Kinderlehrer-Otto (2000) “The heat flow is the gradient flow of H in P 2 ( X ) " Gradient flow means ˙ γ t = − ∇ F ( γ t ) . We need a metric definition of this equation. Nicolas JUILLET Curvature and Diffusion in the Heisenberg group
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