The Laplace-Beltrami operator on rank-varying and non-equiregular SR - - PowerPoint PPT Presentation

The Laplace-Beltrami operator on rank-varying and non-equiregular SR manifolds

(state of the art and new results) Ugo Boscain (CNRS, CMAP, Ecole Polytechnique, Paris) October 12, 2014

Plan of the talk: definition of rank-varying and non-equiregular SR structures definition of the SR Laplacian the choice of an intrinsic volume the singular set acts as a barrier (for the heat and Schroedinger equation) stochastic completeness some related results on conic and anti-conic type surfaces

Classical sub-Riemannian manifolds

Definition A classical sub-Riemannian manifold is a triple (M, , g), where (i) M is a connected smooth 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 LBG condition (iii) gq is a Riemannian metric on q, that is smooth as function of q. →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. G=(dim(1),dim(2), . . .) is the growth vector. →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. (1) The Carnot Carath´ eodory distance d(q1, q2) := inf{ 1

• k
• i=1

u2

i (t) dt | γ(0) = q1, γ(1) = q2, ˙

γ(t) =

k

• i=1

ui(t)Xi(γ(t))}

Rank-varying sub-Riemannian manifolds

Rank-varying sub-Riemannian structures are structures for which also dim()=dim(1) depends on the point. Locally they are defined by a set

• f k LBG vector fields that can become collinear, playing the role of

an orthonormal frame: →(q) =span{X1(q), . . . , Xk(q)} →The Carnot Carath´ eodory distance d(q1, q2) := inf{ 1

• k
• i=1

u2

i (t) dt | γ(0) = q1, γ(1) = q2, ˙

γ(t) =

k

• i=1

ui(t)Xi(γ(t))} Remark One cannot say that there exists a positive definite, symmetric bilinear form g such that gq(Xi(q), Xj(q)) = δij. Since on the singular set some vector fields could be zero. →Example: the Grushin plane (x, y) ∈ R2, X1 = 1

• , X2 =

x

• ,

For x = 0 we would have g(X2(0), X2(0)) = 1 with X2(0) = 0.

A global definition require the use of a fiber bundle : Definition (sub-Riemannian manifold (possibly rank-varying) A (n, k)-sub-Riemannian structure on M is a triple (M, U, f) where: M is a connected smooth manifold of dimension n; U is an Euclidean bundle with base M and Euclidean fiber Uq i.e. Uq is a vector space equipped with a scalar product ., .q. We assume dim(Uq) = k (the rank of the structure); f : U → T M is a smooth map that is a morphism of vector bundles, i.e. the following diagram is commutative U

πU

• ❉

❉ ❉ ❉ ❉ ❉ ❉ ❉

f

T M

π

• M

(2) and f is linear on fibers. Here πU and π are the canonical projections. := {f ◦ σ, σ ∈ Γ(U)} is Lie bracket generated. Here Γ(U) is the set

• f smooth sections of U.

the map σ → f ◦ σ from Γ(U) to Vec(U) is injective.

(n, n) sub-Riemannian structures are called almost-Riemannian structures (generalized Riemannian structures of dimension n having local

• rthonormal frames made by n vector fields that can become collinear but

satisfy the LBG assumption)

Examples

Classical equiregular sub-Riemannian structure: Heisenberg (x, y, z) ∈ R3, X1 =   1 −y/2   , X2 =   1 x/2   , dim(1(q)) = 2, dim(2(q)) = 3. Classical nonequiregular sub-Riemannian structure: Martinet (x, y, z) ∈ R3, X1 =   1 y2/2   , X2 =   1   ,        dim(1(q)) = 2, dim(2(q)) = 3. if y = 0 dim(2(q)) = 2. if y = 0

• dim(3(q)) = 3.

Rank-varying sub-Riemannian structure: Grushin structure (Baouendi, ’67, Grushin ’70, Franchi-Lanconelli ’84) (almost-Riemannian structure) (x, y) ∈ R2, X1 = 1

• , X2 =

x

• ,

   dim(1(q)) = 2. if x = 0 dim(1(q)) = 1. if x = 0

• dim(2(q)) = 2.

Singular set where some of the i change dimension

Hypoelliptic diffusion

Historically (Jerison, Fefferman, Sanchez-Calle Hormander, etc....) hypoelliptic diffusion has been introduced as the diffusion generated by an

• perator L

second order (without zero order) L = n

i=1 gij(q)∂i∂j + n i=1 bi(q)∂i

the matrix gij is positive semidefinite. a “Lie-Bracket generating condition” is verified. For instance

gij(q) = k

h=1 Bi h(q)Bj h(q) with B1(q), . . . , Bk(q) smooth

vector fields that are Lie-Bracket generated L is sub-elliptic

From such an operator one can extract a SR metric as the “inverse” of gij. Typical examples are the operators in Hormander form L =

k

• i=1

X2

i + X0 with

X1, . . . Xk LBG vector fields

Riemannian diffusion

This is not the approach that people have in Riemannian geometry where:

• ne starts from the geometric structure (M, g)

and then extracts the diffusion operator (the Laplace-Beltrami

• perator, that in this case is elliptic) as

∆(φ) = divr(grad(φ)), where the divergence is computed with respect to the Riemannian volume r. This approach has the following advantage: Theorem (Strichartz) Let (M, g) be a complete Riemannian manifold. Then the Laplace-Beltrami

• perator defined on C∞

0 (M) is essentially self-adjoint in L2(M, r) and hence

it is in one to one correspondence with a strongly continuous semigroup etL and it admits a smooth symmetric positive L2(M, r) heat kernel. →thanks to self-adjointness one can also study the Schroedinger equation ∆(φ) = 1 √g ∂i(√ggij∂jφ) =

n

• i=1
• X2

i + divr(Xi)Xi

• ,

where {Xi} is an orthonormal frame

A result of this kind is less evident starting from a diffusion operator, because one does not know a priori a volume w.r.t. which guarantee self-adjointness, unless:

• ne starts from an operator in divergence form,
• ne adjusts the first order terms with the volume,
• ne renounce to a good evolution in L2 (and to consider the

Schroedinger equation) Remark We say that in Riemannian geometry the Laplace-Beltrami

• perator is intrinsic since it depends only on the Riemannian structure and

not choice of coordinates choice of the orthonormal frame choice of the volume (the volume is fixed by the Riemannian structure)

The intrinsic Laplacian in SRG

The most natural construction for a diffusion operator in SR geometry is to follow the Riemannian construction and to define: ∆H(φ) = div(gradH(φ)) the horizontal gradient is grad(φ) = k

i=1 Xi(φ)Xi

the divergence is the standard divergence computed with respect to an “intrinsic” volume that we have to determine This approach has the following advantages: In the equiregular case if the sub-Riemannian manifold is complete as metric space we have self-adjointness and existence and uniqueness of a strongly continuous semigroup (Strichartz). In the rank-varying or nonequiregular case we will see appearing some interesting phenomena in the heat diffusion, deeply connecting to the geometry of the SRM.

Remarks

Remark 1 While in the equiregular case the choice of the volume does not affect certain properties as small time heat kernel asymptotics Weyl law in the rank-varying case or in the non equiregular case the choice of an intrinsic volume will crucially affect these properties. Remark 2 The expression of the intrinsic Laplacian is ∆H =

k

• i=1
• X2

i + div(Xi)Xi

• ,

where {Xi} is an orthonormal frameof the SRM Notice that it is not the sum of squares if we start from a diffusion operator L, we extract the metric and we build the intrinsic Laplacian, we will not obtain in general L. diffusion operator L − → sub-Riemannian structure − → ∆H = L (in the rank-varying or non equir. cases they can be drastically different)

Does an intrinsic volume exist in SRG?

An “intrinsic volume” is a volume that is constructed only from the sub-Riemannian structure and that does not depend on choice of the coordinates choice of the orthonormal frame In the equiregular case there are several constructions of intrinsic volumes: Popp’s volume P (Montgomery’s book) “algebraic”. It is always C∞ Spherical Q-Hausdorff volume S Q- Hausdorff volume H

• thers .....

Relations between these volumes: →It is known that they are commensurable (Mitchell): The Radon-Nikodym derivatives are measurable functions that are locally bounded and locally bounded away from zero →for left-invariant sub-Riemannian structures on Lie groups they are proportional →in the Riemannian case they are proportional →Montgomery question: are they proportional in general?

The Relation between P and S (2011-2013)

In [Agrachev, Barilari, B., 2012]

dS dP = 2Q P-volume of the nilpotent ball; (see also [Ghezzi-Jean 2013]) dS dP is a continuous function;

P and S are proportional iff the nilpotent approximation of the structure does no depend on the point (up to dimension 4 they are proportional but in larger dimension not in general); In the (n − 1, n) case dS

dP is C3 but not C5.

In [Barilari, B., Gauthier 2012] and [B. Gauthier 2013] we proved that in the (n − 2, n) case it is C1 (and C2 out of a set of codimenion 7). Open question Is dS

dP C1?

What about the Hausdorff volume? (related to the study of the isodiameter sets. See Leonardi, Rigot, Vittone)

Popp’s and Hausdorff volumes in the rank-varying and non-equiregular cases

Both Popp’s volume and the Q-spherical Hausdorff’s measure explodes. E.g., The Grushin case X1 = 1

• ,

X2 = x

• P = S = r = 1

|x|dx dy ∆H = X2

1 + X2 2 + divP(X1)X1 + divP(X2)X2 = ∂2 x + x2∂2y − 1

x∂x The Martinet case X1 =   1 y2/2   , X2 =   1   P = S = 1 |y|dx dy dz ∆H = X2

1 + X2 2 + divP(X1)X1 + divP(X2)X2 = (∂2 x + y2

2 ∂z)2 + ∂2

y − 1

y ∂y Notice that in both cases the intrinsic volume explodes and it is not L1

loc

w.r.t. a smooth volume. the intrinsic Laplacian contains some diverging first order terms due to the “divergence of the divergence.”

Does the heat flow through the singular set?

Theorem (B., Laurent, 2013) For a 2D almost-Riemannian manifold, let Ω be a connected bounded component of M \ Z. Assume that HA ∂Ω is smooth; (generic hypothesis) HB for every q ∈ M, (q) + [, ](q) = TqM. (technical hypothesis due to our laziness) ⇒ ∆H =divr(grad((.)) with dom. C∞

0 (Ω) is essen. self–adjoint on L2(Ω, dr).

6 4 2 2 4 10 5 5 10

Singular Set Geodesics and Front for Grushin Starting from a Riemannian point

Geodesics cross the singular set, but not the heat (or a quantum particle)

What can we say beyond 2d almost-Riemannian geometry? What seems to be crucial to have a barrier for the heat low is the non-integrability of Popp’s measure.

I “ordered” the following theorem: Theorem (Ghezzi-Jean 2015) Let W be a connected component of the singular set. Assume that W is a codimension 1 smooth surface. Then both the Popp’s and the Q-spherical Hausdorff measure are not L1

loc in a neighborhood of W . (Q is the

Hausdorff dimension out of the singular set) Thanks to the result of Ghezzi-Jean, we were able to extend this to any dimension and to remove (q) + [, ](q) = TqM. Theorem (B., Prandi, Seri, IHP 09-2014) For an almost-Riemannian manifold, let Ω be a connected bounded component of M \ Z with smooth boundary. Then ∆H =divr(grad((.)) with domain C∞

0 (Ω) is essentially self-adjoint on L2(Ω, dr).

Consequences

Corollary Consider the unique solution u of the Schrdinger equation i∂tu + ∆u = (3) u(0) = u0 ∈ L2(M, dr) (4) with u0 supported in a connected component Ω. Then, u(t) is supported in Ω for any t ≥ 0. The same holds for the solution of the heat or for the solution of the wave equation.

initial condition

singular set (contains all the zeros)

geodesics X2 X1 X2 X1

Consequences

classical particle cross the singular set but not the heat or a quantum particle (due to the explosion of the volume) no semiclassical limit it is not possible to reconstruct the distace from the heat-kernel (No Varadhan-Leandre, Neel-Stroock , Barilari-B-Neel, results to reconstruct the cut locus starting from the heat-kernel) −t log Pt(q1, q2) → d2(q1, q2) 4 (NO)

What is known beside the almost-Riemannian case?

Theorem (B., Laurent, 2013) Consider the sub-Riemannian structure in M = Tx × Ry × Tz for which an

• rthonormal basis is given by

X1 = (1, 0, y2 2 ), X2 = (0, 1, 0). Then the corresponding intrinsic sub-Riemannian Laplacian which is given by ∆H = (X1)2 + (X2)2 − 1 y X2 with domain C∞

0 (M \ {y = 0}) is essentially self-adjoint on L2(M, P),

where P =

1 |y|dx dy dz is the Popp volume.

→the general result in a non-equiregular sub-Riemannian structure is still missed

Stochastic completeness (some partial results)

Since the singular set act as a barrier for the heat flow, it is natural to ask if the heat is “reflected” or “absorbed” by the singularity.

Conic and anti-conic type surfaces

We have: Theorem (B. Prandi 2014) Fix α ∈ R. On R × S1 consider the Grushin structure generated by X1 = 1

• ,

X2 =

• xα
• ,

Then ∆H is essentially self adjoint if and only if α / ∈ (−3, 1). for α ≥ 1 ∆H is stochastically incomplete: Pt1 ≤ 1 for α ≤ −3 ∆H is stochastically complete: Pt1 = 1

• 3
• 1

1

• 3
• 1

1

For α ∈ (−3, −1] there are self adjoint extensions that permit the communication between the two parts but

• nly the “average” can cross the singular set.

this extensions are not Markovian (indeed a quantum particle can cross the singular set but not a the heat). →For α ∈ (−1, 1) there are many self adjoint extensions that permit the communication between the two parts and they are not yet classified.

Stochastic completeness in general

We expect that every time that the heat is “trapped” in a connected bounded component of M \ X then ∆H is stochastically incomplete.

(X2

i + divP(Xi)Xi)

The end

Idea of the proof: the Grushin case

We first compactify in the y variable by considering it on Rx × Ty.

x y

By setting f =

• |x|g

∂2

x + x2∂2 y − 1

x∂x

• n L2( 1

|x|dx dy)

⇓ ∂2

x + x2∂2 y − 3

4 1 x2

• n L2(dx dy)

(5)

By making Fourier transform in y, we are reduced to study the selfadjointness of: ∂2

x − 3

4 1 x2 − k2x2

• n ]0, ∞[

i.e. of −∂2

x + 3

4 1 x2 + k2x2

• n ]0, ∞[

Proposition (Reed-Simon) The operator −∂2

x + c |x|2 defined on L2(]0, +∞[) with domain C∞ 0 (]0, +∞[)

is essentially self-adjoint if and only c ≥ 3

4.

The rest of the proof for an almost Riemannian structure consists in generalizing this result for a normal form around a connected component of the singular set and to treat it as a perturbation of the Grushin case.