On the Hausdorff volume in Sub-Riemannian geometry Davide Barilari - - PowerPoint PPT Presentation

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

On the Hausdorff volume in Sub-Riemannian geometry

Davide Barilari (SISSA, Trieste) Nonlinear Control and Singularities, Porquerolles, France October 25, 2010

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 1 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Joint work with

General results and Corank 1 case: Andrei Agrachev Ugo Boscain Corank 2 case: Jean-Paul Gauthier

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 2 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Outline

1

Sub-riemannian geometry

2

Intrinsic volume and sub-Laplacian

3

Comparison between volumes

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 3 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Outline

1

Sub-riemannian geometry

2

Intrinsic volume and sub-Laplacian

3

Comparison between volumes

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 4 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Sub-Riemannian manifolds

Definition

A sub-Riemannian manifold is a triple S = (M, ∆, ·, ·), where (i) M is a connected smooth manifold of dimension n ≥ 3; (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. (iii) ·, ·q is a Riemannian metric on ∆q, that is smooth as function of q. We assume that the Hörmander condition is satisfied Lieq∆ = TqM, ∀ q ∈ M, where ∆ denotes the set of horizontal vector fields on M, i.e. ∆ = {X ∈ Vec M | X(q) ∈ ∆q, ∀ q ∈ M} .

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 5 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

horizontal curve ∆(q) A Lipschitz continuous curve γ : [0, T] → M is said to be horizontal if ˙ γ(t) ∈ ∆γ(t) for a.e. t ∈ [0, T]. Given an horizontal curve γ : [0, T] → M, the length of γ is ℓ(γ) = T

• ˙

γ, ˙ γ dt.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 6 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Carnot-Caratheodory distance

The distance induced by the sub-Riemannian structure on M is d(q, q′) = inf{ℓ(γ) | γ(0) = q, γ(T) = q′, γ horizontal}. (1) From the Hörmander condition (and connectedness of M) it follows: d(q, q′) < ∞, ∀ q, q′ ∈ M (M, d) is a metric space and d(·, ·) is continuous with respect to the topology of M (Chow’s Theorem) The function d(·, ·) is also called Carnot-Caratheodory distance.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 7 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Orthonormal frame

Locally, the pair (∆, ·, ·) can be given by assigning a set of k smooth vector fields, called a local orthonormal frame, spanning ∆ and that are orthonormal ∆q = span{X1(q), . . . , Xk(q)}, Xi(q), Xj(q) = δij. The problem of finding curves that minimize the length between two given points q0, q1, is rewritten as the optimal control problem ˙ q =

k

• i=1

ui Xi(q) T

• k
• i=1

u2

i → min

q(0) = q0, q(T) = q1

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 8 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Growth vector

Define ∆1 := ∆, ∆2 := ∆ + [∆, ∆], ∆i+1 := ∆i + [∆i, ∆]. If the dimension of ∆i

q, i = 1, . . . , m does not depend on the point q ∈ M, the

sub-Riemannian manifold is called regular. In the regular case the Hörmander condition guarantees that there exist m ∈ N, such that ∆m

q = TqM, for all q ∈ M.

m is called the step of the structure, growth vector of the structure is the sequence G(S) := (dim ∆

• k

, dim ∆2, . . . , dim ∆m

• n

)

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 9 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Basic features in SRG

t2 t

spheres are highly non-isotropic and they are not smooth even for small time In the regular case the Hausdorff dimension of (M, d) is given by the formula (Mitchell) Q =

m

• i=1

iki, ki := dim ∆i − dim ∆i−1. In particular the Hausdorff dimension is always bigger than the topological dimension of M: Q > n

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 10 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Outline

1

Sub-riemannian geometry

2

Intrinsic volume and sub-Laplacian

3

Comparison between volumes

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 11 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Motivation

How to define an intrinsic volume in SRG?

(in the sense that it depends only on the sub-Riemannian structure and not on the choice of the coordinates and of the orthonormal frame)

This question naturally arise if one wants to study the Heat equation in the sub-Riemannian setting, in order to study the interplay between the analysis of the sub-Laplacian operator L and the geometric structure of M. analysis ← → geometry ∂tφ = Lφ

?

← → curvature What we need first is a “geometric” definition of the Laplacian.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 12 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Riemannian case

On a Riemannian manifold the Laplace-Beltrami operator L is defined as follows Lφ = div(gradφ)

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 13 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Riemannian case

On a Riemannian manifold the Laplace-Beltrami operator L is defined as follows Lφ = div(gradφ) the gradient of a function φ is the vector field canonically associated to dφ by the metric, i.e. the unique vector field satisfying gradφ, X = dφ(X), ∀ X ∈ Vec M. If X1, . . . , Xn is a local orthonormal frame gradφ =

n

• i=1

Xi(φ)Xi

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 13 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Riemannian case

On a Riemannian manifold the Laplace-Beltrami operator L is defined as follows Lφ = div(gradφ) the divergence of a vector field X says how much the flow of X change the volume

div X>0 div X<0

If µ is a volume on M, div X is the unique function satisfying LXµ = (div X)µ where LX denotes Lie derivative. To define the Laplacian we need to fix a volume!

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 13 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Volume on a Riemannian manifold

In the Riemannian case the geometric structure defines the canonical volume: if X1, . . . , Xn is a local orthonormal frame, the Riemannian volume µ is the volume associated to the n-form ωµ = dX1 ∧ . . . ∧ dXn, µ(A) =

• A

ωµ With respect to the Riemannian volume we can write div X =

n

• j=1

dXj[Xj, X] and the Laplace-Beltrami operator is written Lφ =

n

• i,j=1

X 2

i φ + (dXj[Xj, Xi]) Xiφ

• Example. n = 2 and [X1, X2] = a1X1 + a2X2

L = X 2

1 + X 2 2 − a2X1 + a1X2

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 14 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Volume on a Riemannian manifold

In the Riemannian case the geometric structure defines the canonical volume: if X1, . . . , Xn is a local orthonormal frame, the Riemannian volume µ is the volume associated to the n-form ωµ = dX1 ∧ . . . ∧ dXn, µ(A) =

• A

ωµ With respect to the Riemannian volume we can write div X =

n

• j=1

dXj[Xj, X] and the Laplace-Beltrami operator is written Lφ =

n

• i,j=1

X 2

i φ + (dXj[Xj, Xi]) Xiφ

• Example. n = 2 and [X1, X2] = a1X1 + a2X2

L = X 2

1 + X 2 2 − a2X1 + a1X2

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 14 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Hausdorff measure

Moreover since (M, d) is a metric space we can define the Hausdorff volume Hn the spherical Hausdorff volume Sn Recall that Hn(Ω) = lim inf

δ→0 {

• i

diam(Ui)n, Ω ⊂

• i

Ui, diam(Ui) < δ}

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 15 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Hausdorff measure

Moreover since (M, d) is a metric space we can define the Hausdorff volume Hn the spherical Hausdorff volume Sn Recall that Sn(Ω) = lim inf

δ→0 {

• i

diam(Ui)n, Ω ⊂

• i

Ui, Ui balls, diam(Ui) < δ}

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 15 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Hausdorff measure

Moreover since (M, d) is a metric space we can define the Hausdorff volume Hn the spherical Hausdorff volume Sn Recall that Sn(Ω) = lim inf

δ→0 {

• diam(Ui)n, Ω ⊂
• i

Ui, Ui balls, diam(Ui) < δ} These measures are proportional: µ = α Hn = α Sn, α = Vol(B1) 2n where Vol(B1) is the volume of a unit Euclidean ball in Rn. No problem which one we choose!

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 15 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Sub-Riemannian case

If we want to repeat in the sub-Riemannian case Lφ = div(gradφ) we need to define what grad is and which volume to be used in div.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 16 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Sub-Riemannian case

Lφ = div(gradφ) the horizontal gradient of a function φ is the horizontal vector field associated to dφ by the metric, i.e. the unique vector field satisfying gradφ, X = dφ(X), ∀ X ∈ ∆. If X1, . . . , Xk, k < n is a local orthonormal frame for ∆, we have gradφ =

k

• i=1

Xi(φ)Xi

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 16 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Sub-Riemannian case

Lφ = div(gradφ) div which volume µ do we use to define it? LXµ = (div X)µ The Hausdorff dimension is Q > n, hence we have → the Hausdorff measure HQ → the spherical Hausdorff measure SQ Moreover we have also an intrinsic volume → Popp’s measures P Are these three volumes equivalent for defining L?

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 16 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Popp’s measure

Since the scalar product is defined only on ∆, a local orthonormal frame X1, . . . , Xk for the structure does not permit us to define directly an intrinsic n-form as in the Riemannian case µ = dX1 ∧ . . . ∧ dXk ∧ ? ∧ . . . ∧ ?, k < n. Hypothesis: the structure is regular (i.e. dim ∆i

q = const. in q)

Case n = 3. If X1, X2 is a local orthonormal frame for the structure it is easy to see that the quantity [X1, X2](mod ∆) is well defined and tensorial. This implies that the wedge product of the dual basis P = dX1 ∧ dX2 ∧ d[X1, X2] depends only on the structure (i.e. is invariant w.r.t. rotation of the

• rthonormal frame)

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 17 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Popp’s measure

Since the scalar product is defined only on ∆, a local orthonormal frame X1, . . . , Xk for the structure does not permit us to define directly an intrinsic n-form as in the Riemannian case µ = dX1 ∧ . . . ∧ dXk ∧ ? ∧ . . . ∧ ?, k < n. Hypothesis: the structure is regular (i.e. dim ∆i

q = const. in q)

Case n = 3. If X1, X2 is a local orthonormal frame for the structure it is easy to see that the quantity [X1, X2](mod ∆) is well defined and tensorial. This implies that the wedge product of the dual basis P = dX1 ∧ dX2 ∧ d[X1, X2] depends only on the structure (i.e. is invariant w.r.t. rotation of the

• rthonormal frame)

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 17 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Popp’s measure

In the general case the regularity assumption makes possible to complete the

• rthonormal frame with commutators of the frame, whose structure depends
• nly on the Lie bracket structure of ∆ and not on the point.

In other words we have the flag ∆ ⊂ ∆2 ⊂ . . . ⊂ ∆m = TM Even if we do not have a way to measures vectors on ∆i, for i > 1, we can do it

• n ∆i/∆i−1, and this is sufficient to define a volume.

Remarks for n = 3 we have m = 2 and ∆2/∆ = span[X1, X2] (mod ∆) P is a smooth volume (it is associated to a smooth n-form)

• n left-invariant sub-Riemannian structures on Lie groups the invariant

volume form is proportional to the left-Haar measure

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 18 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Outline

1

Sub-riemannian geometry

2

Intrinsic volume and sub-Laplacian

3

Comparison between volumes

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 19 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Question (Montgomery)

Is Popp’s measure equal to a constant multiple (perhaps depending on the growth vector) of the Hausdorff measure? Montgomery’s remarks Mitchell Theorem: If µ is a smooth volume on a regular sub-Riemannian structure (e.g. P) then dµ = fµHdHQ

• Radon-Nikodym derivative

where fµH is measurable, locally bounded and locally bounded away from zero (“commensurable") Moreover well known estimates show that SQ is commensurable with HQ SQ

comm.

← → HQ

comm.

← → µ

comm.

← → P

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 20 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Purpose of the talk

We answer to the question of Montgomery for the spherical Hausdorff measure i.e. study fPS defined by dP = fPSdSQ

• 1. is fPS constant?
• 2. if not, what is the regularity of fPS?

Remarks.

• 1. and 2. are trivial for left invariant structures: all SQ, HQ, P are

proportional to the left-Haar measures. In 2. one can replace P with every smooth measure µ. We have the complete answer for all regular SRM up to dimension 5 and for the cases (n − 1, n).

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 21 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Results

Theorem (Continuity)

For a regular sub-Riemannian manifold fPS is continuous

Theorem (Small dimensions)

Let dim M = n ≤ 5. Then if G(S) = (4, 5) then fPS is constant if G(S) = (4, 5) then fPS is C4 but not C5 .

Theorem (Corank 1)

Let G(S) = (n − 1, n), then fPS is C4 but not C5.

Theorem (Corank 2)

Let G(S) = (n − 2, n), then fPS is C1 (at least) but is not smooth.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 22 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Results

Theorem (Continuity)

For a regular sub-Riemannian manifold fPS is continuous

Theorem (Small dimensions)

Let dim M = n ≤ 5. Then if G(S) = (4, 5) then fPS is constant if G(S) = (4, 5) then fPS is C4 but not C5 .

Theorem (Corank 1)

Let G(S) = (n − 1, n), then fPS is C4 but not C5.

Theorem (Corank 2)

Let G(S) = (n − 2, n), then fPS is C1 (at least) but is not smooth.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 22 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Results

Theorem (Continuity)

For a regular sub-Riemannian manifold fPS is continuous

Theorem (Small dimensions)

Let dim M = n ≤ 5. Then if G(S) = (4, 5) then fPS is constant if G(S) = (4, 5) then fPS is C4 but not C5 .

Theorem (Corank 1)

Let G(S) = (n − 1, n), then fPS is C4 but not C5.

Theorem (Corank 2)

Let G(S) = (n − 2, n), then fPS is C1 (at least) but is not smooth.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 22 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Results

Theorem (Continuity)

For a regular sub-Riemannian manifold fPS is continuous

Theorem (Small dimensions)

Let dim M = n ≤ 5. Then if G(S) = (4, 5) then fPS is constant if G(S) = (4, 5) then fPS is C4 but not C5 .

Theorem (Corank 1)

Let G(S) = (n − 1, n), then fPS is C4 but not C5.

Theorem (Corank 2)

Let G(S) = (n − 2, n), then fPS is C1 (at least) but is not smooth.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 22 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Results

Theorem (Continuity)

For a regular sub-Riemannian manifold fµS is continuous.

Theorem (Small dimensions)

Let dim M = n ≤ 5. Then if G(S) = (4, 5) then fµS is smooth. if G(S) = (4, 5) then fµS is C4 but not C5 .

Theorem (Corank 1)

Let G(S) = (n − 1, n), then fµS is C4 but not C5.

Theorem (Corank 2)

Let G(S) = (n − 2, n), then fµS is C1 (at least) but is not smooth.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 22 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Idea of the proof: Nilpotent approximation

Let S = (M, ∆, ·, ·) be a regular sub-Riemannian manifold and µ be a smooth measure on M. the metric tangent space (in the Gromov-Hausdorff sense) of S at the point q, denoted Sq, is called the nilpotent approximation at q of the structure. under the regularity assumption Sq is a Carnot group (i.e. is endowed with a left-invariant sub-Riemannian structure on a n-dimensional vector space) it is well defined the left-invariant measure µq induced by µ on the nilpotent approximation.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 23 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Idea of the proof: An explicit formula for the Radon-Nikodym derivative Let S = (M, ∆, ·, ·) be a regular sub-Riemannian manifold.

Theorem

Let µ a volume on M and µq the induced volume on the nilpotent approximation at point q ∈ M. Then if A ⊂ M is open µ(A) = 1 2Q

• A
• µq(

Bq) dSQ, where Bq is the unit ball in the nilpotent approximation at the point q. q (M, µ) (TqM, ˆ µq) Bq ˆ Bq

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 24 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Idea of the proof: An explicit formula for the Radon-Nikodym derivative Let S = (M, ∆, ·, ·) be a regular sub-Riemannian manifold.

Theorem

Let µ a volume on M and µq the induced volume on the nilpotent approximation at point q ∈ M. Then if A ⊂ M is open µ(A) = 1 2Q

• A
• µq(

Bq) dSQ,

• i.e. fµS(q) = 1

2Q µq( Bq)

• where

Bq is the unit ball in the nilpotent approximation at the point q. q (M, µ) (TqM, ˆ µq) Bq ˆ Bq

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 24 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Why we study the spherical Hausdorff measure?

In sub-Riemannian geometry the isodiameter inequality is not valid. Vol(A) ≤ Vol(B1) diam A 2 n = Vol(B1) 2n (diam A)n i.e. balls of radius r do not maximize the volume among sets of diameter 2r ⇒ Answer to the question of Montgomery for the standard Hausdorff volume HQ is more difficult because in SRG balls are more natural than sets of a certain diameter and maximal volume.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 25 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Corollary

If Sq1 is isometric to Sq2 for any q1, q2 ∈ M, then fPS is constant (and fµS is smooth). In particular this happens if the sub-Riemannian structure is free. (i.e. the growth vector has maximal growth) ˆ S1 q1 q2 ˆ S2

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 26 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Corollary

If Sq1 is isometric to Sq2 for any q1, q2 ∈ M, then fPS is constant (and fµS is smooth). In particular this happens if the sub-Riemannian structure is free. (i.e. the growth vector has maximal growth) In the Riemannian case tangent spaces are all isometric fPS(q) = 1 2Q µq( Bq) − → 1 2n Vol(B1) In a sub-Riemannian manifold, the tangent structure may depend on the point.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 26 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Dimension n ≤ 5

Theorem. Let S = (M, ∆, ·, ·) be a regular sub-Riemannian manifold and Sq its nilpotent approximation near q. Up to a change of coordinates and rotations of the

• rthonormal frame we have the expression for the orthonormal frame of

Sq: Case n = 3. − G(S) = (2, 3). (Heisenberg.) X1 = ∂1, X2 = ∂2 + x1∂3. In this case [X1, X2] = ∂3.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 27 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Case n = 4 − G(S) = (2, 3, 4). (Engel.) X1 = ∂1, X2 = ∂2 + x1∂3 + x1x2∂4. In this case [X1, X2] = ∂3 + x2∂4, [X1, [X1, X2]] = ∂4. − G(S) = (3, 4). (Quasi-Heisenberg.) X1 = ∂1, X2 = ∂2 + x1∂4, X3 = ∂3. In this case [X1, X2] = ∂4.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 28 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Case n = 5 − G(S) = (2, 3, 5). (Cartan.) X1 = ∂1, X2 = ∂2 + x1∂3 + 1 2x2

1 ∂4 + x1x2∂5.

In this case [X1, X2] = ∂3 + x1∂4 + x2∂5, [X1, [X1, X2]] = ∂4, [X2, [X1, X2]] = ∂5. − G(S) = (2, 3, 4, 5). (Goursat rank 2.) X1 = ∂1, X2 = ∂2 + x1∂3 + 1 2x2

1 ∂4 + 1

6x3

1 ∂5.

In this case [X1, X2] = ∂3 + x1∂4 + 1 2x2

1 ∂5,

[X1, [X1, X2]] = ∂4 + x1∂5, [X1, [X1, [X1, X2]]] = ∂5.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 29 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

− G(S) = (3, 5). (Corank 2.) X1 = ∂1 − 1 2x2∂4, X2 = ∂2 + 1 2x1∂4 − 1 2x3∂5, X3 = ∂3 + 1 2x2∂4. In this case [X1, X2] = ∂4, [X2, X3] = ∂5. − G(S) = (3, 4, 5). (Goursat rank 3.) X1 = ∂1 − 1 2x2∂4 − 1 3x1x2∂5, X2 = ∂2 + 1 2x1∂4 + 1 3x2

1 ∂5,

X3 = ∂3. In this case [X1, X2] = ∂4 + x1∂5, [X1, [X1, X2]] = ∂5.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 30 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

G(S) = (4, 5). (Bi-Heisenberg.) X1 = ∂1 − α 2 x2∂5, X2 = ∂2 + α 2 x1∂5, X3 = ∂3 − β 2 x4∂5, αβ = 0, X4 = ∂4 + β 2 x3∂5. In this case [X1, X2] = α ∂5, [X3, X4] = β ∂5. Note: one can normalize one between α and β, but not both. This is the first case where the nilpotent approximation is not unique and could depend on the point.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 31 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

G(S) = (4, 5). (Bi-Heisenberg.) X1 = ∂1 − α 2 x2∂5, X2 = ∂2 + α 2 x1∂5, X3 = ∂3 − β 2 x4∂5, αβ = 0, X4 = ∂4 + β 2 x3∂5. In this case [X1, X2] = α ∂5, [X3, X4] = β ∂5. Popp’s measure is computed P = 1

• α2 + β2 dx1 ∧ . . . ∧ dx5

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 31 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

The (4, 5) case

The control system ˙ q = u1X1 + u2X2 + u3X3 + u4X4 can be rewritten as follows, where q = (x1, . . . , x4, y)

• ˙

xi = ui, ˙ y = xTL u i = 1, . . . , 4, L =     α −α β −β     We compute geodesics with the Pontryagin Maximum Principle. there are no abnormal extremals (contact structure). If X1, . . . , X4 is an orthonormal frame, then geodesics are projection on the q-space of Hamiltonian solutions of H(λ) = 1 2

k

• i=1

λ, Xi(q)2, q = π(λ) parameterization by arclength require H = 1/2.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 32 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

The (4, 5) case

• ˙

xi = ui, ˙ y = xTL u i = 1, . . . , 4, L =     α −α β −β     If we fix the initial point q, geodesics γ(λ0, t) starting from q are the parametrized by an initial covector λ0 ∈ S3 × R. Exp : S3 × R → M, (λ0, t) → γ(λ0, t) is called the exponential map.

Lemma

Geodesics with initial covector λ0 = (u0, r) are optimal until time tcut(λ0) = tconj(λ0) = 2π r max{α, β}

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 32 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

The (n − 1, n) contact case

• ˙

xi = ui, ˙ y = xTL u i = 1, . . . , n − 1, L is skew symmetric If we fix the initial point q, geodesics γ(λ0, t) starting from q are the parametrized by an initial covector λ0 ∈ Sn−2 × R. Exp : Sn−2 × R → M, (λ0, t) → γ(λ0, t) is called the exponential map.

Lemma

Geodesics with initial covector λ0 = (u0, r) are optimal until time tcut(λ0) = tconj(λ0) = 2π r max |eig(L)|

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 33 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

The volume of the Nilpotent Ball in the contact case

The volume of the nilpotent ball is computed as Vol(ˆ Bq) =

• ˆ

Bq

dP Since each geodesic is optimal up to tcut(λ0) we perform the change of variables (x1, . . . , x4, y) → (λ0, t). Then Vol(ˆ Bq) =

• ˆ

Bq

dP =

• S3×R

tcut(λ0)

• Jac(Exp(λ0, t)) dt dλ0

For the (n − 1, n) case one can compute explicitly tcut and Jac(exp(λ, t)). For a smooth one parametric family of nilpotent structures α(q(τ)) β(q(τ)) are smooth, but tcut(q(τ)) is not.

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 34 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

We get more regularity than expected since in contact case tcut(λ0) coincide with the first conjugate time tconj(λ0) i.e. the first time at which the Jacobian of the map (λ0, t) → exp(λ0, t)) is singular Is like to compute d dq t(q) f (q, s) ds = t(q) d dq f (q, s) ds + f (q, t(q)) t′(q) the result is the same for every (n − 1, n) manifold, since in quasi contact case there are no strictly abnormal minimizers

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 35 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Corank 2

In this case the control system can be written      ˙ xi = ui, ˙ y1 = xTL1 u ˙ y2 = xTL2 u i = 1, . . . , k, L1, L2 is skew symmetric Geodesics are parametrized by covectors λ0 = (u0, r1, r2) ∈ Sk−1 × R2

Lemma

Geodesics with initial covector λ0 = (u0, r1, r2) are optimal until time tcut(λ0) = 2π max |eig(r1L1 + r2L2)| In general tcut(λ0) = tconj(λ0) but they are equal if r1L1 + r2L2 has double eigenvalue

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 36 / 37

Sub-riemannian geometry Intrinsic volume and sub-Laplacian Comparison between volumes

Conclusions and open questions

What can be said in corank ≥ 2?

• which relation between tcut and tcon?

If fPS always C 1? What about fPH?

Davide Barilari (SISSA) Hausdorff measure in SR geometry October 25, 2010 37 / 37