Branching geodesics in metric spaces with Ricci curvature
Transcription
Branching geodesics in metric spaces with Ricci curvature
Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces Branching geodesics in metric spaces with Ricci curvature lower bounds Tapio Rajala University of Jyväskylä [email protected] Interactions Between Analysis and Geometry: Analysis on Metric Spaces IPAM, UCLA, Mar 18th, 2013 Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces in metric spaces in geodesic metric spaces The space (P(X ), W2) For µ, ν ∈ P(X ) define Z W2 (µ, ν) = inf X ×X X 1/2 (p1 )# σ = µ d(x, y ) dσ(x, y ) . (p2 )# σ = ν 2 ν σ µ Tapio Rajala X Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces in metric spaces in geodesic metric spaces The space (P(X ), W2) For µ, ν ∈ P(X ) define Z W2 (µ, ν) = inf X ×X X 1/2 (p1 )# σ = µ d(x, y ) dσ(x, y ) . (p2 )# σ = ν 2 σ ν µ Tapio Rajala X Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces in metric spaces in geodesic metric spaces The space (P(X ), W2) For µ, ν ∈ P(X ) define Z W2 (µ, ν) = inf X ×X X 1/2 (p1 )# σ = µ d(x, y ) dσ(x, y ) . (p2 )# σ = ν 2 ν µ Tapio Rajala X Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces in metric spaces in geodesic metric spaces The space (P(X ), W2) Question (Existence of optimal maps) When is the/an optimal plan induced by a map? (σ = (id , T )# µ for some map T : X → X ) The usual steps in the proof are (given two measures µ, ν ∈ P(X )): 1 Prove that any optimal plan from µ to ν is induced by a map. 2 Linear combinations of optimal plans are optimal. 3 The combination of two different optimal maps is not a map. Hence there is only one optimal map. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces in metric spaces in geodesic metric spaces Existence of optimal maps The existence of optimal maps for the quadratic cost has been obtained for example by Brenier (1991) in Rn with µ ≪ Ln . (Earlier steps by Brenier 1987 and Knott & Smith 1984) McCann (1995) in Rn with µ(E ) = 0 for n − 1-rectifiable E . Gangbo & McCann (1996) in Rn with µ(E ) = 0 for all c − c-hypersurfaces E . McCann (2001) for Riemannian manifolds M with µ ≪ vol. Gigli (2011) for Riemannian manifolds M there exists an optimal map from µ to every µ if and only if µ(E ) = 0 for all c − c-hypersurfaces E . Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces in metric spaces in geodesic metric spaces Existence of optimal maps and . . . Bertrand (2008) for finite dimensional Alexandrov spaces with µ ≪ Hd . Gigli (2012) for non-braching CD(K , N)-spaces, N < ∞, with µ ≪ m. For non-branching CD(K , ∞)-spaces with µ, ν ∈ D(Entm ). Rajala & Sturm (2012) for RCD(K , ∞)-spaces with µ, ν ≪ m. Cavalletti & Huesmann (2013) for non-branching MCP(K , N)-spaces with µ ≪ m. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces in metric spaces in geodesic metric spaces The space (P(X ), W2) for (X , d ) geodesic (All the geodesics in this talk are constant speed geodesics parametrized by [0, 1].) If (X , d) is geodesic, then (P(X ), W2 ) is geodesic. A geodesic (µt ) ⊂ Geo(P(X )) can be realized as a probability measure π ∈ P(Geo(X )) in the sense that for all t ∈ [0, 1] we have (et )# π = (µt ). Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces in metric spaces in geodesic metric spaces From Geo(P(X )) to P(Geo(X )) Optimal plan between µ0 and µ1 as a geodesic in P(X ). µ0 µ1 µ1 2 Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces in metric spaces in geodesic metric spaces From Geo(P(X )) to P(Geo(X )) Optimal plan between µ0 and µ1 as a measure in P(Geo(X )). µ1 µ0 Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces in metric spaces in geodesic metric spaces Optimal plans σ vs. Geo(P(X )) vs. P(Geo(X )) Denote by OptGeo(µ0 , µ1 ) ⊂ P(Geo(X )) the set of all π for which (et )# π is a geodesic connecting µ0 and µ1 . The space OptGeo(µ0 , µ1 ) has the most information on transports between µ0 and µ1 : Usually neither of the maps OptGeo(µ0 , µ1 ) → Geo(P(X )) : π 7→ (t 7→ (et )# π), OptGeo(µ0 , µ1 ) → P(X × X )) : π 7→ (e0 , e1 )# π is injective. Moreover, a geodesic (µt )1t=0 does not (in general) define an optimal plan σ, nor does σ define (µt )1t=0 . Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Optimal transport on Riemannian manifolds zero curvature Curvature changes the size of the support of the transported mass. positive curvature Tapio Rajala negative curvature Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Theorem (Otto & Villani 2000, Cordero-Erausquin, McCann & Schmuckenschläger 2001, von Renesse & Sturm 2005) For any smooth connected Riemannian manifold M and any K ∈ R the following properties are equivalent: 1 Ric(M) ≥ K in the sense that Ricx (v , v ) ≥ K |v |2 for all x ∈ M and v ∈ Tx M. Entvol is K -convex on P2 (M). R Here Entvol (ρvol) = M ρ log ρ dvol and K -convexity of Entvol on P2 (M) means that along every geodesic (µt )1t=0 ⊂ P2 (M) we have 2 Entvol (µs ) ≤ (1 − s)Entvol (µ0 ) + sEntvol (µ1 ) K − s(1 − s)W22 (µ0 , µ1 ) 2 for all s ∈ [0, 1]. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Ricci-curvature bounds in metric spaces Definition (Sturm (2006)) (X , d, m) is a CD(K , ∞)-space (K ∈ R) if between any two absolutely continuous measures there exists a geodesic (µt ) ∈ Geo(P(X R )) along which the entropy Entm (ρm) = ρ log ρ dm is K -convex: Entm (µs ) ≤ (1 − s)Entm (µ0 ) + sEntm (µ1 ) K − s(1 − s)W22 (µ0 , µ1 ) 2 for all s ∈ [0, 1]. There is also a slightly stronger definition by Lott and Villani. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Ricci limit spaces Riemannian manifolds with Ric ≥ K CD(K , ∞) of Lott-Villani CD(K , ∞) of Sturm Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Branching geodesics One problematic feature of the definition is that CD(K , ∞)-spaces include also spaces with branching geodesics; like (R2 , || · ||∞ , L2 ). We say that two geodesics γ 1 6= γ 2 , branch if there exists t0 ∈ (0, 1) so that γt1 = γt2 for all t ∈ [0, t0 ]. γ01 = γ02 γ1 γt10 = γt20 γ2 A space where there are no branching geodesics is called non-branching. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Ricci limit spaces Riemannian manifolds with Ric ≥ K CD(K , ∞) + non-branching CD(K , ∞) of Lott-Villani CD(K , ∞) of Sturm Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization without branching Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization without branching Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization without branching Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization without branching Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization with branching Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization with branching Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization with branching Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization with branching Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization in time with branching Entm µt µ0 µs Tapio Rajala µ1 Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization in time with branching Entm µt µ0 µt1 µs Tapio Rajala µt2 µ1 Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization in time with branching Entm µt µ0 µt1 µs Tapio Rajala µt2 µ1 Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization in time with branching Entm µt µ0 µ1 Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization in time with branching Entm µt µ0 µ1 2 Tapio Rajala µ1 Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization in time with branching Entm µt µ0 µ1 4 Tapio Rajala µ1 2 µ3 4 µ1 Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization in time with branching Entm µt µ0 µ1 4 Tapio Rajala µ1 2 µ3 4 µ1 Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization in time with branching Entm µt µ0 µ1 4 Tapio Rajala µ1 2 µ3 4 µ1 Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Localization in time with branching Not only does the new geodesic satisfy the K -convexity inequality between any three times 0 < t1 < s < t2 < 1, but also the measures along the geodesic have bounded densities (under some assumptions on the initial and final measures). This is true for instance if we start with two measures µ0 and µ1 having bounded densities and bounded supports. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Iterating minimization fails for CD(K , N) R In CD(K , N)-spaces, minimizing the (Rényi) entropy X ρ1−1/N dm does not always produce a geodesic along which the inequalities required by CD(K , N)-condition hold. Entm µt µ0 µ1 4 Tapio Rajala µ1 2 µ1 Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Iterating minimization fails for CD(K , N) R In CD(K , N)-spaces, minimizing the (Rényi) entropy X ρ1−1/N dm does not always produce a geodesic along which the inequalities required by CD(K , N)-condition hold. However, if one of the measures is singular with respect to m, we get the correct CD(K , N)-bounds when taking intermediate points towards this measure. Combining this observation with the density bounds gives Theorem CD(K , N) ⇒ MCP(K , N) (by Ohta). Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Definition A space (X , d, m) is said to satisfy the measure contraction property MCP(K , N) (inpthe sense of Ohta) if for every x ∈ X and A ⊂ X (and A ⊂ B(x, π (N − 1)/K ) if K > 0) with 0 < m(A) < ∞ there exists 1 π ∈ GeoOpt δx , m|A m(A) so that dm ≥ (et )# t N βt (γ0 , γ1 )m(A)dπ(γ) . Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities local Poincaré inequality in non-branching MCP(K , N) Theorem (Lott & Villani, von Renesse, Sturm, Hinde & Petersen, Cheeger & Colding) Suppose that (X , d, m) is a nonbranching MCP(K , N)-space with K ∈ R. Then the weak local Poincaré inequality Z Z g dm |u − huiB(x,r ) | dm ≤ C (N, K , r )r − − B(x,2r ) B(x,r ) holds for any measurable function u defined on X , any upper gradient g of u and for each point x ∈ X and radius r > 0. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities local Poincaré inequality in CD(K , ∞) Theorem Suppose that (X , d, m) is a CD(K , ∞)-space (in the sense of Sturm) with K ∈ R. Then the weak local Poincaré inequality Z Z K −r 2 g dm |u − huiB(x,r ) | dm ≤ 4re B(x,2r ) B(x,r ) holds for any measurable function u defined on X , any upper gradient g of u and for each point x ∈ X and radius r > 0. Observe that we do not have average integrals in this theorem. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Proof of the local Poincaré inequality Let us prove the CD(0, ∞) case for simplicity. We have to show that Z Z g dm. |u − huiB(x,r ) | dm ≤ 4r B(x,2r ) B(x,r ) Abbreviate B = B(x, r ) and denote m(B) M = inf a ∈ R : m({u > a}) ≤ . 2 Split the ball B into two Borel sets B + and B − so that B = B + ∪ B − , B + ∩ B − = ∅, m(B + ) = m(B − ) and u(x) ≤ M ≤ u(y ) Tapio Rajala for all x ∈ B − , y ∈ B + . Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Proof of the local Poincaré inequality Let (µt )1t=0 be a geodesic between m(B1 + ) m|B + and m(B1 − ) m|B − along which we have the density bound (writing µt = ρt m) ρt (y ) ≤ 2 m(B) for all t ∈ [0, 1] at m-almost every y ∈ X . Let π be a corresponding measure on the set of geodesics. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Proof of the local Poincaré inequality From u(z) ≤ M ≤ u(y ) for all (z, y ) ∈ B − × B + we get |u(γ(0)) − u(γ(1))| = |u(γ(0)) − M| + |M − u(γ(1))| for π-almost every γ ∈ Geo(X ). Therefore Z |u(γ(0)) − u(γ(1))| dπ(γ) Geo(X ) Z Z |M − u(γ(1))| dπ(γ) |u(γ(0)) − M| dπ(γ) + = Geo(X ) Geo(X ) Z Z 2 2 |u(z) − M| dm(z) + |M − u(z)| dm(z) = m(B) B + m(B) B − Z 2 = |u(z) − M| dm(z). m(B) B Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition branching vs. non-branching local Poincaré inequalities Proof of the local Poincaré inequality ZZ 1 |u − huiB(x,r ) | dm ≤ |u(z) − u(y )| dm(z) dm(y ) m(B) B×B B(x,r ) ZZ 1 (|u(z) − M| + |M − u(y )|) dm(z) dm(y ) ≤ m(B) B×B Z Z |u(γ(0)) − u(γ(1))| dπ(γ) |u(z) − M| dm(z) = m(B) =2 Z Geo(X ) B ≤ 2r m(B) = 2r m(B) Z 1Z ≤ 4r 0 Z Geo(X ) Z 1Z 0 Z 1 g (γ(t)) dt dπ(γ) 0 g (z)ρt (z) dm(z) dt Z g (z) dm(z) dt = 4r X B(x,2r ) Tapio Rajala g dm. B(x,2r ) Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching Ricci limit spaces Riemannian manifolds with Ric ≥ K CD(K , ∞) + non-branching CD(K , ∞) of Lott-Villani CD(K , ∞) of Sturm Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching Question Does there exist a stable definition of Ricci curvature lower bounds that excludes branching? Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching RCD(K , ∞) Definition (Ambrosio, Gigli & Savaré, 2011 (preprint)) A metric measure space (X , d, m) has Riemannian Ricci curvature bounded below by K ∈ R, or RCD(K , ∞) for short, if one of the following equivalent conditions hold: 1 2 3 (X , d, m) isRa CD(K , ∞) space and the Cheeger-energy Ch(f ) = 12 |Df |2w is a quadratic form on L2 (X , m). (X , d, m) is a CD(K , ∞) space and the W2 gradient flow of Entm is additive on P2 (X ). Any µ ∈ P2 (X ) is the starting point of an EVIK gradient flow of Entm . Theorem (Daneri & Savaré, 2008) EVIK implies strong displacement K -convexity. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching RCD(K , ∞) ⇒ essential non-branching Theorem (R. & Sturm, 2012 (preprint)) Strong CD(K , ∞)-spaces are essentially non-branching. In particular RCD(K , ∞)-spaces are essentially non-branching. Corollary (of essential non-branching and Gigli’s result) There exist optimal transport maps in strong CD(K , ∞)-spaces between µ0 , µ1 ≪ m. Definition A space (X , d, m) is called essentially non-branching if for every µ0 , µ1 ∈ P2 (X ) that are absolutely continuous with respect to m we have that any π ∈ OptGeo(µ0 , µ1 ) is concentrated on a set of non-branching geodesics. (Meaning that π(Γ) = 1 for some Γ ⊂ Geo(X ) so that there are no two branching geodesics in Γ.) Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching Ricci limit spaces Riemannian manifolds with Ric ≥ K RCD(K , ∞) CD(K , ∞) + essential non-branching CD(K , ∞) of Lott-Villani CD(K , ∞) of Sturm Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching “Proof” Suppose that the claim is not true so that there exists a measure π that is not concentrated on non-branching geodesics. By restricting the measure π we may assume that there are 0 < t1 < t2 < 1 with |t1 − t2 | small and two sets of geodesics Γ1 , Γ2 so that (et )# π|Γ1 = (et )# π|Γ2 for all t ∈ [0, t1 ] (et )# π|Γ1 ⊥ (et )# π|Γ2 for all t ∈ [t2 , 1] and for all t ∈ [t2 , 1]. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces 0 t1 definition essential non-branching t2 1 Γ1 Γ2 Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching Entm 0 t1 t2 1 Γ1 Γ2 Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching Entm 0 t1 t2 1 Γ1 Γ2 Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching Entm log 2 0 t1 t2 1 Γ1 Γ2 Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching ess. nb. Ricci-limits RCD(K , N) non-branching CD(K , N) CD(K , N) ∃ maps Local Poincaré MCP(K , N) MCP(K , N) Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching Why is ||ρ1/2||∞ ≤ max{||ρ0 ||∞ , ||ρ1 ||∞ } =: M?. ρ0 ρ1 ρ1 2 M M M A X X Tapio Rajala X Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching Consider the curve Γ ∈ P(Geo(X )) between the marginals corresponding to the part of the measure which we want to redistribute along which Entm is displacement convex. We have 1 1 Entm (Γ 1 ) ≤ Entm (Γ0 ) + Entm (Γ1 ) ≤ log M. 2 2 2 On the other hand, by Jensen’s inequality we always have Z Entm (Γ 1 ) = ρ 1 log ρ 1 dm 2 2 E 2 Z Z 1 ≥ m(E ) − ρ 1 dm log − ρ 1 dm ≥ log , m(E ) E 2 E 2 where E = {x ∈ X : ρ 1 (x) > 0} and Γt = ρt m. Thus 2 m(E ) ≥ Tapio Rajala 1 . M Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching This is why ||ρ1/2||∞ ≤ max{||ρ0 ||∞ , ||ρ1 ||∞ }. The CD(K , ∞) condition gives a new well spread midpoint for the high-density part of the old midpoint. ρ0 ρ1 ρ1 2 M M M A X E X X E Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching This is why ||ρ1/2||∞ ≤ max{||ρ0 ||∞ , ||ρ1 ||∞ }. Taking a weighted combination of this new midpoint measure and the old one lowers the entropy. ρ1 ρ0 ρ1 2 M M X M X Tapio Rajala X Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching This is why ||ρ1/2||∞ ≤ max{||ρ0 ||∞ , ||ρ1 ||∞ }. Therefore at the minimum of the entropy among the midpoints we have the density bound. ρ1 ρ0 ρ1 2 M M X M X Tapio Rajala X Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching The rest of the geodesic. When we continue taking minimizers in the next level midpoints the bound is preserved. ρ3 ρ0 ρ1 ρ1 ρ1 4 M 4 2 M M X M X Tapio Rajala M X X X Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching The rest of the geodesic. Finally we end up with a complete geodesic with the density bound. ρ0 ρ1 M t X X Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching Question MCP(K , N) ⇒ Local Poincaré? Question Does there exist optimal maps in CD(K , N)-spaces from every µ ≪ m? (not all plans are given by maps) Question Local-to-global for CD(K , ∞)? Question Are RCD(K , ∞)-spaces non-branching? Question Are RCD(K , ∞)-spaces Ricci-limits? Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), no. 3, 903–991. K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), no. 1, 65–131. K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), no. 1, 133–177. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching T. R., Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations, 44 (2012), 477–494. T. R., Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm, J. Funct. Anal., 263 (2012), no. 4, 896–924. T. R., Improved geodesics for the reduced curvature-dimension condition in branching metric spaces, Discrete Contin. Dyn. Syst., 33 (2013), 3043–3056. Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching L. Ambrosio, N.Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, preprint (2011). L. Ambrosio, N.Gigli, A. Mondino and T.R., Riemannian Ricci curvature lower bounds in metric spaces with σ-finite measure, Trans. Amer. Math. Soc., to appear. N. Gigli, Optimal maps in non branching spaces with Ricci curvature bounded from below, Geom. Funct. Anal. 22 (2012), no. 4, 990–999. T. R. and K.-Th. Sturm, Non-branching geodesics and optimal maps in strong CD(K , ∞)-spaces, preprint (2012). Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds Optimal mass transportation CD(K , ∞)-spaces RCD(K , ∞)-spaces definition essential non-branching Thank you! Tapio Rajala Branching geodesics in spaces with Ricci curvature lower bounds
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