Rectifiability of sets and measurestoro/Courses/15-16/IMPA/Lectur… · Tatiana Toro (University of...

Rectifiability of sets and measures

Tatiana Toro

University of Washington

IMPA

February 17, 2016

Tatiana Toro (University of Washington) Structure of n-uniform measure in Rm February 17, 2016 1 / 23

State of affairs I

Let µ be a Radon measure in Rm such that

0 < θn(µ, a) = limr→0

µ(B(a, r))

rn<∞ for µ− a.e a ∈ Rm

Then for µ-a.e. a ∈ Rm every ν ∈ Tan(µ, a) is n-uniform with 0 ∈ spt ν.

Thus to understand the infinitesimal structure µ we need to understandthe structure of n-uniform measures.


Uniformly distributed measures - Kirchheim-Preiss

A measure µ on Rm is uniformly distributed if there is a functionfµ : (0,∞)→ [0,∞] such that

I µ(B(x , r)) = fµ(r) for all x ∈ sptµ and all r > 0.I fµ(r) <∞ for some r .

For µ uniformly distributed in Rm, x ∈ Rm and 0 < s ≤ r <∞

µ(B(x , r)) ≤ 5m( rs

)mfµ(s). (∗)

Cover B(x , r) ⊂ ∪Ni=1B(zi , s/2) with |zi − zj | ≥ s/2.Since {B(zi , s/4)}Ni=1 are disjoint then N

(s4

)m ≤ (54 r)m

.

µ(B(x , r)) ≤N∑i=1

µ(B(zi , s/2)).

Consider 2 cases: µ(B(zi , s/2)) = 0 or µ(B(zi , s/2)) > 0. Ifµ(B(zi , s/2)) > 0 there is z ∈ sptµ ∩ B(zi , s/2), andB(zi , s/2) ⊂ B(z , s).



A measure µ on Rm is uniformly distributed if there is a functionfµ : (0,∞)→ [0,∞] such that

I µ(B(x , r)) = fµ(r) for all x ∈ sptµ and all r > 0.I fµ(r) <∞ for some r .

For µ uniformly distributed in Rm, x ∈ Rm and 0 < s ≤ r <∞

µ(B(x , r)) ≤ 5m( rs

)mfµ(s).

For µ uniformly distributed in Rm (∗)I fµ(r) <∞ for all r > 0

I

ê−s|x−z|

2

dµ(z) =

ˆ 1

0

µ(B(x ,

√− ln r

s)) dr converges for s > 0,

x ∈ Rm.

I

ê−s|x−z|

2

dµ(z) =

ê−s|y−z|

2

dµ(z) for all x , y ∈ sptµ and s > 0.



Let µ be uniformly distributed in Rm, x0 ∈ sptµ, for s > 0 and x ∈ Rm

F (x , s) =

ˆ (e−s|x−z|

2 − e−s|x0−z|2)dµ(z) is well defined and

independent of x0.

x ∈ sptµ iff F (x , s) = 0 for all s > 0.

If x /∈ sptµ there is sx > 0 so that s > sx , F (x , s) < 0 (∗).

sptµ is a real analytic variety.

There are n ∈ {0, 1, · · · ,m}, c > 0 and G ⊂ Rm an open set suchthat

I G ∩ sptµ is an n-dimensional analytic submanifold of Rm.

I µ(Rm\G ) = Hn(Rm\G ) = 0.

I Rm\G countable union of analytic submanifolds of Rm of dimensionless than n.

I µ(A) = cHn(A ∩ G ∩ sptµ) = cHn(A ∩ sptµ) for A ⊂ Rm Borel.


State of affairs II


0 < θn(µ, a) = limr→0

µ(B(a, r))


Then:

For µ-a.e. a ∈ Rm every ν ∈ Tan(µ, a) is n-uniform thus uniformlydistributed.

For x ∈ G ∩ spt ν (n-analytic submanifold) if λ ∈ Tan(ν, x) then thereexists c > 0 such that λ = cHn Vx where Vx = Txspt ν − x .

Since tangents to tangents are tangents λ ∈ Tan(µ, a).

λ ∈ Fn,m = F = {cHn V : c > 0, V ∈ G (m, n)}, i.e. λ is flat.

For µ-a.e. a ∈ Rm, Tan(µ, a) ∩ F 6= ∅.


Are all n-uniform measures flat? How large can thesingular set be?

Theorem If ν is n-uniform in Rn+1, Σ = spt ν then ν = cHn Σ andmodulo translation and rotation

Preiss For n = 1, 2 Σ = Rn × {0}

Kowalski-Preiss For n ≥ 3,

I Σ = Rn × {0}, orI Σ = {(x1, x2, x3, x4, · · · , xn+1) ∈ Rn+1 : x2

4 = x21 + x2

2 + x23}.

Recent work of D. Nimer addresses the question in higher codimensions.Tatiana Toro (University of Washington) Structure of n-uniform measure in Rm February 17, 2016 7 / 23

How do we prove rectifiability of µ?

Theorem (Preiss) Let µ be a Radon measure in Rm with

0 < θn∗(µ, a) ≤ θn,∗(µ, a) <∞ µ− a.e. a ∈ Rm.

Then the following are equivalent:

µ is n-rectifiable

µ− a.e a ∈ Rm there is Va an n-dimensional space in Rm so that

Tan(µ, a) = {cHn Va : 0 < c <∞}

µ− a.e. a ∈ Rm, Tan(µ, a) ⊂ F .

Does Tan(µ, a) ∩ F 6= ∅ =⇒ Tan(µ, a) ⊂ F for µ-a.e. a ∈ Rm?


Metric on the space of Radon measures in Rm

For r ∈ (0,∞), let

L(r) = {f : Rm → [0,∞), spt f ⊂ Br and Lip f ≤ 1} .

For Radon measures Φ and Ψ on Rm set

Fr (Φ,Ψ) = sup

{∣∣∣∣ˆ f dΦ−ˆ

f dΨ

∣∣∣∣ ; f ∈ L(r)

}.

Fr (Φ) = Fr (Φ, 0) =

ˆ(r − |z |)+ dΦ(z). (∗)

If µ, µi (i ∈ N) Radon measures in Rm, then

µi ⇀ µ iff limi→∞

Fr (µi , µ) = 0 ∀r > 0.


Cones of measures

Let R be the space of Radon measures in Rm. For Φ,Ψ ∈ R define

d(Φ,Ψ) =∞∑i=1

2−i min{1,Fi (Φ,Ψ)}.

(R, d) is a complete separable metric space. (∗)If µ, µi (i ∈ N) Radon measures in Rm, then

µi ⇀ µ iff limi→∞

d(µi , µ) = 0.

M⊂ R, 0 6∈ M is a cone if cΨ ∈M for all Ψ ∈M and c > 0.

A cone M is a d-cone if T0,r [Ψ] ∈M for all Ψ ∈M.

For s > 0 the s-distance between a d-cone M and Φ ∈ R is

ds(Φ,M) = inf

{Fs(

Φ

Fs(Φ),Ψ) : Ψ ∈M and Fs(Ψ) = 1

}if Fs(Φ) 6= 0. If Fs(Φ) = 0 set ds(Φ,M) = 1.


A picture to illustrate ds(Φ,M)

M

1

Φ

ΦFs(Φ)

{Ψ ∈ R : Fs(Ψ) = 1}

ds(Φ,M) = inf{Fs( Φ

Fs(Φ) ,Ψ) : Ψ ∈M and Fs(Ψ) = 1}


Additional properties

If M d-cone and Φ ∈ RI ds(Φ,M) ≤ 1 (∗)I ds(Φ,M) = d1(T0,s [Φ],M)

If µi ⇀ µ and Fs(µ) > 0 then ds(µ,M) = limi→∞

ds(µi ,M).

If µ is a non-zero Radon measure Tan(µ, a) is a d-cone.If ν ∈ Tan(µ, a), T0,r [ν] ∈ Tan(µ, a).

If µ is a non-zero Radon measure {ν ∈ Tan(µ, a) : F1(ν) = 1} isclosed under weak convergence (i.e. in the topology induced by d).

The basis of a d-cone M is the set

{Ψ ∈M : F1(Ψ) = 1}.


Compact basis

Let M be a d-cone in R with a closed basis then M has a compactbasis iff there exists κ > 1 such that

Ψ(B(0, 2r)) ≤ κΨ(B(0, r))

for all Ψ ∈M and r > 0,i.e. the doubling constant is uniform on M.

Let µ ∈ R, a ∈ sptµ if

ca = lim supr→0

µ(B(a, 2r))

µ(B(a, r))<∞,

then Tan(µ, a) has a compact basis. (∗)


Connectivity properties of Tan(µ, a) - Preiss

Let M and F be d-cones in R. Assume that F ⊂M, F relatively closedin M and M has a compact basis. Suppose that there exists ε0 > 0 suchthat if

(∗) dr (Φ,F) < ε0 ∀r ≥ r0 > 0 then Φ ∈ F .

Then for a Radon measure µ and a ∈ sptµ if

Tan(µ, a) ⊂M and Tan(µ, a) ∩ F 6= ∅ then Tan(µ, a) ⊂ F


State of affairs III


0 < θn(µ, a) = limr→0

µ(B(a, r))


Then for µ-a.e. a ∈ Rm:

F ⊂M, where F is the set of n-flat measures in Rm and M is theset of n-uniform measures in Rm with 0 in their support.

F is closed in M, which is a d-cone with compact basis.

Tan(µ, a) ⊂M.

Tan(µ, a) ∩ F 6= ∅.

If (∗) holds then Tan(µ, a) ⊂ F =⇒ µ n-rectifiable


What does (∗) really mean?

Let Φ be n-uniform. Recall

dr (Φ,F = inf

{Fr (

Φ

Fr (Φ),Ψ) : Ψ ∈ F and Fr (Ψ) = 1

}.

Fr (Φ) =

ˆ(r − |y |)+ dΦ =

ˆΦ({y : (r − |y |)+ > t}) dt

=

ˆ r

0Φ({y : |y | < r − t})) dt =

ˆ r

0Φ(Br−t) dt

=

ˆ r

0c(r − t)n dt = c

rn+1

n + 1

dr (Φ,F) = dr (cΦ,F) for c = ωn then Fr (Φ) = ωnrn+1

n+1 .

If Ψ = cHn V , Fr (Ψ) = c ωnrn+1

n+1 = 1 implies c =(ωnrn+1

n+1

)−1.


dr (Φ,F) = infV∈G(m,n)

n + 1

ωnrn+1sup

f ∈L(r)

∣∣∣∣ˆ f dΦ−ˆ

f dHn V

∣∣∣∣Note that for

f (x) = dist 2(x ,V )(r − |x |)+

r2∈ L(r).

n + 1

ωnrn+1

∣∣∣∣ˆ f dΦ−ˆ

f dHn V

∣∣∣∣ ≥ n + 1

ωnrn+1

ˆf dΦ

≥ n + 1

ωnrn+1

ˆBr/2

dist 2(x ,V )r

2r2dΦ

≥ n + 1

2ωnrn+2

ˆBr/2

dist 2(x ,V ) dΦ

If (∗) dr (Φ,F) < ε0 for r ≥ r0 > 0, then

infV∈G(m,n)

n + 1

ωnrn+2

ˆBr/2

dist 2(x ,V ) dΦ < 2ε0.


The conditiondr (Φ,F) < ε0 ∀r ≥ r0

implies that there exists ε1 > 0 such that

infV∈G(m,n)

1

ωnrn+2

ˆBr

dist 2(x ,V ) dΦ < ε1 for r ≥ r0.

Thus for r ≥ r0 there exists Vr ∈ G (m, n) such that

(♠)1

ωnrn+2

ˆBr

dist 2(x ,Vr ) dΦ ≤ ε1

To prove (∗) we need to show that if Φ n-uniform is “close to flat atinfinity” in L2 as in (♠) then Φ is flat.


(♠) yields geometric information

There is a large good subset Gr of spt Φ in Br which is close toVr

By Chebyshev’s inequality if (♠) holds then

Gr = {x ∈ spt Φ ∩ Br : dist (x ,Vr ) ≤ ε1/41 r}

satisfiesΦ(Br\Gr ) ≤ ε1/2

1 Φ(Br ).

r

Vr Gr


The small subset of spt Φ inBr\Gr is also close toVr

r

Vr

yspt Φ

Let r1 = r(1− 2ε1/2n1 ). If y ∈ (Br1\Gr ) ∩ spt Φ,

ρ = min{dist (y ,Gr ), dist (y , ∂Br )} > 0 then Bρ(y) ⊂ Br\Gr

ωnρn = Φ(Bρ(y)) ≤ Φ(Br\Gr ) ≤ ε1/2

1 Φ(Br ) = ωnε1/21 rn.

Thus ρ = dist (y ,Gr ) ≤ ε1/2n1 r . For y ∈ (B

r(1−2ε1/2n1 )\Gr ) ∩ spt Φ,

dist (y ,Vr ) ≤ dist (y ,Gr ) + dist (Gr ,Vr ) ≤ ε1/2n1 r .


Summary

If dr (Φ,F) < ε0 for r ≥ r0 > 0 there is Vr such that

(♠)1

ωnrn+2

ˆBr

dist 2(x ,Vr ) dΦ ≤ ε1

with ε1 = C (n)ε0.

In this case for y ∈ Br(1−2ε

1/2n1 )∩ spt Φ, dist (y ,Vr ) ≤ ε1/2n

1 r .

spt Φ close toVr in L2 sense in Br then spt Φ close Vr in L∞ sense inBr(1−2ε

1/2n1 )


(♠) yields additional geometric information for Φn-uniform

Not only is spt Φ close to Vr but Vr is also close to spt Φ.

V1τ

spt Φ

Can there be holes of size τ?NO if ε1 is small enough.


(♠) yields additional geometric information for Φn-uniform

Lemma: Given τ > 0 there exists ε1 = ε1(τ, n,m) > 0 such that if Φ isn-uniform in Rm with 0 ∈ spt Φ, Φ(B1) = 1 and for some V ∈ G (m, n)

ˆB1

dist 2(x ,V ) dΦ < ε1,

then for all z ∈ V ∩ B1

B(z , τ) ∩ spt Φ 6= ∅.


Rectifiability of sets and measurestoro/Courses/15-16/IMPA/Lectur… · Tatiana Toro (University of...

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