<<

. 11
( 17 .)



>>


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]

<<

. 11
( 17 .)



>>