One- Dimensional Convex Integration

Dimensional Convex Integration

V.Borrelli

Bibliography Introductory Examples Fundamental Lemma C⁰-Density

One-Dimensional Convex Integration

Vincent Borrelli

Université Lyon 1 - France

Dimensional Convex Integration

V.Borrelli

Bibliography Introductory Examples Fundamental Lemma C⁰-Density

Bibliography

Mikhail Gromov

Dimensional Convex Integration

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

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Yakov Eliashberg

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

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David Spring

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

An Elementary Example

Exercice.– Let f₀: [0,1] −→ R³ t 7−→ (0,0,t)

Findf : [0,1]−→^C1 R³such that :

i) ∀t ∈[0,1], |cos(f⁰(t),e₃)|<

ii) kf −f₀k_C0 < δ

where >0 andδ >0 are given.

e₃

e₁

e₂ f ₀

?

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

An Elementary Example

Exercice.– Let f₀: [0,1] −→ R³ t 7−→ (0,0,t)

Findf : [0,1]−→^C1 R³such that :

i) ∀t ∈[0,1], |cos(f⁰(t),e₃)|<

ii) kf −f₀k_C0 < δ

where >0 andδ >0 are given.

e₃

e₁

e₂

f ₀ f

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

Rephrasing...

Condition(i)means that the image off⁰ lies inside the blue cone :

e

R

Ris thedifferential relationof our problem.

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... and new understanding

The mapf must beC⁰near off₀:

f ’₀ f ’

The average off⁰ for eachloopoff isf₀⁰(t): 1

Long(I) Z

I

f⁰(u)du =f₀⁰(t)

(I=the preimage of one loop byf).

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A more general problem

Problem.– LetR ⊂R³be a path-connected subset (=our differential relation) andf₀: [0,1]−→^C1 R³be a map such

∀t ∈[0,1], f₀⁰(t)∈ Conv(R).

Findf : [0,1]−→^C1 R³such that : i) ∀t ∈[0,1], f⁰(t)∈ R ii) kf −f₀k_C0 < δ

withδ >0 given.

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

How to build a solution ?

f ’ ₀

R

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

How to build a solution ?

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

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

h: [0,1] −→ C⁰(R/Z,R)

u 7−→ h_u

such that

∀u∈ [0,1], Z

[0,1]

h_u(s)ds=f₀⁰(u).

h u

f ’ (u)0

s

u

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Construction of the solution : step 2

Then set

f(t) :=f₀(0) + Z t

0

h_u({Nu})du

whereN∈N^∗and{Nu}is the fractional part ofNu.

({Nu}) f’(u):=h u

s

u

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Construction of the solution : step 2

Then set

f(t) :=f₀(0) + Z t

0

h_u({Nu})du

whereN∈N^∗and{Nu}is the fractional part ofNu.

We say thatf is obtained fromf₀by aconvex integration.

({Nu}) f’(u):=h u

s

u

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Fundamental Lemma

Notation.– LetR ⊂Rⁿ andσ ∈ R.We denote by IntConv(R, σ)the interior of the convex hull of the connected component ofRto whichσ belongs.

Definition.– A (continuous) loopg : [0,1]→Rⁿ, g(0) =g(1),strictly surrounds z ∈Rⁿif

IntConv(g([0,1]))⊃ {z}.

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Fundamental Lemma

Fundamental Lemma.–LetR ⊂Rⁿbe an open set,σ∈ R and z∈IntConv(R, σ)There exists a loop h: [0,1]−→ R^C0 with base pointσ that strictly surrounds z and such that :

z = Z 1

0

h(s)ds.

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Fundamental Lemma

Remark.– A priori h∈Ω_σ(R),but it is obvious that we can choosehamong "round-trips"i.ethe space :

Ω^AR_σ (R) ={h∈Ω_σ(R)| ∀s∈[0,1] h(s) =h(1−s)}.

The point is that the above space is contractible. For every u∈[0,1]we then denote byh_u: [0,1]−→ Rthe map defined by

h_u(s) =

h(s) if s∈[0,^u₂]∪[1−^u₂] h(u) if s∈[^u₂,1− ^u₂].

This homotopy induces a deformation retract ofΩ^AR_σ (R)to the constant map

σe: [0,1] −→ 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. Letσ ∈Γ(R)and z ∈Γ(E)such that :

∀p∈[a,b], z(p)∈IntConv(R_p, σ(p)) whereR_p :=π⁻¹(p)∩ R.Then, there exists h: [a,b]×[0,1] ^C

∞

−→ Rsuch that :

h(.,0) =h(.,1) =σ∈Γ^∞(R),

∀p ∈[a,b], h(p, .)∈Ω^AR_σ(p)(R_p) and

∀p∈[a,b], z(p) = Z 1

0

h(p,s)ds.

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Proof

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

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

Proof

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

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

The Convex Integration Process

LetR ⊂Rⁿbe a arc-connected subset,f₀∈C^∞(I,Rⁿ)be a map such thatf₀⁰(I)⊂ IntConv(R).We set

∀t∈I, F(t) :=f₀(0) + Z t

0

h(s,Ns)ds

withN ∈N^∗.

Definition.– We say thatF ∈C^∞(I,Rⁿ)is a solution ofR obtained fromf₀by aconvex integration process.

ObviouslyF⁰(t) =h(t,Nt)∈ Rand thusF is a solution of the differential relationR.

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

C

-Density

One crucial property of the convex integration process is that the solutionF can be made arbitrarily close to the initial mapf₀.

Proposition (C⁰-density).–We have

kF −f₀k_C0 ≤ 1 N

2khk_C0+k∂h

∂tk_C0

wherekgk_C0 =sup_p∈Dkg(p)k

E³ denotes the C⁰norm of a function g:D→E³.

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C

-Density

Remark.– Even iff₀(0) =f₀(1), the mapF obtained by a convex integration fromf₀does not satisfyF(0) =F(1)in general. This can be easily corrected by defining a new map f with the formula

∀t∈[0,1] , f(t) =F(t)−t(F(1)−F(0)).

Proposition (C⁰-density).–We have

kf −f₀k_C0 ≤ 2 N

2khk_C0+k∂h

∂tk_C0

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

C

-Density

Remark.– Even iff₀(0) =f₀(1), the mapF obtained by a convex integration fromf₀does not satisfyF(0) =F(1)in general. This can be easily corrected by defining a new map f with the formula

∀t∈[0,1] , f(t) =F(t)−t(F(1)−F(0)). Proposition (C⁰-density).–We have

kf −f₀k_C0 ≤ 2 N

2khk_C0+k∂h

∂tk_C0

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C

Density, N = 3

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C

Density, N = 5

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C

Density, N = 10

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C

Density, N = 20

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Multi Variables Setting

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

f(c₁, ...,c_m) :=f₀(c₁, ...,c_m−1,0)+

Z cm

0

h(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 a

C^1,bm-density property where the notationC^1,bmmeans that the closeness is measured with the following norm

kfk_C1,bm =max(kfk_C0,k ∂f

∂c₁k_C0, ...,k ∂f

∂cm−1

k_C0).

Proposition (C^1,bm-density).–We have

kf−f₀k_C1,mb =O 1

N

.

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Bibliography Introductory Examples Fundamental Lemma C⁰-Density

Multi Variables Setting

•TheC⁰-density property can be enhanced to a

C^1,bm-density property where the notationC^1,bmmeans that the closeness is measured with the following norm

kfk_C1,bm =max(kfk_C0,k ∂f

∂c₁k_C0, ...,k ∂f

∂cm−1

k_C0).

Proposition (C^1,bm-density).–We have

kf−f₀k_C1,bm =O 1

N

.

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Mikhaïl Gromov