L3: 1D Convex Integration Vincent Borrelli

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L3: 1D Convex Integration

Vincent Borrelli

Institut Camille Jordan - Université Claude Bernard Lyon 1

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General Approach

Problem.– LetR ⊂Rⁿbe a path-connected subset (=our differential relation) andf₀: [0,1]−→^C¹ Rⁿbe a map such

∀t∈[0,1], f₀⁰(t)∈ Conv(R).

FindF : [0,1]−→^C¹ Rⁿsuch that : i) ∀t ∈[0,1], F⁰(t)∈ R ii) kF −f₀k_C0 < δ withδ >0 given.

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How to build a solution ?

f ’ ₀

R

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How to build a solution ?

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How to build a solution ?

f ’ ₀

R

f ’

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Construction of the solution

Step 1.– Choose a continuous family of loops γ : [0,1] −→ C⁰(R/Z,R)

u 7−→ γ(u, .)

such that

∀u∈ [0,1], Z

[0,1]

γ(u,s)ds=f₀⁰(u).

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Construction of the solution

Step 2.– We defineF :=CIγ(f₀,N)to be the map obtained by a convex integrationfromf₀:

F(t) :=f₀(0) + Z t

0

γ(u,Nu)du

whereN ∈N^∗ is a free parameter.

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C

⁰

-Density

•ObviouslyF =CI_γ(f₀,N)fulfills conditioni)since F⁰(t) =γ(t,Nt)∈ R

for allt ∈ [0,1]. The fact thatF also fulfills conditionii)forN large enough will ensue from the following proposition

Proposition (C⁰-density).–Ifγ is of class C¹and satisfies the average condition

∀u∈ [0,1], Z

[0,1]

γ(u,s)ds =f₀⁰(u)

then

kF −f₀k_C0 ≤ 1

N(2kγk_C0+k∂₁γk_C0) wherekγk_C0 = sup(u,s)∈[0,1]²kγ(u,s)k_E3.

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C

⁰

-Density

•ObviouslyF =CI_γ(f₀,N)fulfills conditioni)since F⁰(t) =γ(t,Nt)∈ R

for allt ∈ [0,1]. The fact thatF also fulfills conditionii)forN large enough will ensue from the following proposition

∀u∈ [0,1], Z

[0,1]

then

kF −f₀k_C0 ≤ 1

N(2kγk_C0+k∂₁γk_C0) wherekγk_C0 = sup(u,s)∈[0,1]²kγ(u,s)k_E3.

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C

⁰

-Density

Proof :Lett∈[0,1].Putn=bNtc,I_j = [_N^j,^j+1_N ]for 0≤j ≤n−1, In= [_Nⁿ,t].Since

∀t ∈I, F(t) :=f₀(0) + Z t

0

γ(s,Ns)ds

we obviously have

F(t)−f₀(0) =

n

X

k=0

F^[k]

where

F^[k]= Z

I_k

γ(s,Ns)ds.

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C

⁰

-Density

Since

f₀(t) = f₀(0) + Z t

x=0

∂f₀

∂x(x)dx

= f₀(0) + Z t

x=0

Z 1 u=0

γ(x,u)dudx we also have

f₀(t)−f₀(0) =

n

X

j=0

f^[j]

with

f^[j]= Z

Rj

γ(x,u)dxdu andR_j =I_j×[0,1].

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C

⁰

-Density

We considerj ∈[0,n−1]. By the change of variablesu=Ns−j, we get

F^[j]= Z 1

0

1 Nγ

u+j N ,u

du.

We now define

H_j : R_j → Rⁿ (x,u) 7→ γ(^u+j_N ,u).

In particular,H_j is constant over each horizontal segment inR_j. It ensues that

F^[j]= Z

Rj

H_j(x,u)dxdu implying

|F^[j]−f₀^[j]| ≤ Z

Rj

kγ(u+j

N ,u)−γ(x,u)kdxdu ≤ 1

N²k∂₁γk_∞. The last inequality follows from the mean value theorem and the fact that the area ofR_j = [_N^j,^j+1_N ]×[0,1]is 1/N.

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C

⁰

-Density

We considerj ∈[0,n−1]. By the change of variablesu=Ns−j, we get

F^[j]= Z 1

0

1 Nγ

u+j N ,u

du.

We now define

H_j : R_j → Rⁿ (x,u) 7→ γ(^u+j_N ,u).

In particular,H_j is constant over each horizontal segment inR_j. It ensues that

F^[j]= Z

Rj

H_j(x,u)dxdu implying

|F^[j]−f₀^[j]| ≤ Z

R_j

kγ(u+j

N ,u)−γ(x,u)kdxdu ≤ 1

N²k∂₁γk∞. The last inequality follows from the mean value theorem and the fact that the area ofR = [^j,^j+1]×[0,1]is 1/N.

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C

⁰

-Density

Forj =nwe can use a simpler upper bound : kF^[n]−f₀^[n]k ≤ kF^[n]k+kf₀^[n]k ≤ 2

Nkγk_∞. We finally obtain

kF(t)−f₀(t)k ≤

n−1

X

j=0

kF^[j]−f₀^[j]k+kF^[n]−f₀^[n]k

≤ 1

Nk∂₁γk_∞+ 2 Nkγk_∞.

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Summing up...

•To sum up, we are able to construct a solution of our initial problem as long as we have found a family of loops

γ : [0,1] −→ C⁰(R/Z,R)

u 7−→ γ(u, .)

that satisfies theaverage conditioni. e.

∀u∈ [0,1], Z

[0,1]

•The existence of such a familyγ is theFundamental Lemmaof Convex Integration Theory.

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Summing up...

•To sum up, we are able to construct a solution of our initial problem as long as we have found a family of loops

γ : [0,1] −→ C⁰(R/Z,R)

u 7−→ γ(u, .)

that satisfies theaverage conditioni. e.

∀u∈ [0,1], Z

[0,1]

•The existence of such a familyγ is theFundamental Lemmaof Convex Integration Theory.

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Fundamental Lemma

Notation.– LetR ⊂Rⁿbe a subset ofRⁿ(not necessarily path connected) andσ∈ R.We denote byIntConv(R, σ)the interior of the convex hull of the component ofRto whichσbelongs.

Fundamental Lemma (Gromov, 1969).–LetR ⊂Rⁿbe an open set, σ ∈ Rand z∈IntConv(R, σ)There exists a loopγ :S¹−→ Rwith base pointσ such that :

z = Z 1

0

γ(s)ds.

Proof.– On the blackboard.

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Fundamental Lemma

Notation.– LetR ⊂Rⁿbe a subset ofRⁿ(not necessarily path connected) andσ∈ R.We denote byIntConv(R, σ)the interior of the convex hull of the component ofRto whichσbelongs.

Fundamental Lemma (Gromov, 1969).–LetR ⊂Rⁿbe an open set, σ ∈ Rand z∈IntConv(R, σ)There exists a loopγ :S¹−→ Rwith base pointσ such that :

z = Z 1

0

γ(s)ds.

Proof.– On the blackboard.

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Fundamental Lemma

Remark.– A prioriγ ∈Ωσ(R),but it is obvious that we can chooseγ among back and forth loopsi.ethe space :

Ω^BF_σ (R) ={γ ∈Ωσ(R)| ∀s∈[0,1] γ(s) =γ(1−s)}.

The point is that the above space is contractible. For everyτ ∈[0,1]

we then denote byγ^τ :R/Z−→ Rthe loop defined by

γ^τ(s) =







γ(s) if s∈h 0,τ

2 i

∪h 1− τ

2 i

γ(τ) if s∈hτ 2,1−τ

2 i

.

This homotopy induces a deformation retract ofΩ^BF_σ (R)to the constant loop

eσ: R/Z −→ R s 7−→ σ.

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Parametric Fundamental Lemma

Parametric version of the Fundamental Lemma. –Let

E = [a,b]×Rⁿ−→^π [a,b]be a trivial bundle andR ⊂E be an open set.

LetS∈Γ(R)and z ∈Γ(E)such that :

∀p∈[a,b], z(p)∈IntConv(R_p,S(p)) whereR_p :=π⁻¹(p)∩ R.Then, there existsγ : [a,b]×S¹ ^C

∞

−→ Rsuch that :

γ(.,0) =γ(.,1) =S∈Γ(R),

∀p∈[a,b], γ(p, .)∈Concat(Ω^BF_S(p)(R_p)) and

∀p∈[a,b], z(p) = Z 1

0

γ(p,s)ds.

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Idea of Proof : concatenation of BF loops

Observation.– The parametric lemma still holds if the parameter space[a,b]is replaced by a compact manifoldP.

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Idea of Proof : concatenation of BF loops

Observation.– The parametric lemma still holds if the parameter space[a,b]is replaced by a compact manifoldP.

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Relative Parametric Fundamental Lemma

Parametric version of the Fundamental Lemma (continuation). – Let K be a closed subset of[a,b]. If for some open neighborhood V(K)of K we have

∀p∈V(K), z(p) =S(p)

then the family of loopsγ : [a,b]×S¹−→ Rcan be chosen such that γ(p, .)is the constant loopS(p)

for all p ∈V₁(K)where V₁(K)⊂V(K)is an open neighborhood K.

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Summing up...

•IfRis open then the problem of finding a mapF solvingRandC⁰ close tof₀can be solved by a convex integration. Indeed

Proposition.–Let f =CI_γ(f₀,N)then : i)∀t∈[0,1], ∂_tf(t) =γ(t,Nt)∈ R and ifγ satisfy the average condition :

ii)kf −f₀k_C0 =O(_N¹)

Corollary.–If N is large enough then f =CI_γ(f₀,N)is a solution of the above 1D-problem

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Exercise : the case of closed curves

Exercise.– LetR ⊂Rⁿbe a connected open subset and f₀:S¹ ^C

1

−→Rⁿbe a closed curve such that

∀t∈S¹, f₀⁰(t)∈ Conv(R).

Find a closed curvef :S¹ ^C

1

−→Rⁿsuch that : i) ∀t ∈S¹, f⁰(t)∈ R

ii) kf−f₀k_C0 < δ withδ >0 given.

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Exercise : the case of closed curves (hints)

•Note that, even iff₀is a closed curvef₀(0) =f₀(1), the map F =CI_γ(f₀,N)obtained by a convex integration fromf₀does not satisfyF(0) =F(1)in general.

•One natural choice forf is to take

∀t∈[0,1], f(t) :=F(t)−t(F(1)−F(0)). Sincef(0) =f(1)this defined a closed curve. Observe also that γ(0, .) =γ(1, .)impliesf⁰(0) =f⁰(1).

•From

f⁰(f) =γ(t,Nt)−(F(1)−F(0)) we deduce

kf⁰(t)−γ(t,Nt)k=O 1

N

SinceRis assumed to be open,f⁰(t)∈ RifN is large enough.

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Exercise : the case of closed curves (hints)

•From

N

SinceRis assumed to be open,f⁰(t)∈ RifN is large enough.

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Exercise : the case of closed curves (hints)

•From

N

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Exercise : the case of closed curves (hints)

•Conditionii)will follow from aC⁰-density result for the closed curvef.

∀u∈ [0,1], Z

[0,1]

then

kf −f₀k_C0 ≤ C(kγk_C0,k∂₁γk_C0) N

for some function C (to be determined).

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C

⁰

Density, N = 3

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C

⁰

Density, N = 5

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C

⁰

Density, N = 10

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C

⁰

Density, N = 20

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Multi Variables Setting

•In a multi-variable setting, the convex integration formula takes the following form :

F(c₁, ...,c_m) :=f₀(c₁, ...,c_m−1,0) + Z cm

0

γ(c₁, ...,c_m−1,s,Ns)ds

where(c₁, ...,cm)∈[0,1]^m.

A corrugated plane

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Multi Variables Setting

•TheC⁰-density property can be enhanced to aC^1,^m^b-density property where the notationC^1,b^mmeans that the closeness is measured with the following norm

kFk_C1,bm = max(kFk_C0,k∂F

∂c₁k_C0, ...,k ∂F

∂c_m−1k_C0).

Proposition (C^1,^m^b-density).–Ifγ is of class C¹and satisfies the average condition

∀u ∈ [0,1], Z

[0,1]

γ(c₁, ...,c_m−1,u,s)ds =f₀⁰(c₁, ...,c_m−1,u)

then we have

kF−f₀k_C1,mb =O 1

N

. Proof :left as an exercise.

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Multi Variables Setting

•TheC⁰-density property can be enhanced to aC^1,^m^b-density property where the notationC^1,b^mmeans that the closeness is measured with the following norm

kFk_C1,bm = max(kFk_C0,k∂F

∂c₁k_C0, ...,k ∂F

∂c_m−1k_C0).

Proposition (C^1,b^m-density).–Ifγ is of class C¹and satisfies the average condition

∀u ∈ [0,1], Z

[0,1]

γ(c₁, ...,c_m−1,u,s)ds =f₀⁰(c₁, ...,c_m−1,u)

then we have

kF−f₀k_C1,bm =O 1

N

.

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Whitney-Graustein Theorem

Definition.– AC¹closed curvef :S¹→R²is said to beregular(or to be animmersionof the circle) if for everyt∈S¹we havef⁰(t)6=0.

Definition.– Letf₀,f₁:S¹−→R²be two regular curves. Aregular homotopybetweenf₀andf₁is aC¹map

F : S¹×[0,1] −→ R² (x,s) 7−→ F_s(x) =F(x,s) such thatF₀=f₀,F₁=f₁andF_s is regular.

•The relation of regular homotopy is an equivalence relation whose equivalence classes identify with path connected components of the space of immersionsI(S¹,R²).

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Whitney-Graustein Theorem

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Whitney-Graustein Theorem

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Whitney-Graustein Theorem

Problem.– Classify regular curves up to regular homotopy.

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Whitney-Graustein Theorem

We assume thatS¹=R/Z= [0,1]/∂[0,1]andR²are endowed with an orientation.

Definition.– Letf be a regular closed curve. Theturning number TN(f)off is the number of counterclockwise turns off⁰ around(0,0).

•Therefore the turning number off is given by TN(f) =deg(t) =et(1)−et(0)∈Z whereet: [0,1]−→Ris a lift of the loop

t:= f⁰

kf⁰k : [0,1]−→S¹=R/Z.

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Whitney-Graustein Theorem

We assume thatS¹=R/Z= [0,1]/∂[0,1]andR²are endowed with an orientation.

Definition.– Letf be a regular closed curve. Theturning number TN(f)off is the number of counterclockwise turns off⁰ around(0,0).

•Therefore the turning number off is given by TN(f) =deg(t) =et(1)−et(0)∈Z whereet: [0,1]−→Ris a lift of the loop

t:= f⁰

kf⁰k : [0,1]−→S¹=R/Z.

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Whitney-Graustein Theorem

•Recall that

deg: π₁(S¹) −→ Z [t] 7−→ deg(t) is a bijection.

•Any regular homotopy(ft)_t_∈[0,1]induces a homotopy of the loops (t_t)_t∈[0,1]inS¹. Thus the turning number

t7→TN(f_t) is constant under regular homotopies.

•It ensues that the turning number induces a map TN: π₀(I(S¹,R²)) −→ Z

[f] 7−→ TN(f).

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Whitney-Graustein Theorem

•Recall that

•Any regular homotopy(f_t)_t_∈[0,1]induces a homotopy of the loops (t_t)_t∈[0,1]inS¹. Thus the turning number

[f] 7−→ TN(f).

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Whitney-Graustein Theorem

•Recall that

•Any regular homotopy(f_t)_t_∈[0,1]induces a homotopy of the loops (t_t)_t∈[0,1]inS¹. Thus the turning number

[f] 7−→ TN(f).

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Whitney-Graustein Theorem

•As seen in the figures below, this map is onto :

TN(γ) =−1 TN(f) =0 TN(f) =1 TN(f) =2 TN(f) =3

•It turns out that this map is 1-to-1.

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Whitney-Graustein Theorem

•As seen in the figures below, this map is onto :

TN(γ) =−1 TN(f) =0 TN(f) =1 TN(f) =2 TN(f) =3

•It turns out that this map is 1-to-1.

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Whitney-Graustein Theorem

Hassler Whitney

Whitney-Graustein Theorem (1937). –The turning number TN :π₀(I(S¹,R²))−→Z

induces a bijective map

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Proof of the Whitney-Graustein Theorem

Proof.– It is enough to show the injectivity. Letf₀andf₁be two regular closed curves having the same turning number. We consider the linear interpolation between them :

f_t := (1−t)f₀+t f₁, t ∈ [0,1]

Unless you are extremely lucky, this interpolation will fail to be regular for somet.

•We putR=R²\ {(0,0)}. The subsetRis connected, open and its convex hull isR².

•Observe that ifft is singular at some pointx ∈S¹, i. e ;f_t⁰(x) = (0,0), we obviously have

f_t⁰(x)∈IntConv(R) =R²

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Proof of the Whitney-Graustein Theorem

f_t := (1−t)f₀+t f₁, t ∈ [0,1]

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Proof of the Whitney-Graustein Theorem

f_t := (1−t)f₀+t f₁, t ∈ [0,1]

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Proof of the Whitney-Graustein Theorem

•Sincef₀andf₁have the same TN, there exists a homotopy S: [0,1] −→ C⁰(S¹,R)

t 7−→ S_t

joiningS₀=f₀⁰ toS₁=f₁⁰.

•We use the parametric version of the fundamental lemma with P = [0,1]×S¹to build a family of loops(γt)_t∈[0,1]such that for every p = (t,x)∈P:

1) the average of the loopu7→γ_t(x,u)isf_t⁰(x)i. e. Z 1

0

γ_t(x,u)du=f_t⁰(x)

2) the base point of the loopu7→γt(x,u)isS_t(x). 3)γ₀(x, .) =S₀(x)andγ₁(x, .) =S₁(x).

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Proof of the Whitney-Graustein Theorem

•Sincef₀andf₁have the same TN, there exists a homotopy S: [0,1] −→ C⁰(S¹,R)

t 7−→ S_t

joiningS₀=f₀⁰ toS₁=f₁⁰.

•We use the parametric version of the fundamental lemma with P = [0,1]×S¹to build a family of loops(γt)_t∈[0,1]such that for every p= (t,x)∈P :

1) the average of the loopu7→γt(x,u)isf_t⁰(x)i. e.

Z 1 0

γ_t(x,u)du=f_t⁰(x)

2) the base point of the loopu7→γt(x,u)isS_t(x).

3)γ₀(x, .) =S₀(x)andγ₁(x, .) =S₁(x).

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Whitney-Graustein Theorem

•We consider the family of closed curvesgt :S¹→R²given by g_t(x) :=G_t(x)−x(G_t(1)−G_t(0)) with G_t :=CI_γ_t(f_t,N) IfN is large enough,g_t is regular for everyt∈ [0,1].

•Sinceγ₀(x, .) =S₀(x)andγ₁(x, .) =S₁(x), we have g₀=f₀ and g₁=f₁.

Thusgt is a regular homotopy joiningf₀andf₁.

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Whitney-Graustein Theorem

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Whitney-Graustein Theorem

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Whitney-Graustein Theorem

•An observation : assume that we do not use the relative version of the Parametric Fundamental Lemma. Precisely, assume that

u 7→γ₀(x,u)andu 7→γ₁(x,u)are not constant map. Then the curve g₀(resp.g₁) is not equal tof₀(resp. tof₁). An extra regular homotopy is thus needed to joinf₀tog₀andg₁tof₁.

•This extra homotopy is for free by using the fact that each loop is parametrized back and forth. Indeed...

•Let(g₀^τ)_τ∈_[0,1]be the homotopy defined by

g₀^τ(x) :=G₀^τ(x)−x(G^τ₀(1)−G₀^τ(0)) with G^τ₀:=CI_γ^τ

0(f₀,N) whereτ 7→γ₀^τ is the retraction ofγ0toγ0(0) =σ0(0) =f₀⁰ described at the end of the sectionFundamental Lemma.

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Whitney-Graustein Theorem

0(f₀,N) whereτ 7→γ₀^τ is the retraction ofγ0toγ0(0) =σ0(0) =f₀⁰ described at the end of the sectionFundamental Lemma.

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Whitney-Graustein Theorem

0(f₀,N) whereτ 7→γ₀^τ is the retraction ofγ₀toγ₀(0) =σ₀(0) =f₀⁰ described at the end of the sectionFundamental Lemma.

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Whitney-Graustein Theorem

•For allx ∈S¹we have

(g₀^τ)⁰(x) := (G^τ₀)⁰(x)−(G^τ₀(1)−G^τ₀(0)) with

(G^τ₀)⁰(x) =γ₀^τ(x,Nx)∈ R

•SinceRis open, we deduce form theC⁰-property thatg₀^τ is a regular homotopy joiningf₀tog₀providedN is large enough.

•Obviously the same process also give a regular homotopy joiningf₁ tog₁.

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Whitney-Graustein Theorem

(g₀^τ)⁰(x) := (G^τ₀)⁰(x)−(G^τ₀(1)−G^τ₀(0)) with

(G^τ₀)⁰(x) =γ₀^τ(x,Nx)∈ R

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Whitney-Graustein Theorem

(g₀^τ)⁰(x) := (G^τ₀)⁰(x)−(G^τ₀(1)−G^τ₀(0)) with

(G^τ₀)⁰(x) =γ₀^τ(x,Nx)∈ R

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Beyond the Whitney-Graustein Theorem

Definitions.– The subsetR=R²\ {(0,0}is called thedifferential relationof regular curves.

•The space of all mapsS:S¹→ Ris denotedΓ(R).

•A mapS∈Γ(R)is calledholonomicif there existsf :S¹→R²such thatf⁰ =S.

•In that case the mapf is called asolutionofR.The space of all solutions is denoted bySol(R). Observe thatSol(R) =I(S¹,R²).

•There is an obvious inclusionJ :Sol(R)⊂Γ(R)given byJ(f) =f⁰. Whitney-Graustein Theorem (1937). –The inclusion J induces a bijective map at theπ₀-level :

π0(J) :π0(Sol(R))−→π0(Γ(R))

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Beyond the Whitney-Graustein Theorem

•There is an obvious inclusionJ :Sol(R)⊂Γ(R)given byJ(f) =f⁰.

Whitney-Graustein Theorem (1937). –The inclusion J induces a bijective map at theπ₀-level :

π0(J) :π0(Sol(R))−→π0(Γ(R))

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Beyond the Whitney-Graustein Theorem

•There is an obvious inclusionJ :Sol(R)⊂Γ(R)given byJ(f) =f⁰. Whitney-Graustein Theorem (1937). –The inclusion J induces a bijective map at theπ₀-level :

π0(J) :π0(Sol(R))−→π0(Γ(R))

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Whitney-Graustein Theorem

•In fact more is true. By considering different parametric spacesPin the proof of the Whitney-Graustein Theorem, we can prove the following generalization.

Generalization of the Whitney-Graustein Theorem. –For every k ∈Nthe inclusion J induces a bijective map at theπ_k-level :

πk(J) :πk(Sol(R))−→πk(Γ(R)) In other words, J is a weak homotopy equivalence.

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