On the Hausdorﬀ volume in Sub-Riemannian geometry · 2010-10-26 · Sub-riemannian geometry...

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


Joint work with

General results and Corank 1 case:

Andrei Agrachev

Ugo Boscain

Corank 2 case:

Jean-Paul Gauthier



Outline

1 Sub-riemannian geometry

2 Intrinsic volume and sub-Laplacian

3 Comparison between volumes



Outline






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 thatassociates 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 .



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

0

√〈γ, γ〉 dt.



Carnot-Caratheodory distance

The distance induced by the sub-Riemannian structure on M is

d(q, q′) = infℓ(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 thetopology of M (Chow’s Theorem)

The function d(·, ·) is also called Carnot-Caratheodory distance.



Orthonormal frame

Locally, the pair (∆, 〈·, ·〉) can be given by assigning a set of k smooth vectorfields, called a local orthonormal frame, spanning ∆ and that are orthonormal

∆q = spanX1(q), . . . , Xk(q), 〈Xi (q), Xj (q)〉 = δij .

The problem of finding curves that minimize the length between two given pointsq0, q1, is rewritten as the optimal control problem

q =

k∑

i=1

ui Xi (q)

∫ T

0

√√√√k∑

i=1

u2i → min

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



Growth vector

Define ∆1 := ∆, ∆2 := ∆ + [∆, ∆], ∆i+1 := ∆i + [∆i , ∆].

If the dimension of ∆iq, 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 ∆q

k

, dim ∆2, . . . , dim∆m

qn

)



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 topologicaldimension of M : Q > n



Outline






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 thesub-Riemannian setting, in order to study the interplay between the analysis of thesub-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.



Riemannian case

On a Riemannian manifold the Laplace-Beltrami operator L is defined as follows

Lφ = div(gradφ)



Riemannian case


Lφ = div(gradφ)

the gradient of a function φ is the vector field canonically associated to dφby 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



Riemannian case


Lφ = div(gradφ)

the divergence of a vector field X says how much the flow of X change thevolume

div X>0div 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!



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 thevolume 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 2i φ + (dXj [Xj , Xi ])Xiφ

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

L = X 21 + X 2

2 − a2X1 + a1X2



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 infδ→0

∑

i

diam(Ui)n, Ω ⊂

⋃

i

Ui , diam(Ui ) < δ



Hausdorff measure




Recall that

Sn(Ω) = lim infδ→0

∑

i

diam(Ui )n, Ω ⊂

⋃

i

Ui , Ui balls, diam(Ui ) < δ



Hausdorff measure




Recall that

Sn(Ω) = lim infδ→0

∑

diam(Ui )n, Ω ⊂

⋃

i

Ui , Ui balls, diam(Ui ) < δ

These measures are proportional:

µ = αHn = αSn, α =Vol(B1)

2n

where Vol(B1) is the volume of a unit Euclidean ball in Rn.

No problem which one we choose!



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.



Sub-Riemannian case

Lφ = div(gradφ)

the horizontal gradient of a function φ is the horizontal vector fieldassociated 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



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?



Popp’s measure

Since the scalar product is defined only on ∆, a local orthonormal frameX1, . . . , Xk for the structure does not permit us to define directly an intrinsicn-form as in the Riemannian case

µ = dX1 ∧ . . . ∧ dXk ∧ ? ∧ . . .∧ ?, k < n.

Hypothesis: the structure is regular (i.e. dim ∆iq = const. in q)

Case n = 3. If X1, X2 is a local orthonormal frame for the structure it is easyto see that the quantity [X1, X2](mod ∆) is well defined and tensorial. Thisimplies 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 theorthonormal frame)



Popp’s measure

In the general case the regularity assumption makes possible to complete theorthonormal frame with commutators of the frame, whose structure dependsonly 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 iton ∆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)

on left-invariant sub-Riemannian structures on Lie groups the invariantvolume form is proportional to the left-Haar measure



Outline






Question (Montgomery)

Is Popp’s measure equal to a constant multiple (perhaps depending on the growthvector) of the Hausdorff measure?

Montgomery’s remarks

Mitchell Theorem: If µ is a smooth volume on a regular sub-Riemannianstructure (e.g. P) then

dµ = fµHdHQ

q

Radon-Nikodym derivative

where fµH is measurable, locally bounded and locally bounded away fromzero (“commensurable")

Moreover well known estimates show that SQ is commensurable with HQ

SQ comm.←→ HQ comm.

←→ µcomm.←→ P



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 areproportional 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 thecases (n − 1, n).



Results

Theorem (Continuity)

For a regular sub-Riemannian manifold fPS is continuous

Theorem (Small dimensions)

Let dim M = n ≤ 5. Then

if G(S) 6= (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.



Results

Theorem (Continuity)

For a regular sub-Riemannian manifold fµS is continuous.

Theorem (Small dimensions)

Let dim M = n ≤ 5. Then

if G(S) 6= (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.



Idea of the proof: Nilpotent approximation

Let S = (M , ∆, 〈·, ·〉) be a regular sub-Riemannian manifold and µ be a smoothmeasure 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 aleft-invariant sub-Riemannian structure on a n-dimensional vector space)

it is well defined the left-invariant measure µq induced by µ on the nilpotentapproximation.



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 approximationat 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 , µq)

Bq

Bq



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 approximationat point q ∈ M. Then if A ⊂ M is open

µ(A) =1

2Q

∫

A

µq(Bq) dSQ ,

(i.e. fµS(q) =

1

2Qµq(Bq)

)

where Bq is the unit ball in the nilpotent approximation at the point q.

q

(M , µ)

(TqM , µq)

Bq

Bq



Why we study the spherical Hausdorff measure?

In sub-Riemannian geometry the isodiameter inequality is not valid.

Vol(A) ≤ Vol(B1)

(diamA

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 volumeHQ is more difficult because in SRG balls are more natural than sets of acertain diameter and maximal volume.



Corollary

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

S1

q1q2

S2



Corollary

If Sq1 is isometric to Sq2 for any q1, q2 ∈ M, then fPS is constant (and fµS issmooth). 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

2Qµq(Bq) −→

1

2nVol(B1)

In a sub-Riemannian manifold, the tangent structure may depend on the point.



Dimension n ≤ 5

Theorem.

Let S = (M , ∆, 〈·, ·〉) be a regular sub-Riemannian manifold and Sq its nilpotentapproximation near q. Up to a change of coordinates and rotations of theorthonormal 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.



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.



Case n = 5

− G(S) = (2, 3, 5). (Cartan.)

X1 = ∂1,

X2 = ∂2 + x1∂3 +1

2x

2

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

2x

2

1 ∂4 +1

6x

3

1 ∂5.

In this case

[X1, X2] = ∂3 + x1∂4 +1

2x

2

1 ∂5,

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



− 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

3x

2

1 ∂5,

X3 = ∂3.

In this case

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



G(S) = (4, 5). (Bi-Heisenberg.)

X1 = ∂1 −α

2x2∂5,

X2 = ∂2 +α

2x1∂5,

X3 = ∂3 −β

2x4∂5, αβ 6= 0,

X4 = ∂4 +β

2x3∂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 casewhere the nilpotent approximation is not unique and could depend on the point.



G(S) = (4, 5). (Bi-Heisenberg.)

X1 = ∂1 −α

2x2∂5,

X2 = ∂2 +α

2x1∂5,

X3 = ∂3 −β

2x4∂5, αβ 6= 0,

X4 = ∂4 +β

2x3∂5.

In this case [X1, X2] = α ∂5, [X3, X4] = β ∂5.

Popp’s measure is computed

P =1√

α2 + β2dx1 ∧ . . . ∧ dx5



The (4, 5) case

The control systemq = u1X1 + u2X2 + u3X3 + u4X4

can be rewritten as follows, where q = (x1, . . . , x4, y)

xi = ui ,

y = xTL ui = 1, . . . , 4, L =

0 α 0 0−α 0 0 00 0 0 β0 0 −β 0

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 theq-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.



The (4, 5) case

xi = ui ,

y = xTL ui = 1, . . . , 4, L =

0 α 0 0−α 0 0 00 0 0 β0 0 −β 0

If we fix the initial point q, geodesics γ(λ0, t) starting from q are theparametrized by an initial covector λ0 ∈ S3 × R.

Exp : S3 × R→ M , (λ0, t) 7→ 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 maxα, β



The (n− 1, n) contact case

xi = ui ,

y = xTL ui = 1, . . . , n− 1, L is skew symmetric

If we fix the initial point q, geodesics γ(λ0, t) starting from q are theparametrized by an initial covector λ0 ∈ Sn−2 × R.

Exp : Sn−2 × R→ M , (λ0, t) 7→ 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)|



The volume of the Nilpotent Ball in the contact case

The volume of the nilpotent ball is computed as

Vol(Bq) =

∫

Bq

dP

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

Vol(Bq) =

∫

Bq

dP =

∫

S3×R

tcut (λ0)∫

0

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)) aresmooth, but tcut(q(τ)) is not.



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) 7→ exp(λ0, t)) issingular

Is like to compute

d

dq

∫ t(q)

0

f (q, s) ds =

∫ t(q)

0

d

dqf (q, s) ds + f (q, t(q)) t ′(q)

the result is the same for every (n − 1, n) manifold, since in quasi contactcase there are no strictly abnormal minimizers



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) 6= tconj(λ0) but they are equal if r1L1 + r2L2 has doubleeigenvalue



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?


On the Hausdorﬀ volume in Sub-Riemannian geometry · 2010-10-26 · Sub-riemannian geometry...

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Transcript of On the Hausdorﬀ volume in Sub-Riemannian geometry · 2010-10-26 · Sub-riemannian geometry...