Metric and differentiable structures with Ricci lower bounds Luigi - - PowerPoint PPT Presentation

Metric and differentiable structures with Ricci lower bounds

Luigi Ambrosio

Scuola Normale Superiore, Pisa luigi.ambrosio@sns.it http://cvgmt.sns.it

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 1 / 26

Introduction

In the ’90 Cheeger-Colding studied in detail limits, in the Gromov-Hausdorff sense, of sequences of Riemannian manifolds with given dimension N and uniform lower bound K on Ricci tensor (with more recent contributions by Colding-Naber, Honda). Even though many results (rectifiability, tangent spaces, etc.) are available, these limits can be described only as metric measure spaces, and their properties are proved “by approximation”. Question: is there an intrinsic/richer description of spaces with Ricci lower bounds and dimension upper bounds? Can we develop intrinsic calculus tools (gradient, differential, heat flow,..), independent of the approximation? Can we relate the “Lagrangian” CD(K, N) theory, developed by Lott-Sturm- Villani, to the “Eulerian” cd(ρ, n) theory of Bakry-Emery? By analogy one can think to the purely metric theory of Alexandrov spaces, encoding upper/lower bounds on sectional curvature via triangle comparison and concavity/convexity properties of d2(·, z). However, in the BE and LSV theories, the distance and the measure are interwined in a more subtle way.

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 2 / 26

Introduction

In the ’90 Cheeger-Colding studied in detail limits, in the Gromov-Hausdorff sense, of sequences of Riemannian manifolds with given dimension N and uniform lower bound K on Ricci tensor (with more recent contributions by Colding-Naber, Honda). Even though many results (rectifiability, tangent spaces, etc.) are available, these limits can be described only as metric measure spaces, and their properties are proved “by approximation”. Question: is there an intrinsic/richer description of spaces with Ricci lower bounds and dimension upper bounds? Can we develop intrinsic calculus tools (gradient, differential, heat flow,..), independent of the approximation? Can we relate the “Lagrangian” CD(K, N) theory, developed by Lott-Sturm- Villani, to the “Eulerian” cd(ρ, n) theory of Bakry-Emery? By analogy one can think to the purely metric theory of Alexandrov spaces, encoding upper/lower bounds on sectional curvature via triangle comparison and concavity/convexity properties of d2(·, z). However, in the BE and LSV theories, the distance and the measure are interwined in a more subtle way.

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 2 / 26

Introduction

In the ’90 Cheeger-Colding studied in detail limits, in the Gromov-Hausdorff sense, of sequences of Riemannian manifolds with given dimension N and uniform lower bound K on Ricci tensor (with more recent contributions by Colding-Naber, Honda). Even though many results (rectifiability, tangent spaces, etc.) are available, these limits can be described only as metric measure spaces, and their properties are proved “by approximation”. Question: is there an intrinsic/richer description of spaces with Ricci lower bounds and dimension upper bounds? Can we develop intrinsic calculus tools (gradient, differential, heat flow,..), independent of the approximation? Can we relate the “Lagrangian” CD(K, N) theory, developed by Lott-Sturm- Villani, to the “Eulerian” cd(ρ, n) theory of Bakry-Emery? By analogy one can think to the purely metric theory of Alexandrov spaces, encoding upper/lower bounds on sectional curvature via triangle comparison and concavity/convexity properties of d2(·, z). However, in the BE and LSV theories, the distance and the measure are interwined in a more subtle way.

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 2 / 26

Introduction

In the ’90 Cheeger-Colding studied in detail limits, in the Gromov-Hausdorff sense, of sequences of Riemannian manifolds with given dimension N and uniform lower bound K on Ricci tensor (with more recent contributions by Colding-Naber, Honda). Even though many results (rectifiability, tangent spaces, etc.) are available, these limits can be described only as metric measure spaces, and their properties are proved “by approximation”. Question: is there an intrinsic/richer description of spaces with Ricci lower bounds and dimension upper bounds? Can we develop intrinsic calculus tools (gradient, differential, heat flow,..), independent of the approximation? Can we relate the “Lagrangian” CD(K, N) theory, developed by Lott-Sturm- Villani, to the “Eulerian” cd(ρ, n) theory of Bakry-Emery? By analogy one can think to the purely metric theory of Alexandrov spaces, encoding upper/lower bounds on sectional curvature via triangle comparison and concavity/convexity properties of d2(·, z). However, in the BE and LSV theories, the distance and the measure are interwined in a more subtle way.

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 2 / 26

Introduction

In the ’90 Cheeger-Colding studied in detail limits, in the Gromov-Hausdorff sense, of sequences of Riemannian manifolds with given dimension N and uniform lower bound K on Ricci tensor (with more recent contributions by Colding-Naber, Honda). Even though many results (rectifiability, tangent spaces, etc.) are available, these limits can be described only as metric measure spaces, and their properties are proved “by approximation”. Question: is there an intrinsic/richer description of spaces with Ricci lower bounds and dimension upper bounds? Can we develop intrinsic calculus tools (gradient, differential, heat flow,..), independent of the approximation? Can we relate the “Lagrangian” CD(K, N) theory, developed by Lott-Sturm- Villani, to the “Eulerian” cd(ρ, n) theory of Bakry-Emery? By analogy one can think to the purely metric theory of Alexandrov spaces, encoding upper/lower bounds on sectional curvature via triangle comparison and concavity/convexity properties of d2(·, z). However, in the BE and LSV theories, the distance and the measure are interwined in a more subtle way.

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 2 / 26

Introduction

In the ’90 Cheeger-Colding studied in detail limits, in the Gromov-Hausdorff sense, of sequences of Riemannian manifolds with given dimension N and uniform lower bound K on Ricci tensor (with more recent contributions by Colding-Naber, Honda). Even though many results (rectifiability, tangent spaces, etc.) are available, these limits can be described only as metric measure spaces, and their properties are proved “by approximation”. Question: is there an intrinsic/richer description of spaces with Ricci lower bounds and dimension upper bounds? Can we develop intrinsic calculus tools (gradient, differential, heat flow,..), independent of the approximation? Can we relate the “Lagrangian” CD(K, N) theory, developed by Lott-Sturm- Villani, to the “Eulerian” cd(ρ, n) theory of Bakry-Emery? By analogy one can think to the purely metric theory of Alexandrov spaces, encoding upper/lower bounds on sectional curvature via triangle comparison and concavity/convexity properties of d2(·, z). However, in the BE and LSV theories, the distance and the measure are interwined in a more subtle way.

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 2 / 26

Plan

1

The Eulerian side: Bakry-Émery cd(k, n) theory

2

The Lagrangian side: Lott-Villani and Sturm CD(K, N) theory

3

Adding the “Riemannian” assumption to the CD theory

4

Local structure, geometric/functional inequalities, comparison results

5

Basic equivalence results: heat flow and Sobolev functions

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 3 / 26

Plan

1

The Eulerian side: Bakry-Émery cd(k, n) theory

2

The Lagrangian side: Lott-Villani and Sturm CD(K, N) theory

3

Adding the “Riemannian” assumption to the CD theory

4

Local structure, geometric/functional inequalities, comparison results

5

Basic equivalence results: heat flow and Sobolev functions

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 3 / 26

Plan

1

The Eulerian side: Bakry-Émery cd(k, n) theory

2

The Lagrangian side: Lott-Villani and Sturm CD(K, N) theory

3

Adding the “Riemannian” assumption to the CD theory

4

Local structure, geometric/functional inequalities, comparison results

5

Basic equivalence results: heat flow and Sobolev functions

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 3 / 26

Plan

1

The Eulerian side: Bakry-Émery cd(k, n) theory

2

The Lagrangian side: Lott-Villani and Sturm CD(K, N) theory

3

Adding the “Riemannian” assumption to the CD theory

4

Local structure, geometric/functional inequalities, comparison results

5

Basic equivalence results: heat flow and Sobolev functions

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 3 / 26

Plan

1

The Eulerian side: Bakry-Émery cd(k, n) theory

2

The Lagrangian side: Lott-Villani and Sturm CD(K, N) theory

3

Adding the “Riemannian” assumption to the CD theory

4

Local structure, geometric/functional inequalities, comparison results

5

Basic equivalence results: heat flow and Sobolev functions

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 3 / 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 Pt in an abstract measure space (X, F, m). Then we define the operator ∆ as the infinitesimal generator of Pt, so that

d dtPt f = ∆Pt f. Considering also the bilinear form E canonically associated to

Pt, we introduce a “metric” structure, namely a carré du champ Γ(f), inspired by the calculus identity |∇f|2w = ∇f, ∇(fw) − 1 2∇f 2, ∇w i.e. (by integration on X)

• X

Γ(f)w dm = E(f, fw) − 1 2E(f 2, w). 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 Pt in an abstract measure space (X, F, m). Then we define the operator ∆ as the infinitesimal generator of Pt, so that

d dtPt f = ∆Pt f. Considering also the bilinear form E canonically associated to

Pt, we introduce a “metric” structure, namely a carré du champ Γ(f), inspired by the calculus identity |∇f|2w = ∇f, ∇(fw) − 1 2∇f 2, ∇w i.e. (by integration on X)

• X

Γ(f)w dm = E(f, fw) − 1 2E(f 2, w). 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 Pt in an abstract measure space (X, F, m). Then we define the operator ∆ as the infinitesimal generator of Pt, so that

d dtPt f = ∆Pt f. Considering also the bilinear form E canonically associated to

Pt, we introduce a “metric” structure, namely a carré du champ Γ(f), inspired by the calculus identity |∇f|2w = ∇f, ∇(fw) − 1 2∇f 2, ∇w i.e. (by integration on X)

• X

Γ(f)w dm = E(f, fw) − 1 2E(f 2, w). 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 Pt in an abstract measure space (X, F, m). Then we define the operator ∆ as the infinitesimal generator of Pt, so that

d dtPt f = ∆Pt f. Considering also the bilinear form E canonically associated to

Pt, we introduce a “metric” structure, namely a carré du champ Γ(f), inspired by the calculus identity |∇f|2w = ∇f, ∇(fw) − 1 2∇f 2, ∇w i.e. (by integration on X)

• X

Γ(f)w dm = E(f, fw) − 1 2E(f 2, w). 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, Pt) (equivalently (X, F, m, E)).

• Definition. We say that (X, F, m, Pt) satisfies cd(k, n) if

1 2∆Γ(f) ≥ 1 n(∆f)2 + Γ(f, ∆f) + kΓ(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, Pt) (equivalently (X, F, m, E)).

• Definition. We say that (X, F, m, Pt) satisfies cd(k, n) if

1 2∆Γ(f) ≥ 1 n(∆f)2 + Γ(f, ∆f) + kΓ(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, Pt) (equivalently (X, F, m, E)).

• Definition. We say that (X, F, m, Pt) satisfies cd(k, n) if

1 2∆Γ(f) ≥ 1 n(∆f)2 + Γ(f, ∆f) + kΓ(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, Pt) (equivalently (X, F, m, E)).

• Definition. We say that (X, F, m, Pt) satisfies cd(k, n) if

1 2∆Γ(f) ≥ 1 n(∆f)2 + Γ(f, ∆f) + kΓ(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 (P2(X), W2), where W2 is the Wasserstein distance with cost=distance2, namely W2

2(µ, ν) := inf

• X×X

d2(x, y) dΣ(x, y) : (π1)#Σ = µ, (π2)#Σ = ν

• .

I will not enter into the details, but just mention that, in geodesic metric spaces, constant speed geodesics µt, 0 ≤ t ≤ 1, beween two probability measures µ0, µ1 ∈ P2(X) are “geodesic plans”, i.e. there exists π ∈ P(Geo(X)) with

• φ dµt =
• Geo(X)

φ(γ(t)) dπ(γ) for all t ∈ [0, 1], φ bounded Borel. “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 (P2(X), W2), where W2 is the Wasserstein distance with cost=distance2, namely W2

2(µ, ν) := inf

• X×X

d2(x, y) dΣ(x, y) : (π1)#Σ = µ, (π2)#Σ = ν

• .

I will not enter into the details, but just mention that, in geodesic metric spaces, constant speed geodesics µt, 0 ≤ t ≤ 1, beween two probability measures µ0, µ1 ∈ P2(X) are “geodesic plans”, i.e. there exists π ∈ P(Geo(X)) with

• φ dµt =
• Geo(X)

φ(γ(t)) dπ(γ) for all t ∈ [0, 1], φ bounded Borel. “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 (P2(X), W2), where W2 is the Wasserstein distance with cost=distance2, namely W2

2(µ, ν) := inf

• X×X

d2(x, y) dΣ(x, y) : (π1)#Σ = µ, (π2)#Σ = ν

• .

I will not enter into the details, but just mention that, in geodesic metric spaces, constant speed geodesics µt, 0 ≤ t ≤ 1, beween two probability measures µ0, µ1 ∈ P2(X) are “geodesic plans”, i.e. there exists π ∈ P(Geo(X)) with

• φ dµt =
• Geo(X)

φ(γ(t)) dπ(γ) for all t ∈ [0, 1], φ bounded Borel. “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 Entm(µ) :=

• X ρ log ρ dm

if µ = ρm; +∞

• therwise

along W2-geodesics µt, namely Entm(µt) ≤ (1 − t)Entm(µ0) + t Entm(µ1) − K 2 t(1 − t)W2

2(µ0, µ1).

The roles of d and of m are nicely decoupled. Definition motivated by the classical inequality

• − log Jt(x)

′′

• t=0

≥ |∇φ(x)|2 N + Ricx

• ∇φ(x), ∇φ(x)
• ,

with Tt(x) = expx(−t∇φ(x)), Jt(x) = det∇Tt(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 Entm(µ) :=

• X ρ log ρ dm

if µ = ρm; +∞

• therwise

along W2-geodesics µt, namely Entm(µt) ≤ (1 − t)Entm(µ0) + t Entm(µ1) − K 2 t(1 − t)W2

2(µ0, µ1).

The roles of d and of m are nicely decoupled. Definition motivated by the classical inequality

• − log Jt(x)

′′

• t=0

≥ |∇φ(x)|2 N + Ricx

• ∇φ(x), ∇φ(x)
• ,

with Tt(x) = expx(−t∇φ(x)), Jt(x) = det∇Tt(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 Entm(µ) :=

• X ρ log ρ dm

if µ = ρm; +∞

• therwise

along W2-geodesics µt, namely Entm(µt) ≤ (1 − t)Entm(µ0) + t Entm(µ1) − K 2 t(1 − t)W2

2(µ0, µ1).

The roles of d and of m are nicely decoupled. Definition motivated by the classical inequality

• − log Jt(x)

′′

• t=0

≥ |∇φ(x)|2 N + Ricx

• ∇φ(x), ∇φ(x)
• ,

with Tt(x) = expx(−t∇φ(x)), Jt(x) = det∇Tt(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 Entm(µ) :=

• X ρ log ρ dm

if µ = ρm; +∞

• therwise

along W2-geodesics µt, namely Entm(µt) ≤ (1 − t)Entm(µ0) + t Entm(µ1) − K 2 t(1 − t)W2

2(µ0, µ1).

The roles of d and of m are nicely decoupled. Definition motivated by the classical inequality

• − log Jt(x)

′′

• t=0

≥ |∇φ(x)|2 N + Ricx

• ∇φ(x), ∇φ(x)
• ,

with Tt(x) = expx(−t∇φ(x)), Jt(x) = det∇Tt(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 Entm has to be replaced by the Reny’s N-dimensional entropies RN(µ) := N

• X
• ρ − ρ1−1/N

dm if µ = ρm + µ⊥ 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:

RN(µt) ≤ − τ 1−t

κ

(d(x1, x0))ρ−1/N (x0)+τ t

κ(d(x1, x0))ρ−1/N 1

(x1)

• dΣ(x0, x1).

In the simpler case K = 0, since τ s

0(θ) = s, this is simply convexity of RN.

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 Entm has to be replaced by the Reny’s N-dimensional entropies RN(µ) := N

• X
• ρ − ρ1−1/N

dm if µ = ρm + µ⊥ 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:

RN(µt) ≤ − τ 1−t

κ

(d(x1, x0))ρ−1/N (x0)+τ t

κ(d(x1, x0))ρ−1/N 1

(x1)

• dΣ(x0, x1).

In the simpler case K = 0, since τ s

0(θ) = s, this is simply convexity of RN.

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 Entm has to be replaced by the Reny’s N-dimensional entropies RN(µ) := N

• X
• ρ − ρ1−1/N

dm if µ = ρm + µ⊥ 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:

RN(µt) ≤ − τ 1−t

κ

(d(x1, x0))ρ−1/N (x0)+τ t

κ(d(x1, x0))ρ−1/N 1

(x1)

• dΣ(x0, x1).

In the simpler case K = 0, since τ s

0(θ) = s, this is simply convexity of RN.

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

• f 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 (Xi, di, mi) MGH-converge to (X, d, m) if we can find isometric embeddings ji : Xi → Z, j : X → Z with ji(Xi) → j(X) in the Hausdorff sense and (ji)#mi → 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

• f 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 (Xi, di, mi) MGH-converge to (X, d, m) if we can find isometric embeddings ji : Xi → Z, j : X → Z with ji(Xi) → j(X) in the Hausdorff sense and (ji)#mi → 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

• f 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 (Xi, di, mi) MGH-converge to (X, d, m) if we can find isometric embeddings ji : Xi → Z, j : X → Z with ji(Xi) → j(X) in the Hausdorff sense and (ji)#mi → 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

• f 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 (Xi, di, mi) MGH-converge to (X, d, m) if we can find isometric embeddings ji : Xi → Z, j : X → Z with ji(Xi) → j(X) in the Hausdorff sense and (ji)#mi → 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

• f 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 (Xi, di, mi) MGH-converge to (X, d, m) if we can find isometric embeddings ji : Xi → Z, j : X → Z with ji(Xi) → j(X) in the Hausdorff sense and (ji)#mi → 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|(x) := lim sup

y→x

|f(y) − f(x)| d(y, x) . Then, following Cheeger, we can define a kind of Dirichlet energy, by the L2(X, m) relaxation of |∇f|2: Ch(f) := 1 2 inf

• lim inf

n→∞

• X

|∇fn|2 dm : fn ∈ Lip(X),

• X

|fn − f|2 dm → 0

• .

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

2

• X |∇f|2

∗ dm and standard calculus properties

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|(x) := lim sup

y→x

|f(y) − f(x)| d(y, x) . Then, following Cheeger, we can define a kind of Dirichlet energy, by the L2(X, m) relaxation of |∇f|2: Ch(f) := 1 2 inf

• lim inf

n→∞

• X

|∇fn|2 dm : fn ∈ Lip(X),

• X

|fn − f|2 dm → 0

• .

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

2

• X |∇f|2

∗ dm and standard calculus properties

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|(x) := lim sup

y→x

|f(y) − f(x)| d(y, x) . Then, following Cheeger, we can define a kind of Dirichlet energy, by the L2(X, m) relaxation of |∇f|2: Ch(f) := 1 2 inf

• lim inf

n→∞

• X

|∇fn|2 dm : fn ∈ Lip(X),

• X

|fn − f|2 dm → 0

• .

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

2

• X |∇f|2

∗ dm and standard calculus properties

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|(x) := lim sup

y→x

|f(y) − f(x)| d(y, x) . Then, following Cheeger, we can define a kind of Dirichlet energy, by the L2(X, m) relaxation of |∇f|2: Ch(f) := 1 2 inf

• lim inf

n→∞

• X

|∇fn|2 dm : fn ∈ Lip(X),

• X

|fn − f|2 dm → 0

• .

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

2

• X |∇f|2

∗ dm and standard calculus properties

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

• f the following equivalent conditions:

(i) (X, d, m) is a CD(K, ∞) space and the L2 heat flow Pt is linear; (ii) (X, d, m) is a CD(K, ∞) space and Ch is a quadratic form; (iii) the heat flow t → Pt f, when seen as a curve of measures t → µt = Pt f m, satisfies the EVI (evolution variational inequality) d dt 1 2W2

2(µt, µ)

• + K

1 2W2

2(µt, µ)

• + Entm(µt) ≤ Entm(µ)

for all µ ∈ P2(X). Notice that (iii) combines in a nice way the two basic ingredients of the LSV theory, namely Entm and W2. 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

• f the following equivalent conditions:

(i) (X, d, m) is a CD(K, ∞) space and the L2 heat flow Pt is linear; (ii) (X, d, m) is a CD(K, ∞) space and Ch is a quadratic form; (iii) the heat flow t → Pt f, when seen as a curve of measures t → µt = Pt f m, satisfies the EVI (evolution variational inequality) d dt 1 2W2

2(µt, µ)

• + K

1 2W2

2(µt, µ)

• + Entm(µt) ≤ Entm(µ)

for all µ ∈ P2(X). Notice that (iii) combines in a nice way the two basic ingredients of the LSV theory, namely Entm and W2. 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

• f the following equivalent conditions:

(i) (X, d, m) is a CD(K, ∞) space and the L2 heat flow Pt is linear; (ii) (X, d, m) is a CD(K, ∞) space and Ch is a quadratic form; (iii) the heat flow t → Pt f, when seen as a curve of measures t → µt = Pt f m, satisfies the EVI (evolution variational inequality) d dt 1 2W2

2(µt, µ)

• + K

1 2W2

2(µt, µ)

• + Entm(µt) ≤ Entm(µ)

for all µ ∈ P2(X). Notice that (iii) combines in a nice way the two basic ingredients of the LSV theory, namely Entm and W2. 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

• f the following equivalent conditions:

(i) (X, d, m) is a CD(K, ∞) space and the L2 heat flow Pt is linear; (ii) (X, d, m) is a CD(K, ∞) space and Ch is a quadratic form; (iii) the heat flow t → Pt f, when seen as a curve of measures t → µt = Pt f m, satisfies the EVI (evolution variational inequality) d dt 1 2W2

2(µt, µ)

• + K

1 2W2

2(µt, µ)

• + Entm(µt) ≤ Entm(µ)

for all µ ∈ P2(X). Notice that (iii) combines in a nice way the two basic ingredients of the LSV theory, namely Entm and W2. 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 d dtσ2

K/N

1 2W2(µt, µ)

• + Kσ2

K/N

1 2W2(µt, µ)

• ≤ N

2

• 1 − UN(µ)

UN(µt)

• for all µ ∈ P2(X), with UN a dimensional modification of Entm:

UN(µ) := exp

• − 1

N Entm(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 µt = Pt f m with µt = St f m, where St is the nonlinear diffusion semigroup given by d dtSt f = 1 N ∆(St f)1− 1

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 d dtσ2

K/N

1 2W2(µt, µ)

• + Kσ2

K/N

1 2W2(µt, µ)

• ≤ N

2

• 1 − UN(µ)

UN(µt)

• for all µ ∈ P2(X), with UN a dimensional modification of Entm:

UN(µ) := exp

• − 1

N Entm(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 µt = Pt f m with µt = St f m, where St is the nonlinear diffusion semigroup given by d dtSt f = 1 N ∆(St f)1− 1

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 d dtσ2

K/N

1 2W2(µt, µ)

• + Kσ2

K/N

1 2W2(µt, µ)

• ≤ N

2

• 1 − UN(µ)

UN(µt)

• for all µ ∈ P2(X), with UN a dimensional modification of Entm:

UN(µ) := exp

• − 1

N Entm(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 µt = Pt f m with µt = St f m, where St is the nonlinear diffusion semigroup given by d dtSt f = 1 N ∆(St f)1− 1

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 Pt, t > 0, maps L∞(X, m) to Cb(X);
• Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm):
• ptimal 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 Pt, t > 0, maps L∞(X, m) to Cb(X);
• Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm):
• ptimal 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 Pt, t > 0, maps L∞(X, m) to Cb(X);
• Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm):
• ptimal 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 Pt, t > 0, maps L∞(X, m) to Cb(X);
• Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm):
• ptimal 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 Pt, t > 0, maps L∞(X, m) to Cb(X);
• Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm):
• ptimal 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 Pt, t > 0, maps L∞(X, m) to Cb(X);
• Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm):
• ptimal 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 Pt, t > 0, maps L∞(X, m) to Cb(X);
• Essential nonbranching property (Rajala-Sturm, Gigli-Rajala-Sturm):
• ptimal 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) Pt, t > 0, maps L∞(X, m) to Cb(X, τ); (b) the Biroli-Mosco intrinsic distance dE(x, y) := sup {|f(x) − f(y)| : f ∈ Cb(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) + kΓ(f). holds. Then (X, dE, 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) Pt, t > 0, maps L∞(X, m) to Cb(X, τ); (b) the Biroli-Mosco intrinsic distance dE(x, y) := sup {|f(x) − f(y)| : f ∈ Cb(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) + kΓ(f). holds. Then (X, dE, 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) Pt, t > 0, maps L∞(X, m) to Cb(X, τ); (b) the Biroli-Mosco intrinsic distance dE(x, y) := sup {|f(x) − f(y)| : f ∈ Cb(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) + kΓ(f). holds. Then (X, dE, 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 Pt

This also lead to other characterizations of RCD based on gradient contractivity: |∇Pt f|2 ≤ e−2KtPt |∇f|2 (N = ∞) When N < ∞, setting κ = K/N we need to consider different times s, t (Bolley, Gentil, Guillin, Erbar, Kuwada, Sturm) s2

κ

1 2W2(Pt f m, Ps g m)

• ≤

e−K(t+s)s2

κ

1 2W2(f m, g m)

• +

1 κ

• 1 − e−K(t+s)(√t − √s)2

2(t + s) .

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 15 / 26

Characterizations based on contractivity of Pt

This also lead to other characterizations of RCD based on gradient contractivity: |∇Pt f|2 ≤ e−2KtPt |∇f|2 (N = ∞) When N < ∞, setting κ = K/N we need to consider different times s, t (Bolley, Gentil, Guillin, Erbar, Kuwada, Sturm) s2

κ

1 2W2(Pt f m, Ps g m)

• ≤

e−K(t+s)s2

κ

1 2W2(f m, g m)

• +

1 κ

• 1 − e−K(t+s)(√t − √s)2

2(t + s) .

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 (X, d, m) ∼ (Y, ˜ d, ˜ m) × (R, deu, L 1) with (Y, ˜ d, ˜ m) ∈ RCD(0, N − 1). 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 Xi, each bi-Lipschitz to a subset of Rk(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

• f 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 (X, d, m) ∼ (Y, ˜ d, ˜ m) × (R, deu, L 1) with (Y, ˜ d, ˜ m) ∈ RCD(0, N − 1). 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 Xi, each bi-Lipschitz to a subset of Rk(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

• f 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 (X, d, m) ∼ (Y, ˜ d, ˜ m) × (R, deu, L 1) with (Y, ˜ d, ˜ m) ∈ RCD(0, N − 1). 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 Xi, each bi-Lipschitz to a subset of Rk(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

• f 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 (X, d, m) ∼ (Y, ˜ d, ˜ m) × (R, deu, L 1) with (Y, ˜ d, ˜ m) ∈ RCD(0, N − 1). 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 Xi, each bi-Lipschitz to a subset of Rk(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

• f 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 (X, d, m) ∼ (Y, ˜ d, ˜ m) × (R, deu, L 1) with (Y, ˜ d, ˜ m) ∈ RCD(0, N − 1). 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 Xi, each bi-Lipschitz to a subset of Rk(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

• f 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

• Br(x)

|f − fx,r| dm ≤ reK−r2

B2r(x)

|∇f| dm f ∈ Lip(X). Global Poincaré (spectral gap). (Lott-Villani, Sturm) In CD(K, N) spaces with N > 1, K > 0,

• X

|f|2 dm ≤ N − 1 KN

• X

|∇f|2 dm f ∈ Lip(X). Li-Yau. (Garofalo-Mondino, Jiang, after Baudoin-Garofalo, Bakry-Ledoux) |∇ log Pt f|2 − d dt log Pt f ≤ N 2t (f > 0, in RCD(0, N) spaces) (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

• Br(x)

|f − fx,r| dm ≤ reK−r2

B2r(x)

|∇f| dm f ∈ Lip(X). Global Poincaré (spectral gap). (Lott-Villani, Sturm) In CD(K, N) spaces with N > 1, K > 0,

• X

|f|2 dm ≤ N − 1 KN

• X

|∇f|2 dm f ∈ Lip(X). Li-Yau. (Garofalo-Mondino, Jiang, after Baudoin-Garofalo, Bakry-Ledoux) |∇ log Pt f|2 − d dt log Pt f ≤ N 2t (f > 0, in RCD(0, N) spaces) (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

• Br(x)

|f − fx,r| dm ≤ reK−r2

B2r(x)

|∇f| dm f ∈ Lip(X). Global Poincaré (spectral gap). (Lott-Villani, Sturm) In CD(K, N) spaces with N > 1, K > 0,

• X

|f|2 dm ≤ N − 1 KN

• X

|∇f|2 dm f ∈ Lip(X). Li-Yau. (Garofalo-Mondino, Jiang, after Baudoin-Garofalo, Bakry-Ledoux) |∇ log Pt f|2 − d dt log Pt f ≤ N 2t (f > 0, in RCD(0, N) spaces) (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

• Br(x)

|f − fx,r| dm ≤ reK−r2

B2r(x)

|∇f| dm f ∈ Lip(X). Global Poincaré (spectral gap). (Lott-Villani, Sturm) In CD(K, N) spaces with N > 1, K > 0,

• X

|f|2 dm ≤ N − 1 KN

• X

|∇f|2 dm f ∈ Lip(X). Li-Yau. (Garofalo-Mondino, Jiang, after Baudoin-Garofalo, Bakry-Ledoux) |∇ log Pt f|2 − d dt log Pt f ≤ N 2t (f > 0, in RCD(0, N) spaces) (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

• Br(x)

|f − fx,r| dm ≤ reK−r2

B2r(x)

|∇f| dm f ∈ Lip(X). Global Poincaré (spectral gap). (Lott-Villani, Sturm) In CD(K, N) spaces with N > 1, K > 0,

• X

|f|2 dm ≤ N − 1 KN

• X

|∇f|2 dm f ∈ Lip(X). Li-Yau. (Garofalo-Mondino, Jiang, after Baudoin-Garofalo, Bakry-Ledoux) |∇ log Pt f|2 − d dt log Pt f ≤ N 2t (f > 0, in RCD(0, N) spaces) (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(Br(x)) m(Bs(x)) ≤ VK,N(r) VK,N(s) 0 < s ≤ r. Laplacian comparison. (Gigli) In CD(K, N) spaces (with ˜ τ0,N ≡ N) 1 2∆d2(·, z) ≤ ˜ τK,N

• d(·, z)
• ∀z ∈ X,

in the sense of distributions. Lévy-Gromov. (Cavalletti-Mondino, after Klartag) In essentially nonbranching CD(K, N) spaces (X, d, m), for E ⊂ X one has |∂E| m(X) ≥ |∂B| |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(Br(x)) m(Bs(x)) ≤ VK,N(r) VK,N(s) 0 < s ≤ r. Laplacian comparison. (Gigli) In CD(K, N) spaces (with ˜ τ0,N ≡ N) 1 2∆d2(·, z) ≤ ˜ τK,N

• d(·, z)
• ∀z ∈ X,

in the sense of distributions. Lévy-Gromov. (Cavalletti-Mondino, after Klartag) In essentially nonbranching CD(K, N) spaces (X, d, m), for E ⊂ X one has |∂E| m(X) ≥ |∂B| |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(Br(x)) m(Bs(x)) ≤ VK,N(r) VK,N(s) 0 < s ≤ r. Laplacian comparison. (Gigli) In CD(K, N) spaces (with ˜ τ0,N ≡ N) 1 2∆d2(·, z) ≤ ˜ τK,N

• d(·, z)
• ∀z ∈ X,

in the sense of distributions. Lévy-Gromov. (Cavalletti-Mondino, after Klartag) In essentially nonbranching CD(K, N) spaces (X, d, m), for E ⊂ X one has |∂E| m(X) ≥ |∂B| |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(Br(x)) m(Bs(x)) ≤ VK,N(r) VK,N(s) 0 < s ≤ r. Laplacian comparison. (Gigli) In CD(K, N) spaces (with ˜ τ0,N ≡ N) 1 2∆d2(·, z) ≤ ˜ τK,N

• d(·, z)
• ∀z ∈ X,

in the sense of distributions. Lévy-Gromov. (Cavalletti-Mondino, after Klartag) In essentially nonbranching CD(K, N) spaces (X, d, m), for E ⊂ X one has |∂E| m(X) ≥ |∂B| |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(Br(x)) m(Bs(x)) ≤ VK,N(r) VK,N(s) 0 < s ≤ r. Laplacian comparison. (Gigli) In CD(K, N) spaces (with ˜ τ0,N ≡ N) 1 2∆d2(·, z) ≤ ˜ τK,N

• d(·, z)
• ∀z ∈ X,

in the sense of distributions. Lévy-Gromov. (Cavalletti-Mondino, after Klartag) In essentially nonbranching CD(K, N) spaces (X, d, m), for E ⊂ X one has |∂E| m(X) ≥ |∂B| |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 L2 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 ft = ∆ ft where, by definition, −∆ f = −∆d,m f is the element with minimal norm in the subgradient ∂Ch(f) of Ch at f: ∂Ch(f) :=

• ξ ∈ L2(X, m) : Ch(g) ≥ Ch(f) +
• X

ξ(g − f) dm ∀g

• .

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 L2 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 ft = ∆ ft where, by definition, −∆ f = −∆d,m f is the element with minimal norm in the subgradient ∂Ch(f) of Ch at f: ∂Ch(f) :=

• ξ ∈ L2(X, m) : Ch(g) ≥ Ch(f) +
• X

ξ(g − f) dm ∀g

• .

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 L2 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 ft = ∆ ft where, by definition, −∆ f = −∆d,m f is the element with minimal norm in the subgradient ∂Ch(f) of Ch at f: ∂Ch(f) :=

• ξ ∈ L2(X, m) : Ch(g) ≥ Ch(f) +
• X

ξ(g − f) dm ∀g

• .

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 L2 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 ft = ∆ ft where, by definition, −∆ f = −∆d,m f is the element with minimal norm in the subgradient ∂Ch(f) of Ch at f: ∂Ch(f) :=

• ξ ∈ L2(X, m) : Ch(g) ≥ Ch(f) +
• X

ξ(g − f) dm ∀g

• .

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 L2 distance with the quadratic Wasserstein distance in P2(Rn), 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: Entm(µt) + 1 2 t | ˙ µs|2 + |∇−Entm|2(µs) ds ≤ Entm(µ0) ∀t ≥ 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 L2 distance with the quadratic Wasserstein distance in P2(Rn), 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: Entm(µt) + 1 2 t | ˙ µs|2 + |∇−Entm|2(µs) ds ≤ Entm(µ0) ∀t ≥ 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 L2 distance with the quadratic Wasserstein distance in P2(Rn), 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: Entm(µt) + 1 2 t | ˙ µs|2 + |∇−Entm|2(µs) ds ≤ Entm(µ0) ∀t ≥ 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 L2 distance with the quadratic Wasserstein distance in P2(Rn), 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: Entm(µt) + 1 2 t | ˙ µs|2 + |∇−Entm|2(µs) ds ≤ Entm(µ0) ∀t ≥ 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 ∈ L2(X, m) : f ≥ 0,
• X

f dm = 1

• .

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 ∈ L2(X, m) : f ≥ 0,
• X

f dm = 1

• .

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 ∈ L2(X, m) : f ≥ 0,
• X

f dm = 1

• .

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 ∈ L2(X, m) : f ≥ 0,
• X

f dm = 1

• .

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

• ptimal transport context (AGS) it represents nonnegative solutions to the

continuity equation d dtµt + div(vtµt) = 0 t ∈ (0, 1) as marginals µt of positive finite measures π in C([0, 1]; X) concentrated on solutions to the ODE γ′ = vt(γ). A metric version of this result is possible (Lisini). Kuwada lemma. If µt = Pt f m with f ≥ 0,

• X f dm = 1, then

µt ∈ AC2 [0, 1]; (P2(X), W2)

• with

| ˙ µt|2 ≤

• {Pt f>0}

|∇Pt f|2

∗

Pt f dm for L 1-a.e. t > 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

• ptimal transport context (AGS) it represents nonnegative solutions to the

continuity equation d dtµt + div(vtµt) = 0 t ∈ (0, 1) as marginals µt of positive finite measures π in C([0, 1]; X) concentrated on solutions to the ODE γ′ = vt(γ). A metric version of this result is possible (Lisini). Kuwada lemma. If µt = Pt f m with f ≥ 0,

• X f dm = 1, then

µt ∈ AC2 [0, 1]; (P2(X), W2)

• with

| ˙ µt|2 ≤

• {Pt f>0}

|∇Pt f|2

∗

Pt f dm for L 1-a.e. t > 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

• ptimal transport context (AGS) it represents nonnegative solutions to the

continuity equation d dtµt + div(vtµt) = 0 t ∈ (0, 1) as marginals µt of positive finite measures π in C([0, 1]; X) concentrated on solutions to the ODE γ′ = vt(γ). A metric version of this result is possible (Lisini). Kuwada lemma. If µt = Pt f m with f ≥ 0,

• X f dm = 1, then

µt ∈ AC2 [0, 1]; (P2(X), W2)

• with

| ˙ µt|2 ≤

• {Pt f>0}

|∇Pt f|2

∗

Pt f dm for L 1-a.e. t > 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

• ptimal transport context (AGS) it represents nonnegative solutions to the

continuity equation d dtµt + div(vtµt) = 0 t ∈ (0, 1) as marginals µt of positive finite measures π in C([0, 1]; X) concentrated on solutions to the ODE γ′ = vt(γ). A metric version of this result is possible (Lisini). Kuwada lemma. If µt = Pt f m with f ≥ 0,

• X f dm = 1, then

µt ∈ AC2 [0, 1]; (P2(X), W2)

• with

| ˙ µt|2 ≤

• {Pt f>0}

|∇Pt f|2

∗

Pt f dm for L 1-a.e. t > 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 2

• X

|∇f|2

∗ dm

∀f ∈ D(Ch). 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 + ǫg (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 2

• X

|∇f|2

∗ dm

∀f ∈ D(Ch). 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 + ǫg (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 2

• X

|∇f|2

∗ dm

∀f ∈ D(Ch). 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 + ǫg (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 2

• X

|∇f|2

∗ dm

∀f ∈ D(Ch). 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 + ǫg (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 2

• X

|∇f|2

∗ dm

∀f ∈ D(Ch). 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 + ǫg (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 2

• X

|∇f|2

∗ dm

∀f ∈ D(Ch). 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 + ǫg (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(γǫ) (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(γ1) − f(γ0)| ≤

• γ

G

• n 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 AC2([0, 1]; X). Then, we define |∇f|w as the weak upper gradient G with smallest L2(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(γǫ) (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(γ1) − f(γ0)| ≤

• γ

G

• n 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 AC2([0, 1]; X). Then, we define |∇f|w as the weak upper gradient G with smallest L2(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(γǫ) (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(γ1) − f(γ0)| ≤

• γ

G

• n 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 AC2([0, 1]; X). Then, we define |∇f|w as the weak upper gradient G with smallest L2(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(γǫ) (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(γ1) − f(γ0)| ≤

• γ

G

• n 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 AC2([0, 1]; X). Then, we define |∇f|w as the weak upper gradient G with smallest L2(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(γǫ) (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(γ1) − f(γ0)| ≤

• γ

G

• n 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 AC2([0, 1]; X). Then, we define |∇f|w as the weak upper gradient G with smallest L2(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(γǫ) (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(γ1) − f(γ0)| ≤

• γ

G

• n 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 AC2([0, 1]; X). Then, we define |∇f|w as the weak upper gradient G with smallest L2(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(γǫ) (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(γ1) − f(γ0)| ≤

• γ

G

• n 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 AC2([0, 1]; X). Then, we define |∇f|w as the weak upper gradient G with smallest L2(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(γǫ) (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(γ1) − f(γ0)| ≤

• γ

G

• n 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 AC2([0, 1]; X). Then, we define |∇f|w as the weak upper gradient G with smallest L2(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/3

We say that a (Borel) set Γ of absolutely continuous curves γ : [0, 1] → X is null if π(Γ) = 0 for any test plan π. Here, the class of test plans is simply the collection of all probability measures π in AC2 [0, 1]; X

• satisfying

(et)♯π ≤ Cm ∀t ∈ [0, 1] for some C = C(π) ≥ 0.

• Theorem. [AGS] In any complete and separable metric measure space

(X, d, m) with m finite on bounded sets the minimal relaxed gradient |∇f|∗ and the minimal weak upper gradient |∇f|w coincide m-a.e. in X. Of course, maybe they are both trivial without extra assumptions. In the related context of PI and differentiability spaces, similar identification results between “Eulerian” notions (slopes, A-Kirchheim metric currents) and “Lagrangian” ones (Alberti representation, slopes along curves) have been recently investigated (Bate, Cheeger, Kleiner, Marchese, Preiss, Schioppa...).

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 25 / 26

Metric Sobolev spaces and weak gradients/3

We say that a (Borel) set Γ of absolutely continuous curves γ : [0, 1] → X is null if π(Γ) = 0 for any test plan π. Here, the class of test plans is simply the collection of all probability measures π in AC2 [0, 1]; X

• satisfying

(et)♯π ≤ Cm ∀t ∈ [0, 1] for some C = C(π) ≥ 0.

• Theorem. [AGS] In any complete and separable metric measure space

(X, d, m) with m finite on bounded sets the minimal relaxed gradient |∇f|∗ and the minimal weak upper gradient |∇f|w coincide m-a.e. in X. Of course, maybe they are both trivial without extra assumptions. In the related context of PI and differentiability spaces, similar identification results between “Eulerian” notions (slopes, A-Kirchheim metric currents) and “Lagrangian” ones (Alberti representation, slopes along curves) have been recently investigated (Bate, Cheeger, Kleiner, Marchese, Preiss, Schioppa...).

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 25 / 26

Metric Sobolev spaces and weak gradients/3

We say that a (Borel) set Γ of absolutely continuous curves γ : [0, 1] → X is null if π(Γ) = 0 for any test plan π. Here, the class of test plans is simply the collection of all probability measures π in AC2 [0, 1]; X

• satisfying

(et)♯π ≤ Cm ∀t ∈ [0, 1] for some C = C(π) ≥ 0.

• Theorem. [AGS] In any complete and separable metric measure space

(X, d, m) with m finite on bounded sets the minimal relaxed gradient |∇f|∗ and the minimal weak upper gradient |∇f|w coincide m-a.e. in X. Of course, maybe they are both trivial without extra assumptions. In the related context of PI and differentiability spaces, similar identification results between “Eulerian” notions (slopes, A-Kirchheim metric currents) and “Lagrangian” ones (Alberti representation, slopes along curves) have been recently investigated (Bate, Cheeger, Kleiner, Marchese, Preiss, Schioppa...).

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 25 / 26

Metric Sobolev spaces and weak gradients/3

We say that a (Borel) set Γ of absolutely continuous curves γ : [0, 1] → X is null if π(Γ) = 0 for any test plan π. Here, the class of test plans is simply the collection of all probability measures π in AC2 [0, 1]; X

• satisfying

(et)♯π ≤ Cm ∀t ∈ [0, 1] for some C = C(π) ≥ 0.

• Theorem. [AGS] In any complete and separable metric measure space

(X, d, m) with m finite on bounded sets the minimal relaxed gradient |∇f|∗ and the minimal weak upper gradient |∇f|w coincide m-a.e. in X. Of course, maybe they are both trivial without extra assumptions. In the related context of PI and differentiability spaces, similar identification results between “Eulerian” notions (slopes, A-Kirchheim metric currents) and “Lagrangian” ones (Alberti representation, slopes along curves) have been recently investigated (Bate, Cheeger, Kleiner, Marchese, Preiss, Schioppa...).

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 25 / 26

Metric Sobolev spaces and weak gradients/3

We say that a (Borel) set Γ of absolutely continuous curves γ : [0, 1] → X is null if π(Γ) = 0 for any test plan π. Here, the class of test plans is simply the collection of all probability measures π in AC2 [0, 1]; X

• satisfying

(et)♯π ≤ Cm ∀t ∈ [0, 1] for some C = C(π) ≥ 0.

• Theorem. [AGS] In any complete and separable metric measure space

(X, d, m) with m finite on bounded sets the minimal relaxed gradient |∇f|∗ and the minimal weak upper gradient |∇f|w coincide m-a.e. in X. Of course, maybe they are both trivial without extra assumptions. In the related context of PI and differentiability spaces, similar identification results between “Eulerian” notions (slopes, A-Kirchheim metric currents) and “Lagrangian” ones (Alberti representation, slopes along curves) have been recently investigated (Bate, Cheeger, Kleiner, Marchese, Preiss, Schioppa...).

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 25 / 26

Metric Sobolev spaces and weak gradients/3

We say that a (Borel) set Γ of absolutely continuous curves γ : [0, 1] → X is null if π(Γ) = 0 for any test plan π. Here, the class of test plans is simply the collection of all probability measures π in AC2 [0, 1]; X

• satisfying

(et)♯π ≤ Cm ∀t ∈ [0, 1] for some C = C(π) ≥ 0.

• Theorem. [AGS] In any complete and separable metric measure space

(X, d, m) with m finite on bounded sets the minimal relaxed gradient |∇f|∗ and the minimal weak upper gradient |∇f|w coincide m-a.e. in X. Of course, maybe they are both trivial without extra assumptions. In the related context of PI and differentiability spaces, similar identification results between “Eulerian” notions (slopes, A-Kirchheim metric currents) and “Lagrangian” ones (Alberti representation, slopes along curves) have been recently investigated (Bate, Cheeger, Kleiner, Marchese, Preiss, Schioppa...).

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 25 / 26

Thank you for the attention! Slides available upon request

Luigi Ambrosio (SNS) Ricci lower bounds Bristol, March 2016 26 / 26