Proposition 3.17 Let φ be a Hamiltonian di¬eomorphism with compact support in

A and F its generating function, that is dF = φ— » ’ », where d» = ω. Consider

a Hamiltonian ¬‚ow t ’ Ht , with compact support in A, such that the time one map

equals φ. Then the following inequality holds:

1

| F (x) ’ c | dx : c ∈ R ¤ vol(A)

C inf Ht dt

∞,A

A 0

with C > 0 an universal constant equal to

vol(B CC (0, 1))

C=

vol(B E (0, 1)

3.7 Invariants of volume preserving maps

Theorem 3.8 gives the strucure of a volume preserving locally bi-Lipschitz homeo-

morphism. We have denoted by Hom(H(n), vol, Lip) the class of these maps. Any

˜

φ ∈ Hom(H(n), vol, Lip) can be written as

˜¯

φ(x, x) = (φ(x), x + F (x))

¯

where φ ∈ Sympl(R2n , Lip) is a locally bi-Lipschitz symplectomorphism of R2n . For a

set B ‚ R2n we denote by vol(B) it™s Lebesgue measure. For a set C ‚ R l(C) will

be it™s length (one dimensional Lebesgue measure). An immediate consequence of the

structure theorem 3.8 is given further.

˜

Corollary 3.18 Let A ‚ H(n) be a H2n+2 measurable set such that H2n (A) <

˜

˜

+∞, where A denotes the orthogonal projection of A on R2n . Then for any φ ∈

Hom(H(n), vol, Lip) we have:

˜˜

˜

2n+2 (A) H2n+2 (φ(A))

˜ =H

w(A) =

vol(A) vol(φ(A))

51

3 THE HEISENBERG GROUP

˜ ˜

We call w(A) the width of A.

˜

More general, for any i ≥ 1 de¬ne the i-height of A to be

1

i 1/i

˜ ˜

l( x ∈ R : (x, x) ∈ A )

hi (A) = ¯ ¯ dx

vol(A)i A

˜˜

˜

Then hi (A) = hi (φ(A)).

Proof. Any symplectomorphism is volume preserving. The thesis follows from this

and the structure theorem 3.8.

˜

Suppose A is open, bounded and classically smooth. In the case of the Heisenberg

group H(1) Pansu [27] proved the following isoperimetric inequality:

4

˜ ˜

4 3 3

H (A) ¤ C H (‚ A

The isoperimetric inequality is a very important subject in analysis in metric spaces.

It has been intensively studied in it™s disguised form (Sobolev inequalities) in many

papers. To have a grasp of what is hidden here the reader can consult Varopoulos

& al. [32], Hajlasz, Koskela [14] and Garofalo, Nhieu [10]. In order to give a decent

exposition of this subject we would need a good notion of perimeter. In this ¬rst part

we don™t enter much into the measure theoretic aspects though.

Let us work with objects which have classical regularity, as in Gromov [12] section

2.3. There we ¬nd a more general isoperimetric inequality, true in particular in any

˜

Carnot group, provided that A is open, bounded, with (classically) smooth boundary.

If we denote by Q the homogeneous dimension of the Carnot group N (which equals

its Hausdor¬ dimension) then Gromov™s isoperimetric inequality is:

Q

˜ ˜ Q’1

Q Q’1

H (A) ¤ C H (‚ A)

Recall that for Heisenberg group H(n) the homogeneous dimension is Q = 2n + 2.

We should use further the group Dif f 2 (H(n), vol) instead of Hom(H(n), vol, Lip),

because we work with classical regularity . We kindly ask the reader to believe that a

˜

good de¬nition of the perimeter of the set A allows us to use the group Hom(H(n), vol, Lip).

˜

We shall denote the perimeter function by P er. In ”smooth” situations P er(‚ A) =

˜

HQ’1 (‚ A).

˜

We can de¬ne then the isoperimeter of A to be:

2n+2

˜˜

P er ‚ φ(A)

˜

isop(A)2n+1 = inf

(H2n (A))2n+1

˜

with respect to all φ ∈ Hom(H(n), vol, Lip). This is another invariant of the group of

volume preserving locally bi-Lipschitz maps.

Yet another invariant is the isodiameter. This is de¬ned as

˜ ˜ 2n+2

˜ = inf (diam φ(A))

isod(A)

H2n (A)

52

4 SUB-RIEMANNIAN LIE GROUPS

˜ ˜

with respect to all φ ∈ Hom(H(n), vol, Lip). Because the diameter of A is greater than

˜

the square root of the ∞ height of A, it follows that

˜ ˜

h∞ (A)n+1 w(A)

˜

isod(A) ≥

˜

H2n+2 (A)

All these invariants scale with dilatations like this (denote generically by inv any

of the mentioned invariants):

˜ ˜

inv δµ A = µ2 A

The groupoid Hom(H(n), vol, Lip) with objects open sets H(n) and arrows volume

preserving locally bi-Lipschitz maps between then, can be considered instead of the

˜

group Hom(H(n), vol, Lip). Then the heights are no longer invariants. Indeed, set A

to be the disjoint union of two isometric sets. Then we can keep one of these sets ¬xed

and put on the top of it the other one, using only translations. The width will change.

We can de¬ne other invariants though, using the heights and the number of connected

components, for example.

By using lifts of symplectic ¬‚ows one can de¬ne symplectic invariants. Conversely,

with any symplectic capacity comes an invariant of Hom(H(n), vol, Lip). Indeed, let c

˜˜

be a capacity. Then c(A) = c(A) is an invariant.

Is there any general de¬nition of a Hom(H(n), vol, Lip) invariant which has as

particular realization heights and symplectic capacities?

How about the invariants of the group Hom(N, vol, Lip), where N is a general

Carnot group? We leave this for further study.

4 Sub-Riemannian Lie groups

In this section we shall look closer to the last example of section 1.5. Let G be a real

connected Lie group with Lie algebra g and D ‚ g a vector space which generates the

algebra. This means that the sequence

V1 = D , V i+1 = V i + [D, V i ]

provides a ¬ltration of g:

V 1 ‚ V 2 ‚ ... V m = g (4.0.1)

4.1 Nilpotentisation

The ¬ltration has the straightforward property: if x ∈ V i and y ∈ V j then [x, y] ∈ V i+j

(where V k = g for all k ≥ m and V 0 = {0}). This allows to construct the Lie algebra

n(g, D) = •m Vi , Vi = V i /V i’1

i=1

ˆ

with Lie bracket [ˆ, y ]n = [x, y], where x = x + V i’1 if x ∈ V i \ V i’1 .

xˆ ˆ

Proposition 4.1 n(g, D) with distinguished space V1 = D is the Lie algebra of a

Carnot group of step m. It is called the nilpotentisation of the ¬ltration (4.0.1).

53

4 SUB-RIEMANNIAN LIE GROUPS

Remark that n(g, D) and g have the same dimension, therefore they are isomorphic as

vector spaces.

We set the distribution induced by D to be

Dx = T Lx D

where x ∈ G is arbitrary and Lx : G ’ G, Lx y = xy is the left translation by x. We

shall use the same notation D for the induced distribution.

Let {X1 , ..., Xp } be a basis of the vector space D. We shall build a basis of g which

will give a vector spaces isomorphism with n(g, D).

A word with letters A = {X1 , ..., Xp } is a string Xh(1) ...Xh(s) where h : {1, ..., s} ’

{1, ..., p}. The set of words forms the dictionary Dict(A), ordered lexicographically. We

set the function Bracket : Dict(A) ’ g to be

Bracket(Xh(1) ...Xh(s) ) = [Xh(1) , [Xh(2) , [..., Xh(s) ]...]

For any x ∈ Bracket(Dict(A)) let x ∈ Dict(A) be the least word such that Bracket(ˆ) =

ˆ x

x. The collection of all these words is denoted by g .

ˆ

The length l(x) = length(ˆ) is well de¬ned. The dictionary Dict(A) admit a

x

¬ltration made by the length of words function. In the same way the function l gives

the ¬ltration

V 1 © Bracket(Dict(A)) ‚ V 2 © Bracket(Dict(A)) ‚ ... Bracket(Dict(A))

ˆ ˆ

Choose now, in the lexicographic order in g, a set B such that B = Bracket(B) is a

ˆ

basis for g. Any element X in this basis can be written as

X = [Xh(1) , [Xh(2) , [..., Xh(s) ]...]

such that l(X) = s or equivalently X ∈ V s \ V s’1 .

It is obvious that the map

˜

X ∈ B ’ X = X + V l(X)’1 ∈ n(g, D)

˜ ˜

is a bijection and that B = Xj : j = 1, ..., dim g is a basis for n(g, D). We can

˜

identify then g with n(g, D) by the identi¬cation Xj = Xj .

Equivalently we can de¬ne the nilpotent Lie bracket [·, ·]n directly on g, with the

use of the dilatations on n(g, D).

Instead of the ¬ltration (4.0.1) let us start with a direct sum decomposition of g

g = • m Wi , V i = • i Wj (4.1.2)

i=1 j=1

such that [V i , V j ] ‚ V i+j . The chain V i form a ¬ltration like (4.0.1).

We set

m

µi xi

δµ (x) =

i=1

m

for any µ > 0 and x ∈ g, which decomposes according to (4.1.2) as x = i=1 xi .

54

4 SUB-RIEMANNIAN LIE GROUPS

Proposition 4.2 The limit

’1

[x, y]n = lim δµ [δµ x, δµ y] (4.1.3)

µ’0

exists for any x, y ∈ g and (g, [·, ·]n ) is the Lie algebra of a Carnot group with dilatations

δµ .

i l

Proof. Let x = j=1 xj and y = k=1 yk . Then

i+l s

s

[δµ x, δ ’ µy] = µ [xj , yk ]p

s=2 j+k=s p=1

’1

We apply δµ to this equality and we obtain:

i+l s

’1

µs’p [xj , yk ]p

’ µy] =

δµ [δµ x, δ

s=2 j+k=s p=1

When µ tends to 0 the expression converges to the limit

i+l

[x, y]n = [xj , yk ]s (4.1.4)

s=2 j+k=s

For any µ > 0 the expression

[x, y]µ = δµ [δµ x, δ ’ µy]