Rectifiable currents

In this section we define the class of rectifiable currents. We first give an intrinsic definition and then, as in the classical theory, we c o m p a r e it with a p a r a m e t r i c one adopted, with minor variants, in [20].

We say t h a t an 7-/k-measurable set

S c E

is

countably 7-lk-rectifiable

if there exist sets As C R k and Lipschitz functions

fs:Ai--+ E

such t h a t

It is not hard to prove t h a t any countably 7-/k-rectifiable set is separable; by the com- pleteness assumption on E the sets As can be required to be closed, or compact.

LEMMA 4.1.

Let S c E be countably ~k-rectifiable. Then there exist finitely or eountably many compact sets Ks C R k and bi-Lipschitz maps f~: Ki-~ S such that fi (Ks) are pair-wise disjoint and 7-t k ( s \ u ~ f i ( K~ ) ) = 0.

Proof.

By L e m m a 4 of [38] we can find c o m p a c t sets Ki C R k and bi-Lipschitz m a p s

fs: Ks--+E

such t h a t

SC U~

f i ( K s ) , up to 7-/k-negligible sets. Then, setting B 0 = K 0 and

B ~ : = K i \ I - I ( s N U f j ( K j ) ) E B ( R k)

f o r a l l i ) l , j < i

we represent 7-/k-almost all of S as the disjoint union of fs (Bs). For any i E N , representing /:k-almost all of Bs by a disjoint union of c o m p a c t sets the proof is achieved. []

Definition

4.2 (rectifiable currents). Let k~>l be integer and let T E M k ( E ) ; we say t h a t T is rectifiable if

(a) IITII is concentrated on a countably ~k-rectifiable set;

(b) IITll vanishes on 7-/k-negligible Borel sets.

We say t h a t a rectifiable current T is integer-rectifiable if for any ~ E Lip(E, R k) and any open set

A c E

we have

~#(Tt_A)=~O~

for some

OELI(R k,

Z).

T h e collections of rectifiable and integer-rectifiable currents will be respectively de- noted by T~k(E) and

Ik(E).

T h e space of integral currents I k ( E ) is defined by

I k ( E )

:= I k ( E ) N N k ( E ) .

We have proved in the previous section t h a t condition (b) holds if either k = 1, 2 or T is normal. We will also prove in T h e o r e m 8.8 (i) t h a t condition (a) can be weakened by requiring t h a t T is concentrated on a Borel set, a-finite with respect to 7 / n - l , and

CURRENTS IN METRIC SPACES 23 that, for normal currents T, the integer rectifiability of all projections ~ #

(TLA)

implies the integer rectifiability of T.

In the case k = 0 the definition above can be easily extended by requiring the existence of countably many points xh c E and 0hC R (or ~h ~ Z, in the integer case) such t h a t

T ( f ) = E Ohf(Xh)

for all

f e B ~ ( E ) . h

It follows directly from the definition t h a t

7~k(E)

and :/:k(E) are Banach subspaces of Mk (E).

We will also use the following rectifiability criteria, based on Lipschitz projections, for 0-dimensional currents; the result will be extended to k-dimensional currents in The- orem 8.8.

THEOREM 4.3.

Let

S E M o ( E ) .

Then

(i)

Seh[o(E) if and only if S(XA)cZ for any open set ACE;

(ii)

SEZo(E) if and only if

~#SET_~(R)

for any

~ELip(E);

(iii)

if E = R y for some N, then SCno(E) if and only if

~#SET~0(R)

for any

~ e L i p ( E ) .

Proof.

(i) If

S(XA )

is integer for any open set A, we set E := {x e E : IISll (Be (x))/> 1 for all 0 > 0}

and notice t h a t E is finite and that, by a continuity argument, S L E E I o ( E ) . If x ~ E we can find a ball B centered at x such t h a t ]]SI](B)<I; as

S(XA)

is an integer for any open set

A c B ,

it follows that

S(XA)=O,

and hence ]]SI](B)=0. A covering argument proves that

]]SII(K)=O

for any compact set

K C E \ E ,

and Lemma 2.9 implies that S is supported on E.

(ii) Let

A c E

be an open set and let ~ be the distance function from the complement of A. Since

S ( x A ) = ~ # s ( x ( o , ~ ) ) E z the statement follows from (i).

(iii) The statement follows by L e m m a 4.4 below. []

LEMMA 4.4.

Let # be a signed measure in R y. Set

Q = Q N x ( Q M ( 0 , oc))N

and consider the countable family of Lipsehitz maps

fx,~(Y) =max)~iixi--yi[, i~N y c R N,

24 L. A M B R O S I O A N D B. K I R C H H E I M

where

(x, A)

runs through Q.

Then ,eno(a N) if and only if

f~,x##CT40(R)

for all (x, A)c Q.

Proof.

We can assume with no loss of generality that # has no atom and denote by N" ]]~ the lo~-norm in R g. Assume # to be a counterexample to our conclusion and let K~< N be the smallest dimension of a coordinate-parallel subspace of R N charged by I#], i.e. K is the smallest integer such that there exist

x~ R N, I c

{ 1, ..., N} with cardinality

N - K

such that

I#l(Pi(x~

where

o for any i C I }.

P 4 z ~ := {x e R ~ : x~ = z ,

Since # has no atom, K > 0. Replacing # by - # if necessary, we find r > 0 and xl E QN such t h a t

# ( M ) > 3~ where M :----

P1(x~

] I y - z 1I]o~ < 1).

Next we choose k sufficiently large such t h a t

I # ] ( M ) < ~ w i t h ~ / I : = { y e R g : d i s t ~ ( y , M ) e ( 0 , 2 / k ) } .

Modifying x 1 only in the i t h coordinates for i E I we can, without changing M, in addition assume that

I ( x ~

for all

i e t .

We define A c ( Q N ( 0 , cc)) N by A~=k if

iEI,

and Ai = 1 otherwise. Observe t h a t

M C f~-llA ([0, 1)) C MUl~r,

Let T be the countable set on which /5=f~1,~## is concentrated. Due to our minimal choice of K we have

[#[(MNf~,~(s))=O

for any s E R ; hence our choice of 7~f gives

[#I(f~IIA(TM[0, 1)))~< [#[(f~]l~([0, 1 ) ) \ M ) < s, and we obtain t h a t [/5[([0, 1))<r On the other hand,

/5((0, 1]) -- #(/~-,~x ([0, 1))) >~ # ( M ) -

I I(M)

^>/2~.

This contradiction finishes our proof. []

It is also possible to show t h a t this kind of statement fails in any infinite-dimensional situation, for instance when E is L 2. In fact, it could be proved that given any sequence of Lipschitz functions on a Hilbert space, we can always find a continuous probability measure on it whose images under all these maps are purely atomic.

Now we show t h a t rectifiable currents have a parametric representation, as sums of images of rectifiable Euclidean currents (see also [20]).

C U R R E N T S I N M E T R I C S P A C E S 25 THEOREM 4.5 (parametric representation).

Let TEMk(E). Then, TCT~k(E) (resp.

TCZk(E) ) if and only if there exist a sequence of compact sets Ki, functions 0i

E L l ( R k )

(resp. O{ELI(R k,

Z))

with

spt

OiCKi, and bi-Lipschitz maps fi: Ki-+E, such that

T = E f i # [ O i ] and M(fi#[Oi~)=M(T).

i = 0 i = 0

Moreover, if E is a Banach space, T can be approximated in mass by a sequence of normal currents.

Proof.

One implication is trivial, since

fi#IOi~

is rectifiable, being concentrated on f~(Ki) (the absolute-continuity property (b) is a consequence of the fact t h a t f - l :

fi (Ki)--+Ki

is a Lipschitz function), and T~k (E) is a Banach space. For the integer case, we notice that

Ti=fi#[Oi~

is integer-rectifiable if 0i takes integer values, because for any

~ E L i p ( E , R k) and any open set

A c E ,

setting h = ~ o f i :

K~-+R k

and

A ' = f ( I ( A ) ,

we have

~#(TiLA)=h#([Oi~LA')= I E Oi(x)sign(det Vh(x)) 1

x E h - l ( y ) A A '

as a simple consequence of the Euclidean area formula.

Conversely, let us assume that T is rectifiable, let S be a countably 7-/k-rectifiable set on which IITII is concentrated, and let Ki and fi be given by Lemma 4.1. Let

g~=f~-lELip(Si,Ki),

with

Si=f~(K~),

and set

Ri=g~#(Tt-Si);

since IIRill vanishes on

"r/k-negligible sets, by Theorem 3.7 there exists an integrable function 0i vanishing outside of Ki such that R~=I0i~, with integer values if

TCIk(E).

Since

f~ogi(x)=x

on

Si,

the locality property (3.6) of currents implies

TLSi = (fiogi)#(TLSi) = f{#ni = fi#[Oi~.

Adding with respect to i, the desired representation of T follows. Finally, if E is a Banach space we can assume (see [37]) that fi are Lipschitz functions defined on the whole of R k and, by a rescaling argument, t h a t L i p ( f i)~< 1; for c > 0 given, we can choose 0~ E B V ( R k) such that fRk 10i--0~1

dx<c2-i

to obtain that the normal current T = E ~

fi#[O~]

satisfies []

The following theorem provides a canonical (and minimal) set

ST

on which a recti- fiable current T is concentrated.

THEOREM 4.6.

Let TET~k(E) and set

ST := {X e E : O*k(llTII, X) > 0). (4.2)

26 L. A M B R O S I O A N D B. K I R C H H E I M

Then ST is countably 7-lk-rectifiable, and

[[T[I

is concentrated on ST; moreover, any Borel set S on which IlTII is concentrated contains ST, up to 7-lk-negligible sets.

Proof.

Let S be a countably 7-/k-rectifiable set on which ReTie is concentrated; by the Radon Nikodym theorem we can find a nonnegative function

OELI(7-~ki_s)

such t h a t L[T[[

=07tk[_S.

By Theorem 5.4 of [7] we obtain that Ok(liT[I,

x)=O(x)

for 7-/k-a.e.

x e S ,

while (1.3) gives Ok(llTII, x ) = 0 for 7-/k-a.e.

x e E \ S .

This proves that

ST=SN{O>O},

up to 7-/k-negligible sets, and since [[T[[ is concentrated on SN {0>0} the proof is achieved. []

Definition

4.7 (size of a rectifiable current). The

size

of

TET~k(E)

is defined by

S(T) := ~(sr)

where

ST

is the set described in Theorem 4.6.

Dans le document by Currents in metric spaces (Page 22-26)