Curvature and Diffusion in the Heisenberg group Nicolas JUILLET - - PowerPoint PPT Presentation

Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group

Nicolas JUILLET

IRMA, Strasbourg

Paris, IHP , October 2014

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group

Outline

1

Notations and definitions

2

Curvature

3

Diffusion

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Notations and definitions

The subRiemannian H.

A basis for left-invariant vector fields is X = E1 − y 2 E3 , Y = E2 + x 2 E3 and Z = [X,Y] = E3. A curve is horizontal if ˙

γ ∈ Vect(X,Y) for any t. Actually ˙ γ(t) = a(t)X(γ(t))+ b(t)Y(γ(t)).

It has norm |˙

γ| =

• a2(t)+ b2(t) and the length of γ is

|˙

γ|.

The diffusion operator is ∆ = X2 + Y2. The horizontal gradient is

∇f = XfX+ YfY.

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Notations and definitions

The Riemannian Hε for ε > 0

Let ε > 0 and Hε := (R3,dε,L3). An orthonormal basis at point (x,y,u) is X = E1 − y 2 E3 , Y = E2 + x 2 E3 and

εZ = ε[X,Y] = εE3.

If

˙ γ(t) = (a(t)X+ b(t)Y+ c(t)Z)(γ(t))

then |˙

γ(t)|ε =

• a2 + b2 + c2

ε2 and lengthε(γ) =

a2 + b2 + c2

ε2 (t)dt.

The gradient is ∇εf = ∇f +(ε2Zf)Z.

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Notations and definitions

Curves in P2(X).

For a metric measure space (X,d,ν), two theories make use of curves in the space

• f probability measures P2(X) endowed with the transport distance W.

Ricci bounds: geodesic curves. (Lott-Villani and Sturm) Heat diffusion: curves with maximal slope. (Ambrosio-Gigli-Savaré)

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Notations and definitions

Comparaison of the Wasserstein distances

Let W be the L2-minimal metric with respect to the Carnot-Carathéodory metric dCC and Wε with respect to dε. Wε ≤ W,

P2(H) = P2(Hε) as topological spaces.

Lipschitz curves (resp. absolutely continuous) of P2(H) are Lipschitz (resp. absolutely continuous) in P2(Hε).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Notations and definitions

Geodesic curves

Theorem (Ambrosio-Rigot, 2004) Let µ0,µ1 ∈ P2(H) such that µ0 is absolutely continuous. Then there is a unique

• ptimal coupling π. It is π = (Id⊗T)#µ0 for some map T. Moreover there is a unique

geodesic between p and T(p) (µ0-almost surely). In fact T(p) = p.expH(∇ψ(p),Zψ(p)). Here ψ : H → R depends on µ0,µ1. The unique geodesic (µs)s∈[0,1] between µ0 and µ1 is defined by µs = (Ts)#µ0 where Ts(p) = p.expH(s∇ψ(p),sZψ(p)).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Notations and definitions

Absolutely continuous curves

Theorem Let (µt)t∈[0,T] be an absolutely continuous curve of P2(H). Then for almost every t > 0 there exists a field of horizontal vectors vt ∈ TanµtP2(H) such that the continuity equation is satisfied dµt/dt + div(vtµt) = 0. Moreover

|vt|2dµt = limh→0+ h−1W(µt,µt+h) =: Speed(µt).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Notations and definitions

The relative entropy

The relative entropy of µ = ρL is given by H(µ) = H(µ | L) =

• ρln(ρ)(x)dL(x).

Big entropy: µ concentrated on a few space. Small entropy: µ take a lot of space.

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

Ricci curvature bounds on a Riemannian manifolds

Theorem Let (M,g) be a Riemannian manifold of dimension N and K ∈ R. Then the following properties are equivalent.

1

∀(x,v) ∈ TM, Ricci(v,v) ≥ Kg(v,v)

2

∀f ∈ C∞(M),∀x ∈ M, Γ2(f)(x) ≥ KΓ(f)(x)+(∆f(x))2/N

3

(M,dg,volg) satisfies the Curvature Dimension Condition CD(K,N),

4

(M,dg,volg) satisfies the Measure Contraction Property MCP(K,N)

Under this condition, one can deduce a lot of analytico-geometric results.

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

MCP is satisfied

Jacobian estimate For any e ∈ H. The contraction map M s

e : f → M s(e,f) is differentiable with

Jac(M s

e )(f) ≥ s5.

Equality case : e and f are on a horizontal line. As a consequence (H,dc,L3) satisfies MCP(0,5): Measure Contraction Property MCP(0,N) for (X,d,ν): for every point e ∈ X and for all s ∈ [0,1],

M s

e #ν ≤ sNν.

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

K-convexity and Ricci curvature

Theorem (Cordero–McCann–Schmuckenschläger and Sturm–von Renesse) Let M be a Riemannian manifold. It has Ricci curvature bounded from below by K ∈ R if and only if for every geodesic

(µt)t∈[0,T] in P2(M) of speed 1 the function

t ∈ [0,T] → H(µt | VolM)− K.t2 2 ∈ R is convex. This theorem has been turned into the definition of CD(K,+∞) for metric measure spaces: A space satisfies CD(K,+∞) if H(.|ν) is K-convex along every geodesic curve of (P2(X),Wd).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

The curvature dimension condition is not satisfied.

Theorem (J.) In P2(H) the entropy H with respect to L is not K-convex along geodesics (for any K ∈ R). The curvature-dimension conditions introduced by Lott–Villani and Sturm are not satisfied

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

The general theorem is Theorem (J. 2009) In (Hn,L2n+1,dc), the Heisenberg group with the Lebesgue measure and the Carnot-Carathéodory distance, MCP(K,N) is true if and only if N ≥ 2n + 3 and K ≤ 0, CD(K,N) is false for every (K,N).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

Sketch of proof

p p′ 0H a b b′ a′

E F Let F be a small ball such that 0H and the center of the ball are on a “bad" geodesic. For E we take the “geodesic inverse" of F.

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

Sketch of proof

p p′ 0H a b b′ a′

E F Let F be a small ball such that 0H and the center of the ball are on a “bad" geodesic. For E we take the “geodesic inverse" of F. It turns out that L(E) = L(F). We want to prove L

• M 1/2(E,F)
• < L(F) because it is a contradiction to the

Brunn-Minkowski inequality.

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

Sketch of proof

p p′ 0H a b b′ a′ For each e ∈ E, the contracted set M 1/2(e,F) is a sort of ellipsoid that contains 0H. The volume of such an ellipsoid is 2−5L(F).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

Sketch of proof

p p′ 0H a b b′ a′ For each e ∈ E, the contracted set M 1/2(e,F) is a sort of ellipsoid that contains 0H. The volume of such an ellipsoid is 2−5L(F).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

Sketch of proof

p p′ 0H a b b′ a′ For each e ∈ E, the contracted set M 1/2(e,F) is a sort of ellipsoid that contains 0H. The volume of such an ellipsoid is 2−5L(F).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

Sketch of proof

p p′ 0H a b b′ a′ The midset M 1/2(E,F) is made of the reunion of these ellipsoids.

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Curvature

Sketch of proof

p p′ 0H a b b′ a′ All of them contains 0H. Then M 1/2(E,F) is an ellipsoid of size 2. Its volume is 23

·2−5L(F)

• = L(F)

4

.

Then L(M 1/2) < L(F) = L(E).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Diffusion

After Jordan-Kinderlehrer-Otto (2000) “The heat flow is the gradient flow of H in P2(X)" Gradient flow means ˙

γt = −∇F(γt). We need a metric definition of this equation.

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Diffusion

Gradient flow of the relative entropy.

An absolutely continuous curve (µt)t≥0 is a gradient flow of H if t → H(µt) is decreasing. for almost every t > 0,

   ∂tH(µt) = −Speed(µt)· Slope(H)(µt)

Speed(µt) = Slope(H)(µt). Theorem (J. 2014) If the curve (µt)t>0 is a gradient flow of H then µt = ρtL3 and

∂ ∂t ρt = ∆ρt.

The converse is true.

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Diffusion

Is Slope(µ) the same as

• I(µ) =

|∇ρ|2/ρdL ?

In the Riemannian setting we have Slopeε(µ) =

√

I +ε2J where J(µ) = (Zρ)2/ρdL

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Diffusion

HWI inequalities in measures with finite Riemannian slope

We can go from the HWI inequality H(ν) ≥ H(µ)−

• Iε(µ)Wε(µ,ν)+ 1

2 ·

−1

2ε2

• Wε(µ,ν)2

that is available in Hε for every ε > 0 to H(ν) ≥ H(µ)−

• I(µ)W(µ,ν)− C(µ)W(µ,ν)3/2,

but only if Iε(µ) = I(µ)+ε2J(µ) is finite (and C = (J/2I)+ 1/4).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Diffusion

Slopeε =

√

I +ε2J is an upper gradient of H on P2(Hε)

With the following Riemannian result Upper gradient on the Riemannian Hε Let (µt)t≥0 be an absolutely continuous curve of P2(Hε). If t1

t0

Slopeε(H)(µt)Speedε(µt) < +∞ then t → H(µt) is absolutely continuous. Moreover

|H(µt0)− H(µt1)| ≤

t1

t0

• Iε(µt)Speedε(µt)dt.

We can prove that

√

I (≤ Slope(H)) is an upper gradient of H in P2(H).

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group

Curvature and Diffusion in the Heisenberg group Diffusion

Theorem (Ambrosio-Gigli-Savaré 2014)

µt = ρtν is a gradient flow of H(.|ν) in P2(X) if and only if ft is a gradient flow of the

Dirichlet-Cheeger energy in L2(X,ν). It holds without any assumption on (X,d,ν) but the heat flow can be degenerate, non-unique, non-linear. A good additional assumption is that the space satisfies RCD∗(K,N). A large part of the theory holds for Carnot groups

Nicolas JUILLET Curvature and Diffusion in the Heisenberg group