The intrinsic hypoelliptic Laplacian on sub-Riemannian manifolds Ugo - - PowerPoint PPT Presentation

The intrinsic hypoelliptic Laplacian on sub-Riemannian manifolds

Ugo Boscain (CNRS, CMAP, Ecole Polytechnique, Paris) Robert Neel, (Lehigh University) Luca Rizzi, (CMAP, Ecole Polytechnique, Paris) March 11, 2015

Plan of the Talk

definition of the Laplacian in a Riemannian manifold

macroscopic Laplacian ∆ (as divergence of the gradient) microscopic Laplacian L (as limit of geodes. rand. walks)

we have that ∆ = L up to constants. sub-Riemannian manifolds (M, , g)

macroscopic Laplacian ∆ω (as divergence of the horizontal gradient) →we need a volume ω microscopic Laplacian L▽ (as limit of geodesic random walks) →we need a measure on the space of geodesics. We will focus on random walks induced by a complement ▽ to .

(q) ▽(q)

Q1 Under which conditions on (ω, ▽) we have ∆ω = L▽? Q2 Since there exists a canonical choice of ω (the Popp volume), does it exist ▽ s.t. ∆ω = L▽ ? Is it unique?

The Riemannian macroscopic Laplacian

M: smooth, connected, orientable, geodesically-complete n-dim Riemannian manifold. The infinitesimal conservation condition for a smooth scalar quantity φ (the temperature, a concentration, the probability density of a randomly-moving particle, etc.) that flows via a flux F (which says how much of φ is crossing a unit of surface and in a unit of time) is expressed via the equation : ∂tφ + div(F) = 0 here div(·) is computed w.r.t. the Riemannian volume R. If one postulates F = − grad(φ), one obtains the Riemannian heat equation ∂tφ = ∆φ, (1) where ∆(·) = div(grad(·)), is the Laplace-Beltrami operator. Since equation (1) has been obtained thinking to φ as to a fluid without a microscopic structure in the following we refer to it as to the macroscopic heat equation and to the corresponding operator ∆ as to the macroscopic Laplacian.

Remarks

The divergence form of ∆ with the geodesic completeness of the manifold implies that [Strichartz]:

∆ is self adjoint (evolution well defined in L2(M, R)) it admits a smooth positive symmetric smooth heat kernel

φ(q, t) =

• M

Pt(q, ¯ q)φ0(¯ q) dR If {X1, . . . Xn} is a local orthonormal frame of the Riemannian manifold we have the nice formulas grad(φ) =

n

• i=1

Xi(φ)Xi, ∆(φ) =

n

• i=1

(X2

i + div(Xi)Xi)(φ).

The Riemannian microscopic view point

We consider a particle that at time zero is in q0; at time δt jumps on a point q1 of the sphere of radius ε centered in q0, uniformly on the directions, by following a geodesic; at time 2δt jumps on a point q2 of the sphere of radius ε centered in q1 uniformly on the directions, by following a geodesic; .......

q2 M q0 q1

If φ is the density of probability of finding the particle in q we have that: how much φ is increasing at a point q in time δt is proportional to the difference between the average of φ in a sphere of radious ε centered in q and the value of φ(q, t). φ(q, t + δt) − φ(q, t) =

Sn−1 φ

• expq(ε, θ), t
• dθ − φ(q, t)
• .

Dividing by δt we obtain φ(q, t + δt) − φ(q, t) δt = 1 δt

Sn−1 φ

• expq(ε, θ), t
• dθ − φ(q, t)
• .

Taking the parabolic scale (↔ infinite velocity of propagation) δt = ε2/α, and letting δt → 0, ∂tφ = Lφ (2) where Lφ(q, t) = lim

ε→0

α ε2

• Sn−1 φ
• expq(ε, θ), t
• dθ − φ(q, t)
• .

(3) In the following we refer to the equation (2) as to the “microscopic heat equation” and to L as to the “microscopic Laplacian”.

in Riemannian ∆ is proportional to L

If we take α = 2√n, then we have ∆ = L. All of this can be viewed in the following way. On one side, the microscopic perspective is a good interpretation of the macroscopic heat equation. On the other side, the microscopic Laplacian L is a good operator because it is essentially self-adjoint with respect to a volume (the Riemannian one): we can study the evolution equation in L2(M, ω). Indeed we have convergence of the random process.

A remark on the Riemannian volume

Recall that the Riemannian volume can be defined equivalently as: the volume R such that R(X1, . . . , Xn) = 1, for any oriented local

• rthonormal frame {X1, . . . , Xn},

as the n-dimensional Hausdorff or spherical-Hausdorff volume (up to some constant). The above construction gives an alternative way of characterizing the Riemannian volume: it is the unique volume (up to constant rescaling) such that the microscopic Laplacian can be written in divergence form. These facts are much less trivial in the sub-Riemannian context. Indeed in sub-Riemannian geometry there are several notions of intrinsic volumes and the definition of the microscopic Laplacian requires additional structure.

The Laplace Beltrami vs the sum of squarres

Let X1, . . . Xn be an orthonormal frame. We have two possibilities: geodesic random walk (Ito-on-manifolds view point):

q2 M q0 q1

we get L = ∆ = n

i=1(X2 i + div(Xi)Xi)

random walk along the integral curves of X1, . . . Xn (Stratonovich view point):

q0 M q1 q2

1 4 1 4 1 4

we get L = n

i=1 X2 i (it is not intrinsic)

Sub-Riemannian manifolds

Definition A Sub-Riemannian manifold is a triple (M, , g), where (i) M is a smooth, connected, orientable manifold of dimension n; (ii) is a smooth distribution of constant rank k < n, i.e. a smooth map that associates to q ∈ M a k-dimensional subspace q of TqM satisfying the H¨

• rmander condition

(iii) gq is a Riemannian metric on q, that is smooth as function of q.

• x3

x1 x2

→Define 1(q) := (q), i+1 := i + [i, ]. If dim(i), i = 1, . . . , m do not depend on the point, it is called equiregular otherwise non-equiregular. →Locally, (, g) can be given by assigning a set of k smooth vector fields (called a local orthonormal frame) s.t. q = span{X1(q), . . . , Xk(q)}, gq(Xi(q), Xj(q)) = δij. (4)

The Carnot Carath´ eodory distance is d(q1, q2) := inf{ T

• k
• i=1

u2

i (t) dt | γ(0) = q1, γ(T ) = q2, ˙

γ(t) =

k

• i=1

ui(t)Xi(γ(t))} Thanks to the Hormander condition, the control system ˙ q(t) =

k

• i=1

ui(t)Xi(q(t)) is completely controllable and (M, d) is a metric space having the same topology as the original topology of M (Chow theorem)

Candidate minimizers are computed via the Pontryagin Maximum Principle: Normal extremals Projections on the manifold of solutions of the Hamiltonian system having as Hamiltonian H(q, p) = 1 2

k

• i

p, Xi(q)2 Arclength parameterized normal extremals belongs to H = 1

2

Abnormal extremals satisfying p(t), Xi(q(t)) ≡ 0, i = 1 . . . k. →normal extremals are geodesics →abnormal extremals can be or not geodesics

The Macroscopic Laplacian

Fix a volume ω is fixed (the intrinsic definition of ω is a subtle question). The conservation of the heat is written similarly to the Riem. case. ∂tφ + divω(F) = 0 But now one should postulate that the flux is proportional to the horizontal gradient. The horizontal gradient gradH(·) of a C∞ function φ is defined similarly to the Riemannian gradient but is a vector field belonging to the distribution: gq(v, gradH(φ)q) = dqφ(v), ∀v ∈ q We have then for the macroscopic heat equation in the sub-Riemannian context ∂tφ = ∆ωφ where ∆ω(·) = divω(gradH(·)) is the horizontal Laplacian. In terms of a local orthonormal frame: gradH(φ) =

k

• i=1

Xi(φ)Xi, ∆ω(φ) =

k

• i=1

(X2

i + divω(Xi)Xi)(φ)

What is the right volume?

In the Riemannian context, ∆ = L is correct only if ∆ = divR(grad(·)) Q What is the right volume in SRG playing the role of the Riemannian one? One needs some algorithm to assign (intrinsically), with any sub-Riemannian structure on M, a volume form ωM. Definition An intrinsic volume is a map that associates, with any (orientable) (sub)-Riemannian structure (M, , g) a volume form ωM on M such that if φ : M → N is a sub-Riemannian isometry between (M, M, gM) and (N, N, gN), then φ∗ωN = ωM. Surprisingly, even in the Riemannian case, there are many intrinsic definitions of volume. The classical Riemannian volume is the unique volume form R such that R(X1, . . . , Xn) = 1 for any orthonormal (oriented) frame. But R′(X1, . . . , Xn) = 1 + κ2, where κ is the scalar curvature is intrinsic too. The first, loosely speaking, is more “intrinsic” than the second. In fact, it is true that both depend only on the metric invariant of the structure, but R′ involves second order informations about the structure.

Then we need a more precise definition, to rule out R′ and select the classical Riemannian volume. Definition (rough version) We say that an intrinsic volume is N-intrinsic if its value at q depends only

• n the invariants of the nilpotent approximation of the structure at q.

→The nilpotent approximation is the “metric tangent space” to the sub-Riemannian manifold. It is obtained by a nonistrotropic blow up

• procedure. It is a Carnot Group. In the Riemannian case it is the

Euclidean space. →In the Riemannian case, there is only one nilpotent approximation which is Rn, and it has no invariants. Hence there there is a unique N-intrinsic volume, the Riemannian one. This rules out R′. →Unfortunately, in the sub-Riemannian case, nilpotent approximations may be different at different points.

Definition (equi-nilpotentizable sub-Riemannian manifolds) A sub-Riemannian manifold such that nilpotent approximations at different points are isometric, is called equi-nilpotentizable. Examples of equi-nilpotentizable sub-Riemannian manifolds Riemannian manifolds Carnot groups equiregular sub-Riemannian structures in dimension less than or equal to four (in particular 3D contact sub-Riemannian manifolds) Examples of non equi-nilpotentizable sub-Riemannian manifolds generic structures of rank 4 in dimension 5. generic contact manifolds of dimension greater than or equal to 5 and by generic quasi-contact sub-Riemannian manifolds in dimension greater than or equal to 6.

a N-invariant volume

An N-intrinsic volume, that works for any equiregular SR manifold is the Popp volume P introduced by Montgomery Proposition Let (M, , g) be a equi-nilpotentizable (sub)-Riemannian manifold. Then Popp’s volume P is the unique N-intrinsic volume, up to a multiplicative constant. →For non-equi-nilpotentizable cases there are other N-intrinsic volumes as for instance the spherical Hausdorff volume In particular since the divergence of a vector field does not change if the volume is multiplied by a constant, we have the following, Proposition Let (M, , g) be a equi-nilpotentizable (sub)-Riemannian manifold. Then there exists a unique macroscopic Laplacian built w.r.t. an N-intrinsic volume. In other words when the nilpotent approximation has no parameters, there is a unique N-intrinsic Laplacian. When Proposition 2 applies, we call the macroscopic Laplacian the unique N-intrinsic Laplacian and we we indicate it as ∆P.

The sub-Riemannian microscopic Laplacian

Several difficulties

• 1. while in the Riemannian context the geodesics starting from a given

point can always be parametrized by the direction of the initial velocity, i.e., by the points of a (n − 1)-dimensional sphere, in the sub-Riemannian context the geodesics starting from a given point are always parameterized by a non-compact set, namely by the points of a cylinder

q0 = H−1(1/2) ∩ T ∗ q0M = {p ∈ T ∗ q0M | H(q0, p) = 1

2} having the topology of Sk−1 × Rn−k. How to define an intrinsic finite volume on

q0, and thus a probability measure, is a non-trivial

question.

• 2. while in Riemannian geometry, for sufficiently small ε the Riemannian

sphere S(q0, ε) of radius ε centered at q0 coincides with the endpoints

• f the geodesics starting from q0 and having length ε, in

sub-Riemannian geometry this is not true: for a fixed a point q0, there are geodesics starting from q0 that lose optimality arbitrarily close to

• q0. Hence one has to decide whether to average only on the

sub-Riemannian sphere (i.e., on geodesics that are optimal for up to length ε) or on the sub-Riemannian front (i.e., all geodesics of length ε).

In any case, all these problems are indeed encompassed in the specification

• f a measure (possibly singular) on
• q0. Once such a measure µq0 on

q0

is fixed, we define the microscopic Laplacian as Lµφ(q0) = lim

ε→0

α ε2

q0

φ

• expq0(ε, λ), t
• dµq0(λ) − φ(q0, t)
• ,

The main question

Once a volume on M is chosen (hence a a macroscipic Laplacian is defined) and a measure on the cylinder is fixed (hence a microscopic Laplacian is defined), it is natural to ask Q: Under which conditions on ω and µ do we have that ∆ω and Lµ coincide? In other words, we would like to know when a macroscopic Laplacian admits a microscopic interpretation and when a microscopic Laplacian can be written in divergence form (and hence is symmetric) w.r.t. some volume

• n the space.

→In question Q the freedom that we have to define µ is too much. Indeed assigning ω means assigning each point of the manifold a number, while assigning µ means assigning each point of the manifold a measure on the cylinder.

Measures induced by a splitting

For q ∈ M, consider a splitting TqM = q ⊕ ▽q. By duality T ∗

q M = ⊥ q ⊕ ▽⊥ q .

(q)⊥ (q) ▽(q) ▽(q)⊥

Now we can define a Euclidean structure on ▽⊥

q by identifying it with q

Then the cylinder of initial covectors splits as

q = Sk−1 q

× ⊥

q .

(5) We stress that this identification depends on the choice of ▽. Definition A probability measure induced by a choice of ▽ is any measure on

q

• btained as the product

µ▽ = µSk−1

x

µ⊥

q

(6)

• f a uniform probability measure µSk−1 on Sk−1

q

=

q ∩ ▽⊥ q and a finite

probability measure µ⊥

q on ⊥

q .

It turns out that the resulting operator Lµ▽ does not depend on the choice

• f the compactly supported probability measure on the subspace of vertical

covectors, but only on the choice of the complement ▽. Theorem Let µ▽ any measure induced by the choice of a complement ▽ ⊂ T M. Then Lµ▽ depends only on ▽. Moreover, let X1, . . . , Xk a local orthonormal frame for , and Xk+1, . . . , Xn a local frame for ▽. Then Lµ▽ =

k

• i=1

X2

i + k

• i,j=1

cj

jiXi,

(7) where the structural functions cℓ

ij ∈ C∞(M) are defined by

[Xi, Xj] = n

ℓ=1 cℓ ijXℓ,

i, j = 1, . . . , n. Remark The expression of L▽ is the same if one averages only on the horizontal geodesics (i.e. taking a measure of the type µ▽ = µSk−1

x

δ⊥

x ) or

averaging on all possible geodesics.

We can reformulate question Q in terms of ▽: Q1: What relation between ▽ and ω is needed in order to have ∆ω = L▽? and in particular Q1a: Given a volume ω, does it exist ▽ such that ∆ω = L▽? Q1b: If so, is it unique? This question is interesting since on any sub-Riemannian structure there is a smooth, N-intrinsic volume, the Popp’s volume. Then an answer to Q1a gives a way to extract a canonical complement ▽ from the canonical Popp’s volume. A counting argument suggests that such question should have an affirmative since a volume form is essentially given by a non-zero function while a complement is given by (n − k)k functions. However the answer is more complicated because some integrability condition should be satisfied.

A more specific question is the following: Q1c: Let ▽ a complement, and let g▽ a smooth scalar product on ▽. Then the orthogonal direct sum g ⊕ g▽ is a Riemannian extension of the sub-Riemannian structure. Let ω▽ be the corresponding Riemannian

• volume. Is it true that ∆ω▽ = L▽ ?

This last question is even more interesting when it is possible to find an “intrinsic” Riemannian extension, i.e. some choice of ▽ and g▽ that depends only on the sub-Riemannian structure (M, , g). Intrinsic Riemannian extensions can be made in several cases, as for instance: contact case (via the Reeb vector field) quasi-contact case (without crossing of the eigenvalues) However, in general they are not known and it is believed that they do not exists.

Theorem For any complement ▽ and volume ω, the macroscopic operator ∆ω and the microscopic operator L▽ have the same principal symbol, and have no constant term. Moreover L▽ = ∆ω if and only if χ(▽,ω) := L▽ − ∆ω =

k

• i=1

n

• j=k+1

cj

jiXi + gradH θ = 0

(8) where θ = log ω(X1, . . . , Xn) and cℓ

ij are the structural functions associated

with the given local frame. This condition in a less intrinsic form for an averages only on the horizontal geodesics has been obtained also by Thallmaier-Grong [2014] and Markina-Laetsch [2014].

Results in the Contact Case

A sub-Riemannian structure (M, , g) is contact if there exists a smooth

• ne-form η such that = ker η. Moreover, dη| is non-degenerate. This

forces dim M = 2d + 1. Theorem For any volume ω there exists a unique complement ▽ such that L▽ = ∆ω. In particular this is true for the intrinsic Riemannian extension given by the Reeb vector field. →Positive answer to Q1a, Q1b, Q1c

The quasi-contact case: NEGATIVE ANSWERS

A sub-Riemannian structure (M, , g) is quasi-contact if n is even and there exists a smooth one-form η (the quasi-contact form) such that = ker η. This time, however, 0 := ker dη| is non-trivial and we assume that dim 0 = 1. The first negative answer is to Q1b (uniqueness): Theorem If for some ω, there is ▽ such that L▽ = ∆ω, then ▽ is never unique.

Surprisingly, we not only lose uniqueness (w.r.t. the contact case) but also

• existence. Consider the quasi-contact structure on M = R4, with

coordinates (x, y, z, w) defined by = ker η with η = g √ 2

• dw − y

2dx + x 2 dy

• ,

with g = ez. (9) The metric is defined by the following global orthonormal frame for X = 1 √g

• ∂x + 1

2y∂w

• ,

Y = 1 √g

• ∂y − 1

2x∂w

• ,

Z = 1 √g ∂z (10) Choose ω = P, the Popp’s volume, that is P = g5/2

√ 2 dx ∧ dy ∧ dz ∧ dw

(11) We prove that there exist no compatible complement ▽, in the sense that, for any ▽, L▽ = ∆P (12) It turns out that this is the typical (generic) picture in quasi-contact: Theorem Hence Q1a: is generically false for the Popp volume. →very strong in dimension 4: in this case the Popp volume is the unique N-intrinsic Laplacian

On a quasi-contact manifold it is possible to build (under certain assumptions) an analogue of the Reeb vector field, that provides a standard Riemannian extension. It turns out that, in terms of Z, and with our normalization, Popp’s volume is such that P(X1, . . . , Xk, Z) = 1. (13) Then the Riemannian volume of this Riemannian extension is Popp’s

• volume. Thus the above non-existence result provides also an answer to

Q1b: the macroscopic operator ∆ω provided by the quasi-Reeb Riemannian extension is not the microscopic operator L▽ provided by the quasi-Reeb complement.

Conclusions

∆ω = L▽ ? Structure Riemannian yes: for the Riemannian volume (no need of a complement ▽ ) Contact yes: for any ω there exists ▽ in particular it is true for the canonical Riemannian extension given by Reeb vector field Quasi-contact No for the Popp volume in particular it is not true for the canonical Riemannian extens. given by the quasi-Reeb vector field Carnot Groups yes for the Haar=Popp measure, but ▽ is not unique Hence in conclusion it is not so easy to have a microscopic interpretation of the macroscopic Laplacian to have a microscopic Laplacian that is self-adjoint

END