Calculus, heat flow and curvature-dimension bounds in ...

Calculus, heat flow and curvature-dimensionbounds in metric measure spaces

Luigi Ambrosio

Scuola Normale Superiore, [email protected]

http://cvgmt.sns.it

Luigi Ambrosio (SNS) Calculus in metric measure spaces ICM, August 2018 1 / 40

1 Motivations

2 Weakly differentiable functions

3 Optimal Transport and Heat Flow

4 Curvature/dimension bounds

1 Motivations

Goal and motivations

Metric measure space (m.m.s.): (X , d,m), with (X , d) complete and m Borelmeasure, nonnegative, finite on bounded sets.

Goal: Can we extend to these abstract structures the classical notions of calculus(gradient, differential, Laplacian,.....), thus giving a precise meaning to ODEs,PDEs, and to differential geometric properties?

Motivations: Metric and Differential Geometry, Data Sets, Networks, Geometryof diffusion Operators,...

A basic example of metric measure spaceWeighted Riemannian manifold (Mn, g,m) with• d = dg Riemannian distance;• m = e−V volMn .The addition of the weight function e−V makes the theory richer of applicationsand more “stable”. The Gauss space

(Rn, deu,

1(2π)n/2 e−|x |

2/2L n).(limit of n-dimensional linear projections of normalized surface measures on√

m Sm as m→∞) and the spaces([0, π], deu, (sin t)N−1L 1)

are typical models of the theory.Luigi Ambrosio (SNS) Calculus in metric measure spaces ICM, August 2018 4 / 40

(Rn, deu,

1(2π)n/2 e−|x |

(Rn, deu,

1(2π)n/2 e−|x |

(Rn, deu,

1(2π)n/2 e−|x |

Eulerian/Lagrangian dualityMany recent developments ultimately depend on the identification of conceptsstated in Eulerian terms (dealing with gradients, Laplacians, Hessians,...) withthose stated in Lagrangian terms (dealing with 1d curves). The relevance of theseidentifications is not sufficiently recognized when everything is smooth.

First example. One can compute the modulus of ∇f , either by

|∇f |(x) = lim supy→x

|f (y)− f (x)||y − x |

or, in Lagrangian terms, by

|∇f |(x) = sup ∂v f (x) : |v | = 1

= sup dx f (γt) : γt = x , Lip(γ) ≤ 1 .

|f (y)− f (x)||y − x |

|∇f |(x) = sup ∂v f (x) : |v | = 1

|f (y)− f (x)||y − x |

|∇f |(x) = sup ∂v f (x) : |v | = 1

|f (y)− f (x)||y − x |

|∇f |(x) = sup ∂v f (x) : |v | = 1

Eulerian/Lagrangian dualitySecond example: The ODE γt = b(t , γt), γ0 = x corresponds, in Eulerian terms,to the continuity equation

(PDE) ∂tu + div (bu) = 0 (t , x) ∈ R× X

because the flow map X (t , x) := γt of the ODE provides the explicit formula

u(t ,X (t , x)) =u(0, x)

det∇xX (t , x)(t , x) ∈ R× X .

The study of the connections between (ODE) and (PDE), first when b(t , ·) is notsmooth (DiPerna-Lions), then in general metric measure structures (A-Trevisan),led to basic well-posedness and stability results.

Eulerian/Lagrangian dualitySecond example: The ODE γt = b(t , γt), γ0 = x corresponds, in Eulerian terms,to the continuity equation

(PDE) ∂tu + div (bu) = 0 (t , x) ∈ R× X

because the flow map X (t , x) := γt of the ODE provides the explicit formula

u(t ,X (t , x)) =u(0, x)

det∇xX (t , x)(t , x) ∈ R× X .

The study of the connections between (ODE) and (PDE), first when b(t , ·) is notsmooth (DiPerna-Lions), then in general metric measure structures (A-Trevisan),led to basic well-posedness and stability results.

Eulerian/Lagrangian duality

Third example: The incompressible Euler equations in fluid-mechanics arewritten as

∂tv + (v · ∇)v = −∇p, divxv = 0

and, in Lagrangian terms, as

γt = −∇p(t , γt).

Here, when v and p are not smooth, the equivalence of the descriptions/notionsof weak solution is much more subtle and much less understood (Brenier,Shnirelman, Sheffer, De Lellis-Székelyhidi, Isett, Buckmaster-Vicol,...).

Eulerian/Lagrangian duality

Third example: The incompressible Euler equations in fluid-mechanics arewritten as

∂tv + (v · ∇)v = −∇p, divxv = 0

and, in Lagrangian terms, as

γt = −∇p(t , γt).

Here, when v and p are not smooth, the equivalence of the descriptions/notionsof weak solution is much more subtle and much less understood (Brenier,Shnirelman, Sheffer, De Lellis-Székelyhidi, Isett, Buckmaster-Vicol,...).

Weakly differentiable functionsWeak notions of derivative go back to the 1-d Calculus of Variations (Vitali, Tonelli,Lebesgue,...)

min∫ b

aL(t , γt , γt) dt : γa = A, γb = B

Weakly differentiable functionsWeak notions of derivative go back to the 1-d Calculus of Variations (Vitali, Tonelli,Lebesgue,...)

min∫ b

aL(t , γt , γt) dt : γa = A, γb = B

and to Dirichlet’s principle, for the solution of−∆u = 0:

min∫

|∇u|2 dx : u = g on ∂Ω

This led to the modern theory of Sobolevspaces, now a basic ingredient of PDE’s theoryand Geometric Analysis.

∂ΩΩ

Weakly differentiable functions

Many authors contributed to the development of this theory (Levi, Leray, Morrey,Evans, Schwartz, Sobolev, Fuglede,...) and now we recognize that, in theEuclidean space Rn, the approaches

• distributional derivatives• approximation by smooth functions• good behaviour along almost all lines (curves)

are equivalent.

Distributional derivatives. f ∈W 1,p(Rn) if f ∈ Lp(Rn) and∫Rn

f∇φ dx = −∫Rn

Fφ dx ∀φ ∈ C∞c (Rn)

for some vector field F ∈ Lp(Rn;Rn) (the weak derivative).

Approximation by smooth functions. f ∈ H1,p(Rn) if f ∈ Lp(Rn) and there existfi ∈ C∞(Rn), F ∈ Lp(Rn;Rn) with

limi→∞

∫Rn|fi − f |p + |∇fi − F |p dx = 0.

Distributional derivatives. f ∈W 1,p(Rn) if f ∈ Lp(Rn) and∫Rn

f∇φ dx = −∫Rn

Fφ dx ∀φ ∈ C∞c (Rn)

for some vector field F ∈ Lp(Rn;Rn) (the weak derivative).

Approximation by smooth functions. f ∈ H1,p(Rn) if f ∈ Lp(Rn) and there existfi ∈ C∞(Rn), F ∈ Lp(Rn;Rn) with

limi→∞

∫Rn|fi − f |p + |∇fi − F |p dx = 0.

Good behaviour along p-almost all curves. (Fuglede) f ∈ BL1,p(Rn) iff ∈ Lp(Rn) and there exists F ∈ Lp(Rn;Rn) with

f (γ1)− f (γ0) =

F for p-almost every curve γ : [0,1]→ Rn.

Beppo Levi first introduced this concept in 1901, restricting to straight lines inthe coordinate directions. Here the precise meaning of “p-almost every curve”appeals to the notion of p-Modulus, a capacitary notion (Ahlfors-Beurling).

f (γ1)− f (γ0) =

Understanding properly what we mean by “smooth function”, “vector field” ,“gradient” (Weaver, Cheeger, Hajlasz, Koskela-MacManus, Shanmugalingam,...)in the m.m.s. setting, leads to the following result:

Theorem.In any m.m.s. (X , d,m), the three definitions, properly adapted, are equivalentand define the same “gradient”.

The proof in full generality of this statement (A-Gigli-Savaré, Di Marino) requirestools (Hopf-Lax semigroup, superposition principle) providing “bridges” betweenthese viewpoints, as well as key ideas from Optimal Transport.

Background on Optimal Transport

The theory of Optimal Transport, pionereed by Monge (1781) and Kantorovich(1939) has been mostly “confined” for many years in the areas of Probability,Statistics, Linear Programming, Optimization.

In the last two decades, many more connections and applications emerged, inPDE’s, Functional and Geometric inequalities, and (even more recently) MetricGeometry, thanks to:

• new “nonlinear” interpolation between probability measures (Mc Cann)

• new “geometry” of the space of probability measures (Brenier, Otto).

Monge’s formulation

Let µ, ν ∈P(X ) and c : X × X → [0,∞) a Borel cost function.

Monge’s formulation

Let µ, ν ∈P(X ) and c : X × X → [0,∞) a Borel cost function.

Minimize∫

X c(x ,T (x)) dµ(x) among alltransport maps T : X → X pushing µ toν (in short T#µ = ν), i.e.

µ(T−1(E)) = ν(E) ∀E ∈ B(X ).

Tµ ν

T−1(E) E

Kantorovich’s formulation

Minimize∫

X 2 c(x , y) dΣ(x , y) among all couplingsΣ ∈P(X × X ) of µ and ν, i.e.

Σ(A× X ) = µ(A), Σ(X × B) = ν(B).

So, Σ(A× B) is the amount of mass at A,shipped to B.

In Pp(X ) :=µ :

∫X dp(·, x) dµ <∞

, 1 ≤ p <∞, the choice c = dp induces the

“nonlinear” Lp distances (Kantorovich-Rubinstein, Wasserstein)

Wp(µ, ν) :=

∫X 2

dp(x , y) dΣ(x , y)

which metrize weak convergence in P(X ), plus convergence of p-th moments.Luigi Ambrosio (SNS) Calculus in metric measure spaces ICM, August 2018 15 / 40

Kantorovich’s formulation

Minimize∫

X 2 c(x , y) dΣ(x , y) among all couplingsΣ ∈P(X × X ) of µ and ν, i.e.

Σ(A× X ) = µ(A), Σ(X × B) = ν(B).

So, Σ(A× B) is the amount of mass at A,shipped to B.

In Pp(X ) :=µ :

∫X dp(·, x) dµ <∞

, 1 ≤ p <∞, the choice c = dp induces the

“nonlinear” Lp distances (Kantorovich-Rubinstein, Wasserstein)

Wp(µ, ν) :=

∫X 2

dp(x , y) dΣ(x , y)

which metrize weak convergence in P(X ), plus convergence of p-th moments.Luigi Ambrosio (SNS) Calculus in metric measure spaces ICM, August 2018 15 / 40

McCann’s displacement interpolationIn Rn, if T is an optimal transport map from µ ∈P2(Rn) to ν ∈P2(Rn), then withTt = (1− t)Id + tT , one has the displacement interpolation

µt := (Tt)#µ ∈ Geo(P2(Rn)

)and Tt is optimal from µ to µt .

With this new interpolation strategy in P2(Rn), McCann proved that Rényi’sentropy

Rn(µ) := −∫Rn%1−1/n dx µ = %L n

is displacement convex and used this fact to establish a finer version of theBrunn-Minkowski inequality(

L n(A + B))1/n ≥

(L n(A)

)1/n+(L n(B)

McCann’s displacement interpolationIn Rn, if T is an optimal transport map from µ ∈P2(Rn) to ν ∈P2(Rn), then withTt = (1− t)Id + tT , one has the displacement interpolation

µt := (Tt)#µ ∈ Geo(P2(Rn)

)and Tt is optimal from µ to µt .

With this new interpolation strategy in P2(Rn), McCann proved that Rényi’sentropy

Rn(µ) := −∫Rn%1−1/n dx µ = %L n

is displacement convex and used this fact to establish a finer version of theBrunn-Minkowski inequality(

L n(A + B))1/n ≥

(L n(A)

)1/n+(L n(B)

Dynamic formulationMore generally, we can focus on geodesic spaces (X , d), with

Geo(X ) :=

constant speed geodesics γ : [0,1]→ X.

The problem with c = d2 can now be written in terms of geodesic plans η, namelyprobability measures on Geo(X ):

Minimize∫

Geo(X)

length2(γ) dη(γ) among all η ∈P(Geo(X )) from µ to ν.

µ = (e0)#η ν = (e1)#η(et )#η

γ(0)γ(t)

The optimal transport problemcanonically induces probabilitymeasures on Geo(X ). Conversely,geodesics in P2(X ) are all induced byprobability measures in Geo(X )!

Geo(X ) :=

Minimize∫

Geo(X)

µ = (e0)#η ν = (e1)#η(et )#η

γ(0)γ(t)

Geo(X ) :=

Minimize∫

Geo(X)

µ = (e0)#η ν = (e1)#η(et )#η

γ(0)γ(t)

Otto calculus

µ′t

P2(Rn)TµP2(Rn)

µ′t

Otto calculusHaving in mind the continuity equation ∂tµ + div (vµ) = 0, Otto linked infinitesimalvariations s ∈ TµP2(Rn) to gradient velocities v = ∇φ and defined a (formal)Riemannian metric

gµ(s, s′) :=

∫Rn〈∇φ,∇φ′〉dµ − div (µ∇φ) = s, −div (µ∇φ′) = s′.

It turns out (Benamou-Brenier) that the induced Riemannian distance

d2g(µ0, µ1) := min

∫Rn|v(t , ·)|2 dµt dt : ∂tµ + div (vµ) = 0

is precisely W 2

2 (µ, ν)!

Otto calculusHaving in mind the continuity equation ∂tµ + div (vµ) = 0, Otto linked infinitesimalvariations s ∈ TµP2(Rn) to gradient velocities v = ∇φ and defined a (formal)Riemannian metric

gµ(s, s′) :=

∫Rn〈∇φ,∇φ′〉dµ − div (µ∇φ) = s, −div (µ∇φ′) = s′.

It turns out (Benamou-Brenier) that the induced Riemannian distance

d2g(µ0, µ1) := min

∫Rn|v(t , ·)|2 dµt dt : ∂tµ + div (vµ) = 0

is precisely W 2

2 (µ, ν)!

Heat FlowThe heat equation

∂tu = ∆u, u ≥ 0, u(0, ·) = u0,

u0(x) dx = 1

can be understood from many and equally important points of view.The probabilistic one views u(t , ·)L n as the law of the standard Brownian motionX u0

t starting from u0:∫Rn

u(t , x)φ(x) dx = E(φ(X u0

∀φ ≥ 0.

The functional-analytic one (which does not need the sign and normalizationrestrictions) is related to the theory of gradient flows and semigroups.

∂tu = ∆u, u ≥ 0, u(0, ·) = u0,

u0(x) dx = 1

∀φ ≥ 0.

∂tu = ∆u, u ≥ 0, u(0, ·) = u0,

u0(x) dx = 1

∀φ ≥ 0.

Heat Flow

Given a Riemannian manifold (M, g) andS : M→ R, the metric g converts dxSinto a vector ∇S(x)

g(∇S(x), v) = dxS(v) ∀v ∈ TxM

and gives meaning to the (downward)gradient flow equation γt = −∇S(γt).

Then, the functional-analytic interpretation of the heat equation is that u(t , ·) is thegradient flow in the L2 metric of Dirichlet’s energy:

D(u) :=12

∫Rn|∇u|2 dx .

Heat Flow

Given a Riemannian manifold (M, g) andS : M→ R, the metric g converts dxSinto a vector ∇S(x)

g(∇S(x), v) = dxS(v) ∀v ∈ TxM

and gives meaning to the (downward)gradient flow equation γt = −∇S(γt).

Then, the functional-analytic interpretation of the heat equation is that u(t , ·) is thegradient flow in the L2 metric of Dirichlet’s energy:

D(u) :=12

∫Rn|∇u|2 dx .

Heat FlowHowever, at the end of the ’90, Jordan-Kinderlehrer-Otto realized that we canobtain another “Lagrangian” representation of the heat flow as the gradient flow ofthe Entropy Functional

S(µ) :=

∫Rn % log % dx if µ = %L n;

+∞ otherwise

in P2(Rn) with respect to the distance W2!

One of the ways to realize this is to use the (formal) Riemannian structure ofP2(Rn), which provides the formula

∇W S(µ) = ∇ log % if µ = %L n.

Heat FlowHowever, at the end of the ’90, Jordan-Kinderlehrer-Otto realized that we canobtain another “Lagrangian” representation of the heat flow as the gradient flow ofthe Entropy Functional

S(µ) :=

∫Rn % log % dx if µ = %L n;

+∞ otherwise

in P2(Rn) with respect to the distance W2!

One of the ways to realize this is to use the (formal) Riemannian structure ofP2(Rn), which provides the formula

∇W S(µ) = ∇ log % if µ = %L n.

Heat Flow

This discovery led to many more interpretations of conservative PDE’s

∂tρ = div(ρ∇U ′(ρ) + ρ∇V + ρ∇(W ∗ ρ)

)(including Fokker-Planck, porous medium, interacting particle systems, etc.)as gradient flows of suitable “entropies”, generating an enormous literature onstability, trends of convergence to equilibrium,...

A key property, which will play also a role in the abstract and geometric setting, isthat now the roles of d and m are nicely decoupled, unlike what happens for D:• d enters only in the computation of the optimal transport distance W2,• m enters only in the definition of the entropy S.

Heat Flow

∂tρ = div(ρ∇U ′(ρ) + ρ∇V + ρ∇(W ∗ ρ)

Heat Flow

∂tρ = div(ρ∇U ′(ρ) + ρ∇V + ρ∇(W ∗ ρ)

Back to metric measure spacesIn m.m.s. (X , d,m) the JKO interpretation makes perfectly sense, now with theRelative Entropy Functional (Kullback-Leibler divergence)

Entm(µ) :=

∫X % log % dm if µ = %m;

+∞ otherwiseµ ∈P2(X )

(and using a “metric” notion of gradient flow, due to De Giorgi).

But, also the L2-gradient flow interpretation makes sense, replacing Dirichlet’senergy D with the so-called Cheeger energy:

Ch(u) := inf

lim infh→∞

∫X|∇uh|2 dm : uh ∈ Lip(X , d), ‖uh − u‖2 → 0

Back to metric measure spacesIn m.m.s. (X , d,m) the JKO interpretation makes perfectly sense, now with theRelative Entropy Functional (Kullback-Leibler divergence)

Entm(µ) :=

∫X % log % dm if µ = %m;

+∞ otherwiseµ ∈P2(X )

(and using a “metric” notion of gradient flow, due to De Giorgi).

But, also the L2-gradient flow interpretation makes sense, replacing Dirichlet’senergy D with the so-called Cheeger energy:

Ch(u) := inf

lim infh→∞

∫X|∇uh|2 dm : uh ∈ Lip(X , d), ‖uh − u‖2 → 0

Back to metric measure spacesBy “differentiating” Ch one obtains also a laplacian ∆ = ∆d,m, consistent withWitten’s laplacian ∆gu − g(∇V ,∇u) of weighted Riemannian manifolds andFinsler’s laplacian (thus, possibly nonlinear), defining the heat flow as the solutionto

∂tu = ∆u, u(0, ·) = u0 ∈ L2(X ,m).

For u0 ≥ 0 with∫

u0 dm = 1, we can consider at the same time the heat flow u(t , ·)starting from u0 and the W2-gradient flow µt of S starting from µ0 = u0m.

Theorem. (A-Gigli-Savaré)Under a mild regularity assumption on Entm (fulfilled in all CD(K ,∞) m.m.s.), onehas µt = utm for all t ≥ 0.

∂tu = ∆u, u(0, ·) = u0 ∈ L2(X ,m).

Curvature/dimension boundsBounds on the Ricci tensor are at the heart of many functional/geometricinequalities, in the broad field of comparison geometry, i.e. comparing with themodel N-dimensional spaces with constant curvature. In particular, we deal withthe lower bound Ricm := Ricg +∇2V ≥ K g, for m = e−V vol, and the upper boundN on dimension.Example 1. (Spectral gap and Poincaré inequality)∫

X(f − f )2 dm ≤ N − 1

∫X|∇f |2 dm, K > 0.

Example 2. (Gaussian isoperimetric inequality)

Area(∂A) ≥√

KI(m(A)

)A ⊂ X closed, K > 0,

where I is the isoperimetric profile of the Gaussian space(R, deu,

1√2π

e−|x |2/2L 1).

X(f − f )2 dm ≤ N − 1

∫X|∇f |2 dm, K > 0.

Area(∂A) ≥√

KI(m(A)

1√2π

e−|x |2/2L 1).

X(f − f )2 dm ≤ N − 1

∫X|∇f |2 dm, K > 0.

Area(∂A) ≥√

KI(m(A)

1√2π

e−|x |2/2L 1).

X(f − f )2 dm ≤ N − 1

∫X|∇f |2 dm, K > 0.

Area(∂A) ≥√

KI(m(A)

1√2π

e−|x |2/2L 1).

X(f − f )2 dm ≤ N − 1

∫X|∇f |2 dm, K > 0.

Area(∂A) ≥√

KI(m(A)

1√2π

e−|x |2/2L 1).

Curvature/dimension boundsExample 3. (Levy-Gromov inequality) If K > 0,

m(X )≥ |∂B||S|

, where m+(E) := lim infr↓0

m(Er )−m(E)

is the Minkowski content of E and B is a spherical cap in S = SN√K/(N−1)

|B|/|S| = m(E)/m(X ).

Synthetic theoriesThe need of synthetic theories, going beyond the (weighted) Riemannian setting,emerges mainly from two independent directions: geometry of diffusion operators,limits of Riemannian manifolds.

Diffusion operators. Given a diffusion operator L in D(L) ⊂ L2(X ,m), onedefines the carré du champ

Γ(u, v) :=12(L(uv)− u Lv − v Lu

) (= g(∇u,∇v) if L = ∆g

)and then the metric structure provided by the intrinsic distance dL:

dL(x , y) := sup |f (x)− f (y)| : f ∈ D(L), Γ(f , f ) ≤ 1 .

Witten Laplacian Lf = ∆gf − g(∇V ,∇f ) dL = dg.Subelliptic Laplacian Lf =

∑i X 2

i f dL =Carnot-Carathéodory distance.Luigi Ambrosio (SNS) Calculus in metric measure spaces ICM, August 2018 28 / 40

Γ(u, v) :=12(L(uv)− u Lv − v Lu

) (= g(∇u,∇v) if L = ∆g

∑i X 2

Γ(u, v) :=12(L(uv)− u Lv − v Lu

) (= g(∇u,∇v) if L = ∆g

∑i X 2

Γ(u, v) :=12(L(uv)− u Lv − v Lu

) (= g(∇u,∇v) if L = ∆g

∑i X 2

Synthetic theoriesGromov-Hausdorff convergence for metric spaces (Gromov, ’81), and measuredGromov-Hausdorff convergence (Fukaya, ’87) are by now classical analytic toolsin metric and Riemannian spaces.Limits of Riemannian manifolds. By Gromov’s precompactness theorem,geometrically relevant m.m.s. can arise as (measured) Gromov-Hausdorff limitsof Riemannian manifolds Mn, for instance when

Ricg(Mn) ≥ K g, Diam Mn ≤ D.

Example. (collapsed limit)

Synthetic theoriesIn the ’90, Cheeger-Colding developed a remarkable and systematic programaimed at the description of the so-called Ricci limit spaces (with more recentcontributions by Colding-Naber, Cheeger-Jiang-Naber), which includes:• stratification of m-almost all of X in j-dimensional “regular” pieces Rj , 1 ≤ j ≤ n;• in the non-collapsed case, m = cH n and dim (singular set) ≤ (n − 2).

The synthetic theory aims at the description of amore general class of objects “from inside”, i.e.without referring to a smooth approximation (ifany).Compare with Alexandrov’s purely metric theoryof upper and lower bounds on sectionalcurvature, based on triangle comparison.

p0 q p1

p0 qp1

Curvature-dimension: the Bakry-Émery theoryIn the (weighted) Riemannian context, Ricci curvature appears in two basicformulas, an Eulerian and a Lagrangian one.Bochner’s identity.

∆g|∇f |2 − 〈∇f ,∇∆gf 〉 = |Hess (f )|2 + Ricg(∇f ,∇f ).

In the context of diffusion operators L, this leads to the Bakry-Émery BE(K ,N)condition

LΓ(f , f )− Γ(f ,Lf ) ≥ (Lf )2

N+ K Γ(f , f ).

When L is Witten’s laplacian, this is equivalent to a lower bound on the(V ,N)-modified Ricci tensor

Ricg +∇2V − 1N − n

∇V ⊗∇V .

LΓ(f , f )− Γ(f ,Lf ) ≥ (Lf )2

N+ K Γ(f , f ).

Ricg +∇2V − 1N − n

∇V ⊗∇V .

LΓ(f , f )− Γ(f ,Lf ) ≥ (Lf )2

N+ K Γ(f , f ).

Ricg +∇2V − 1N − n

∇V ⊗∇V .

LΓ(f , f )− Γ(f ,Lf ) ≥ (Lf )2

N+ K Γ(f , f ).

Ricg +∇2V − 1N − n

∇V ⊗∇V .

Curvature-dimension: the Bakry-Émery theory

The “calculus” encoded in BE(K ,N) works even in structures very far fromRiemannian manifolds, as (infinite-dimensional) Gaussian spaces.This leads to synthetic proofs of many functional/geometric inequalities, very oftenwith sharp constants (Bakry, Ledoux, Gentil, Bobkov, Baudoin, Garofalo,...).In the case N =∞, gradient contractivity and Logarithmic Sobolev Inequality

|∇Pt f |2 ≤ e−2KtPt |∇f |2,∫

Xf 2 log f 2 dm ≤ 2

∫X|∇f |2 dm

as well as Transport inequalities, Gaussian isoperimetric inequalities, etc.

In the case N <∞, many dimensional inequalities as Sobolev inequalities, Li-Yauparabolic inequalities, sharp two-sided heat kernel estimates, etc.

|∇Pt f |2 ≤ e−2KtPt |∇f |2,∫

∫X|∇f |2 dm

|∇Pt f |2 ≤ e−2KtPt |∇f |2,∫

∫X|∇f |2 dm

|∇Pt f |2 ≤ e−2KtPt |∇f |2,∫

∫X|∇f |2 dm

Ricci curvature and displacement convexityIn the Riemannian setting, the link between Ricci curvature and displacement con-vexity in P2(X ) goes back to the work of Cordero-McCann-Schmuckenschläger,Otto-Villani, Sturm-Von Renesse.It relies on a suitable (K ,N)-concavity inequality satisfied by the Jacobian functionJ (·, x) along geodesics

J (s, x) := det[∇xTs(x)

]with Ts(x) := exp(s∇φ(x)).

∇φ(x)

For instance, in the limit as N →∞, the inequality

logJ (s, x) ≥ s logJ (1, x) + (1− s) logJ (0, x) + Ks(1− s)

2|∇φ|2(x).

∇φ(x)

2|∇φ|2(x).

∇φ(x)

2|∇φ|2(x).

Curvature-dimension: the Lott-Villani and Sturm theoryIn the nonsmooth world, the basic idea (Lott-Villani, Sturm) is to average theseinequalities along the geodesics selected by the Optimal Transport problem withcost=distance2.

CD(K ,N) condition. It is defined through a (K ,N)-convexity inequality, alongconstant speed geodesics of (P2(X ),W2) involving Rényi’s entropies

UN,m(µ) = −∫

X%1−1/N dm if µ = %m ∈P2(X ).

In the simpler case N = ∞ one uses their (rescaled) limit, the relative entropyw.r.t. m:

Entm(µ) :=

∫X% log % dm if µ = %m ∈P2(X ).

UN,m(µ) = −∫

X%1−1/N dm if µ = %m ∈P2(X ).

Entm(µ) :=

∫X% log % dm if µ = %m ∈P2(X ).

UN,m(µ) = −∫

X%1−1/N dm if µ = %m ∈P2(X ).

Entm(µ) :=

∫X% log % dm if µ = %m ∈P2(X ).

Advantages of the CD theory

• Stability w.r.t. measured Gromov-Hausdorff convergence and its many variants,ensured by(a) the decoupling of the roles of d and m;(b) the variational (OT based) principle in the selection of geodesics.

• While the Bakry-Emery theory is more “Riemannian”, the CD theory is able tocover more general classes of spaces, as Finsler manifolds.• As for the Bakry-Emery theory, it provides synthetic proofs of many functionalinequalities and comparison results: Bishop-Gromov, Bonnet-Myers, spectralgap, Laplacian comparison, etc. (Lott-Villani, Sturm, Gigli, Cavalletti-Mondino,...)

Two striking results of the CD theoryRepresenting the heat flow as W2-gradient flow of Entm gives:

Theorem. (Gigli-Mondino-Savaré)Assume that (X n, dn,mn) are CD(K ,∞) and m-GH converge to (X , d,m). Thenthe Cheeger energies Chn, the heat flows Pn

t , the Laplacians ∆n all converge toCh, Pt , ∆ respectively.

The second result follows by an adaptation of the localization technique (Klartag,Gromov-Milman, Kannan-Lovàsz-Simonovitz) to m.m.s.:

Theorem. (Cavalletti-Mondino)The Levy-Gromov inequality holds in CD(K ,N) non-branching spaces (even,when K < 0, with a very general class of model spaces singled out by E.Milman).

Adding the “Riemannian” assumptionWe say that (X , d,m) is infinitesimally Hilbertian if Cheeger’s Dirichlet energy isquadratic, namely

Ch(f + g) + Ch(f − g) = 2Ch(f ) + 2Ch(g) ∀f , g.

This class includes Riemannian manifolds and Alexandrov spaces, it rules outFinsler spaces.

RCD(K ,N) = CD(K ,N) + Infinitesimally Hilbertian (Gigli, 2012)

Theorem. (A.-Gigli-Savaré)The RCD(K ,N) condition is stable under m-GH convergence. Moreover,RCD(K ,∞) is encoded by a differential inequality satisfied by the heat flow Pt :

W 22 (Pt%m, ν) ≤ Entm(ν)− Entm(Pt%m) +

W 22 (Pt%m, ν) ∀ν ∈P2(X ).

Equivalence of BE and RCD

Since the BE and CD theories both provide synthetic upper bounds on dimensionand lower bounds on Ricci curvature, it is natural to guess that they should bemore closely related, at least under the Riemannian assumption.

Thm. (A.-Gigli-Savaré, Erbar-Kuwada-Sturm, A.-Mondino-Savaré)In the class of infinitesimally Hilbertian m.m.s. the BE(K ,N) and the CD(K ,N)theories are (essentially) equivalent.

The advantages of this identification are obvious: one can use the strength ofboth theories, playing at the Eulerian and Lagrangian levels at the same time.

Further developments and open problems

• Equivalence of BE(K ,N)(?) and CD(K ,N) in the non-Riemannian setting;

• Bridge the gap between Ricci limit and RCD(K ,N) spaces;

• Sub-Riemannian geometries, after Baudoin-Garofalo;

• Time-dependent metric measure structures and evolution problems;

• and many more......

ConclusionBesides Nicola Gigli (SISSA, Trieste) and Giuseppe Savaré (Pavia),

let me thank the young SNS students Giada Franz and Daniele Semola, whohelped me with the slides, and

Thank you all for the attention

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