The H-principle for Isometric Embeddings

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The H-principle for

Isometric Embeddings

V.Borrelli

Isometric Maps How to deal with the non- ampleness ? The one dimensional case How to deal with a closed relation ?

The H-principle for Isometric Embeddings

Vincent Borrelli

Université Lyon 1

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The H-principle for

V.Borrelli

Isometric Maps

Definition.– A map f : (M ⁿ , g) −→ ^C

¹

E ^q is an isometry if f ^∗ h., .i = g.

The length of curves is preserved by an isometric map.

In a local coordinate system x = (x ₁ , ..., x _n ) : 1 ≤ i ≤ j ≤ n, h ∂f

∂x _i (x ), ∂f

∂x _j (x )i = g _ij (x ) The Janet dimension :

s _n = n(n + 1)

2 .

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The H-principle for

V.Borrelli

Isometric Maps

Definition.– A map f : (M ⁿ , g) −→ ^C

¹

E ^q is an isometry if f ^∗ h., .i = g.

The length of curves is preserved by an isometric map.

In a local coordinate system x = (x ₁ , ..., x _n ) : 1 ≤ i ≤ j ≤ n, h ∂f

∂x _i (x ), ∂f

∂x _j (x )i = g _ij (x )

The Janet dimension :

s _n = n(n + 1)

2 .

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H-principle for Isometric Embeddings

V.Borrelli

Isometric Maps

Definition.– A map f : (M ⁿ , g) −→ ^C

¹

E ^q is an isometry if f ^∗ h., .i = g.

The length of curves is preserved by an isometric map.

In a local coordinate system x = (x ₁ , ..., x _n ) : 1 ≤ i ≤ j ≤ n, h ∂f

∂x _i (x ), ∂f

∂x _j (x )i = g _ij (x ) The Janet dimension :

s _n = n(n + 1)

2 .

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The H-principle for

V.Borrelli

Nash-Kuiper Theorem

Definition.– A map f : (M ⁿ , g) −→ ^C

¹

E ^q is called strictly short if f ^∗ h., .i < g.

John Forbes Nash Nicolaas Kuiper

Theorem (1954-55-86).– Let M ⁿ be a compact Riemannian manifold and f ₀ : (M ⁿ , g) ^C

1

−→ E ^q be a strictly short

embedding. Then, for every > 0, there exists a C ¹

isometric embedding f : (M ⁿ , g) −→ E ^q such that

kf − f ₀ k _C

0

≤ .

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V.Borrelli

Nash-Kuiper Theorem

Definition.– A map f : (M ⁿ , g) −→ ^C

¹

E ^q is called strictly short if f ^∗ h., .i < g.

John Forbes Nash Nicolaas Kuiper

Theorem (1954-55-86).– Let M ⁿ be a compact Riemannian manifold and f ₀ : (M ⁿ , g) −→ ^C

¹

E ^q be a strictly short

embedding. Then, for every > 0, there exists a C ¹

isometric embedding f : (M ⁿ , g) −→ E ^q such that

kf − f ₀ k _C

0

≤ .

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The H-principle for

V.Borrelli

Nash-Kuiper Spheres

?

Nash-Kuiper Spheres.– Let 0 < r < 1. There exists a

C ¹ -isometric embedding of the unit sphere of E ³ in a ball of

radius r .

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V.Borrelli

Sphere Eversion

Sphere Eversion.– The sphere S ² ⊂ E ³ can be turned

inside out by a regular homotopy of isometric C ¹

immersions.

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The H-principle for

V.Borrelli

Flat Tori

Definition.– Any quotient E ² /Λ of the Euclidean 2-space by a lattice Λ ⊂ E ² is called a flat torus

Flat Tori.– Any flat torus E ² /Λ admits a C ¹ isometric

embedding in E ³ .

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V.Borrelli

Flat Tori

Definition.– Any quotient E ² /Λ of the Euclidean 2-space by a lattice Λ ⊂ E ² is called a flat torus

Flat Tori.– Any flat torus E ² /Λ admits a C ¹ isometric

embedding in E ³ .

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The H-principle for

V.Borrelli

Our Goal

Goal of this talk.– To recover the Nash-Kuiper result on C ¹ isometric embeddings from the machinery of the Gromov Integration Theory.

Our main ingredient.– The Gromov Theorem : Let R ⊂ J ¹ (M, N) be an open and ample differential relation. Then R satisfies the parametric h-principle i. e.

J : Sol(R) −→ Γ(R) is a weak homotopy equivalence.

Our main obstacles.–

• The isometric relation is not ample

• The isometric relation is closed

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V.Borrelli

Decomposition

• For simplicity M ⁿ = [0, 1] ⁿ .

• The image of the metric distorsion

∆ := g − f ₀ ^∗ h., .i _E

^q

lies inside the positive cone M of inner products of E ⁿ .

• There exist S ≥ ⁿ⁽ⁿ⁺¹⁾ ₂ linear forms ` ₁ , . . . ` _S of E ⁿ such that

g − f ₀ ^∗ h., .i _E

^q

=

S

X

j=1

ρ _j ` _j ⊗ ` _j

where ρ _j > 0.

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The H-principle for

V.Borrelli

Adapting the Gromov machinery

The strategy is to do the successive convex integrations along the S directions corresponding to the S linear forms

` ₁ , ..., ` _S .

rather than

along the n directions of the coordinates in [0, 1] ⁿ .

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V.Borrelli

Adapting the Gromov machinery

• This will produce S intermediary maps f ₁ , ..., f _S such that

g − f ₁ ^∗ h., .i _E

^q

≈ρ ₂ ` ² ₂ +ρ ₃ ` ² ₃ +...+ρ _S ` ² _S g − f ₂ ^∗ h., .i _E

^q

≈ ρ 3 ` ² ₃ +...+ρ _S ` ² _S

.. . .. .

g − f _S−1 ^∗ h., .i _E

^q

≈ ρ _S ` ² _S g − f _S ^∗ h., .i _E

^q

≈ 0.

• The map f := f _S is then a solution of R e = Op(R).

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The H-principle for

V.Borrelli

Adapting the Gromov machinery

We have

f _j ^∗ h., .i _E

^q

− f _j−1 ^∗ h., .i _E

^q

=(g − f _j−1 ^∗ h., .i _E

^q

) − (g − f _j ^∗ h., .i _E

^q

)

≈ ρ _j ` _j ⊗ ` _j .

Hence, the fundamental problem is the following : Fundamental Problem.– Given a positive function ρ, a linear form ` 6= 0 and an embedding f ₀ how to build an other embedding f such that

f ^∗ h., .i _E

^q

≈ µ

where µ := f ₀ ^∗ h., .i _E

^q

+ ρ ` ⊗ ` ?

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V.Borrelli

The one dimensional case

One dimensional fundamental problem.– Let

f ₀ : [0, 1] −→ E ^q be an embedding, ρ a positive function,

` 6= 0 a linear form on R , how to build an other embedding f : [0, 1] −→ E ^q such that

∀u ∈ [0, 1], kf ⁰ (u)k ² ≈ kf ₀ ⁰ (u)k ² + ρ(u)` ² (∂ _u ) ?

• For short, we set r (u) :=

q

kf ₀ ⁰ (u)k ² + ρ(u)` ² (∂ u ).

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The H-principle for

V.Borrelli

Choosing the Loops

with

∀s ∈ R / Z , h _u (s) := r (u)e ^iα(u) ^cos(2πs)

and α(u) > 0 is such that Z 1

0 r (u)e ^iα(u) ^cos(2πs) ds = f ₀ ⁰ (u).

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V.Borrelli

Choosing the Loops

with

∀s ∈ R / Z , h _u (s) := r (u)e ^iα(u) ^cos(2πs) and α(u) > 0 is such that

Z 1 0

r (u)e ^iα(u) ^cos(2πs) ds = f ₀ ⁰ (u).

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The H-principle for

V.Borrelli

Our Convex Integration Process

• The convex integration formula : f (t) := f ₀ (0) +

Z t

0 r (u)e ^iα(u) ^{cos 2π} N u du.

where e ^iθ := cos θ t + sin θ n with t := ^f

0 0

kf

₀⁰

k and n is a unit normal to the curve f ₀ .

Lemma.– The map f solves the one dimensional

fundamental problem. Its speed kf ⁰ k is equal to the given function r = (kf ₀ ⁰ k ² + ρ` ² (∂ c ))

¹²

. Moreover

kf − f ₀ k _C

0

= O 1

N

and if N is large enough f is an embedding.

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V.Borrelli

Our Convex Integration Process

• The convex integration formula : f (t) := f ₀ (0) +

Z t

0 r (u)e ^iα(u) ^{cos 2π} N u du.

where e ^iθ := cos θ t + sin θ n with t := ^f

0 0

kf

₀⁰

k and n is a unit normal to the curve f ₀ .

Lemma.– The map f solves the one dimensional

fundamental problem. Its speed kf ⁰ k is equal to the given function r = (kf ₀ ⁰ k ² + ρ` ² (∂ c ))

¹²

. Moreover

kf − f ₀ k _C

0

= O 1

N

and if N is large enough f is an embedding.

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The H-principle for

V.Borrelli

Our Convex Integration Process

A short curve f

₀

(black) and the curve f obtained with the one

dimensional convex integration formula (grey, N = 9 and N = 20).

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The H-principle for

V.Borrelli

A technical difficulty

• We assume for simplicity that ker ` = Span(e ₂ , ..., e _n ) and

`(e ₁ ) = 1.

• Let s ∈ [0, 1] and c = (c ₂ , ..., c _n ) ∈ [0, 1] ⁿ⁻¹ , we set f (s, c) := f ₀ (0, c) +

Z s

0 r (u, c)e ^iα(u,c) ^{cos 2πNu} du

with r = p

µ(e ₁ , e ₁ ) = p

kdf ₀ (e ₁ )k ² + ρ.

• The map f is not a solution of our Fundamental Problem. We do not have

kf ^∗ h., .i _E

^q

− µk _C

0

= O 1

N

with µ := f ₀ ^∗ h., .i _E

^q

+ ρ ` ⊗ `.

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The H-principle for

V.Borrelli

A technical difficulty

• We assume for simplicity that ker ` = Span(e ₂ , ..., e _n ) and

`(e ₁ ) = 1.

• Let s ∈ [0, 1] and c = (c ₂ , ..., c _n ) ∈ [0, 1] ⁿ⁻¹ , we set f (s, c) := f ₀ (0, c) +

Z s

0 r (u, c)e ^iα(u,c) ^{cos 2πNu} du

with r = p

µ(e ₁ , e ₁ ) = p

kdf ₀ (e ₁ )k ² + ρ.

• The map f is not a solution of our Fundamental Problem.

We do not have

kf ^∗ h., .i _E

^q

− µk _C

0

= O 1

N

with µ := f ₀ ^∗ h., .i _E

^q

+ ρ ` ⊗ `.

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The H-principle for

V.Borrelli

A technical difficulty

• By C ^1, ^ˆ ¹ -density we have

df (e _j ) = df ₀ (e _j ) + O( 1 N )

• Thus, for all 1 < i , 1 < j, we have

(f ^∗ h., .i _E

^q

)(e _i , e _j ) = hdf (e _i ), df (e _j )i _E

^q

= hdf ₀ (e _i ), df ₀ (e _j )i _E

^q

+ O 1

N

= µ(e _i , e _j ) + O 1

N

• Therefore

k(f ^∗ h., .i _E

^q

− µ)(e _i , e _j )k _C

0

= O 1

N

for all 1 < i, 1 < j.

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The H-principle for

V.Borrelli

A technical difficulty

• By C ^1, ^ˆ ¹ -density we have

df (e _j ) = df ₀ (e _j ) + O( 1 N )

• Thus, for all 1 < i, 1 < j , we have

(f ^∗ h., .i _E

^q

)(e _i , e _j ) = hdf (e _i ), df (e _j )i _E

^q

= hdf ₀ (e _i ), df ₀ (e _j )i _E

^q

+ O 1

N

= µ(e _i , e _j ) + O 1

N

• Therefore

k(f ^∗ h., .i _E

^q

− µ)(e _i , e _j )k _C

0

= O 1

N

for all 1 < i, 1 < j.

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V.Borrelli

A technical difficulty

• By C ^1, ^ˆ ¹ -density we have

df (e _j ) = df ₀ (e _j ) + O( 1 N )

• Thus, for all 1 < i, 1 < j , we have

(f ^∗ h., .i _E

^q

)(e _i , e _j ) = hdf (e _i ), df (e _j )i _E

^q

= hdf ₀ (e _i ), df ₀ (e _j )i _E

^q

+ O 1

N

= µ(e _i , e _j ) + O 1

N

• Therefore

k(f ^∗ h., .i _E

^q

− µ)(e _i , e _j )k _C

0

= O 1

N

for all 1 < i , 1 < j.

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The H-principle for

V.Borrelli

A technical difficulty

• By definition of f we have

df _(s,c) (e ₁ ) = r (s, c)e ^iα(s,c) ^{cos 2πNs} .

• Thus

(f ^∗ h., .i _E

^q

)(e ₁ , e ₁ ) = hdf (e ₁ ), df (e ₁ )i _E

^q

= r ² (s, c)

= µ(e ₁ , e ₁ )

• Therefore

k(f ^∗ h., .i _E

^q

− µ)(e ₁ , e ₁ )k _C

0

= 0.

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The H-principle for

V.Borrelli

A technical difficulty

• By definition of f we have

df _(s,c) (e ₁ ) = r (s, c)e ^iα(s,c) ^{cos 2πNs} .

• Thus

(f ^∗ h., .i _E

^q

)(e ₁ , e ₁ ) = hdf (e ₁ ), df (e ₁ )i _E

^q

= r ² (s, c)

= µ(e ₁ , e ₁ )

• Therefore

k(f ^∗ h., .i _E

^q

− µ)(e ₁ , e ₁ )k _C

0

= 0.

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The H-principle for

V.Borrelli

A technical difficulty

• By definition of f we have

df _(s,c) (e ₁ ) = r (s, c)e ^iα(s,c) ^{cos 2πNs} .

• Thus

(f ^∗ h., .i _E

^q

)(e ₁ , e ₁ ) = hdf (e ₁ ), df (e ₁ )i _E

^q

= r ² (s, c)

= µ(e ₁ , e ₁ )

• Therefore

k(f ^∗ h., .i _E

^q

− µ)(e ₁ , e ₁ )k _C

0

= 0.

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The H-principle for

V.Borrelli

A technical difficulty

• The problem arises with the mixted term hdf (e ₁ ), df (e _j )i _E

^q

, j > 1.

• Indeed, in the one hand

(f ^∗ h., .i _E

q

)(e ₁ , e _j ) = hdf (e ₁ ), df (e _j )i _E

q

= hdf (e ₁ ), df ₀ (e _j )i _E

^q

+ O 1

N

= hr e ^iF , df ₀ (e _j )i _E

^q

+ O 1

N

= hr cos( F )t, df ₀ (e _j )i _E

^q

+hr sin( F )n, df ₀ (e _j )i _E

^q

+ O

1 N

= r cos( F )

kdf ₀ (e ₁ )k hdf ₀ (e ₁ ), df ₀ (e _j )i _E

^q

+O

1 N

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The H-principle for

V.Borrelli

A technical difficulty

• The problem arises with the mixted term hdf (e ₁ ), df (e _j )i _E

^q

, j > 1.

• Indeed, in the one hand

(f ^∗ h., .i _E

^q

)(e ₁ , e _j ) = hdf (e ₁ ), df (e _j )i _E

^q

= hdf (e ₁ ), df ₀ (e _j )i _E

^q

+ O 1

N

= hr e ^iF , df ₀ (e _j )i _E

^q

+ O 1

N

= hr cos( F )t, df ₀ (e _j )i _E

^q

+hr sin( F )n, df ₀ (e _j )i _E

^q

+ O

1 N

= r cos( F )

kdf ₀ (e ₁ )k hdf ₀ (e ₁ ), df ₀ (e _j )i _E

^q

+O

1 N

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The H-principle for

V.Borrelli

A technical difficulty

• In the other hand, since `(e _j ) = 0, we have µ(e ₁ , e _j ) = hdf ₀ (e ₁ ), df ₀ (e _j )i _E

q

.

• Therefore

k(f ^∗ h., .i _E

^q

− µ)(e ₁ , e _j )k _C

0

6= O 1

N

unless hdf ₀ (e _i ), df ₀ (e _j )i _E

^q

≡ 0.

Claim.– This difficulty vanishes if the convex integration is done along the integral lines of a vector field W such that

∀j ∈ {2, ..., m}, µ(W , e _j ) = 0

i. e. W is µ-orthogonal to ker `.

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The H-principle for

V.Borrelli

A technical difficulty

• In the other hand, since `(e _j ) = 0, we have µ(e ₁ , e _j ) = hdf ₀ (e ₁ ), df ₀ (e _j )i _E

q

.

• Therefore

k(f ^∗ h., .i _E

^q

− µ)(e ₁ , e _j )k _C

0

6= O 1

N

unless hdf ₀ (e _i ), df ₀ (e _j )i _E

^q

≡ 0.

Claim.– This difficulty vanishes if the convex integration is done along the integral lines of a vector field W such that

∀j ∈ {2, ..., m}, µ(W , e _j ) = 0

i. e. W is µ-orthogonal to ker `.

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V.Borrelli

A technical difficulty

• In the other hand, since `(e _j ) = 0, we have µ(e ₁ , e _j ) = hdf ₀ (e ₁ ), df ₀ (e _j )i _E

q

.

• Therefore

k(f ^∗ h., .i _E

^q

− µ)(e ₁ , e _j )k _C

0

6= O 1

N

unless hdf ₀ (e _i ), df ₀ (e _j )i _E

^q

≡ 0.

Claim.– This difficulty vanishes if the convex integration is done along the integral lines of a vector field W such that

∀j ∈ {2, ..., m}, µ(W , e _j ) = 0

i. e. W is µ-orthogonal to ker `.

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The H-principle for

V.Borrelli

Adjusting the convex integration formula

Proposition.– The resulting map f solves the fundamental problem. Precisely

kf ^∗ h., .i _E

^q

− µk = O 1

N

where µ = f ₀ ^∗ h., .i _E

^q

+ ρ ` ⊗ `. Moreover 1) kf − f ₀ k _C

0

= O _N ¹

, 2) kdf − df ₀ k _C

0

≤ ^Cte _N + √

7ρ

¹²

|`(W )|,

and if N is large enough, f is an embedding.

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V.Borrelli

Thickening the Differential

Relation

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The H-principle for

V.Borrelli

Thickening the Differential

Relation

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V.Borrelli

Thickening the Differential

Relation

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The H-principle for

V.Borrelli

Thickening the Differential Relation

Question.– Is the limit

−→0 lim f an isometric map (if it exists) ?

Answer.– No !

−→0 lim f = f ₀ .

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V.Borrelli

Thickening the Differential Relation

Question.– Is the limit

−→0 lim f an isometric map (if it exists) ? Answer.– No !

−→0 lim f = f ₀ .

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The H-principle for

V.Borrelli

Approximating the Differential

Relation

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V.Borrelli

Approximating the Differential

Relation

(43)

The H-principle for

V.Borrelli

Iterated Convex Integrations

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V.Borrelli

Iterated Convex Integrations

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The H-principle for

V.Borrelli

Iterated Convex Integrations

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The H-principle for

V.Borrelli

Thickening the Differential Relation

Question.– Is the limit

k−→+∞ lim f _k an isometric map (if it exists) ?

Answer.– Yes ! Let us see why

i) it is C ⁰ converging, ii) it is C ¹ converging. Consequently

f ∞ := lim

k−→+∞ f _k is a C ¹ isometric map.

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The H-principle for

V.Borrelli

Thickening the Differential Relation

Question.– Is the limit

k−→+∞ lim f _k an isometric map (if it exists) ? Answer.– Yes !

Let us see why

i) it is C ⁰ converging, ii) it is C ¹ converging.

Consequently

f ∞ := lim

k−→+∞ f _k is a C ¹ isometric map.

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V.Borrelli

C ⁰ Convergence

It is enough to control the difference kf _k − f _k−1 k _C

0

.

We set

f _∞ = lim

k→+∞ f _k .

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The H-principle for

V.Borrelli

C ¹ convergence

It is enough to control the difference kdf _k − df k−1 k _C

0

.

kdf _k − df _k−1 k _C

0

≤ C ^te q

dist ( R e _k ₋₁ , R e _k )

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V.Borrelli

C ¹ convergence

It is enough to control the difference kdf _k − df k−1 k _C

0

.

kdf _k − df _k−1 k _C

0

≤ C ^te q

dist ( R e _k−1 , R e _k )

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The H-principle for

V.Borrelli

The H-principle for Isometric Embeddings

The H-principle for Isometric Embeddings

Vincent Borrelli

Université Lyon 1

Isometric Maps

Definition.– A map f : (M n , g) −→ C

E q is an isometry if f ∗ h., .i = g.

The length of curves is preserved by an isometric map.

In a local coordinate system x = (x 1 , ..., x n ) : 1 ≤ i ≤ j ≤ n, h ∂f

∂x i (x ), ∂f

∂x j (x )i = g ij (x ) The Janet dimension :

s n = n(n + 1)

2 .

Isometric Maps

Definition.– A map f : (M n , g) −→ C

E q is an isometry if f ∗ h., .i = g.

The length of curves is preserved by an isometric map.

In a local coordinate system x = (x 1 , ..., x n ) : 1 ≤ i ≤ j ≤ n, h ∂f

∂x i (x ), ∂f

∂x j (x )i = g ij (x )

The Janet dimension :

s n = n(n + 1)

2 .

Isometric Maps

Definition.– A map f : (M n , g) −→ C

E q is an isometry if f ∗ h., .i = g.

The length of curves is preserved by an isometric map.

In a local coordinate system x = (x 1 , ..., x n ) : 1 ≤ i ≤ j ≤ n, h ∂f

∂x i (x ), ∂f

∂x j (x )i = g ij (x ) The Janet dimension :

s n = n(n + 1)

2 .

Nash-Kuiper Theorem

Definition.– A map f : (M n , g) −→ C

E q is called strictly short if f ∗ h., .i < g.

John Forbes Nash Nicolaas Kuiper

Theorem (1954-55-86).– Let M n be a compact Riemannian manifold and f 0 : (M n , g) C

−→ E q be a strictly short

embedding. Then, for every > 0, there exists a C 1

isometric embedding f : (M n , g) −→ E q such that

kf − f 0 k C

≤ .

Nash-Kuiper Theorem

Definition.– A map f : (M n , g) −→ C

E q is called strictly short if f ∗ h., .i < g.

John Forbes Nash Nicolaas Kuiper

Theorem (1954-55-86).– Let M n be a compact Riemannian manifold and f 0 : (M n , g) −→ C

E q be a strictly short

embedding. Then, for every > 0, there exists a C 1

isometric embedding f : (M n , g) −→ E q such that

kf − f 0 k C

≤ .

Nash-Kuiper Spheres

?

Nash-Kuiper Spheres.– Let 0 < r < 1. There exists a

C 1 -isometric embedding of the unit sphere of E 3 in a ball of

radius r .

Sphere Eversion

Sphere Eversion.– The sphere S 2 ⊂ E 3 can be turned

inside out by a regular homotopy of isometric C 1

immersions.

Flat Tori

Definition.– Any quotient E 2 /Λ of the Euclidean 2-space by a lattice Λ ⊂ E 2 is called a flat torus

Flat Tori.– Any flat torus E 2 /Λ admits a C 1 isometric

embedding in E 3 .

Flat Tori

Definition.– Any quotient E 2 /Λ of the Euclidean 2-space by a lattice Λ ⊂ E 2 is called a flat torus

Flat Tori.– Any flat torus E 2 /Λ admits a C 1 isometric

embedding in E 3 .

Our Goal

Goal of this talk.– To recover the Nash-Kuiper result on C 1 isometric embeddings from the machinery of the Gromov Integration Theory.

Our main ingredient.– The Gromov Theorem : Let R ⊂ J 1 (M, N) be an open and ample differential relation. Then R satisfies the parametric h-principle i. e.

J : Sol(R) −→ Γ(R) is a weak homotopy equivalence.

Our main obstacles.–

• The isometric relation is not ample

• The isometric relation is closed

Decomposition

• For simplicity M n = [0, 1] n .

• The image of the metric distorsion

∆ := g − f 0 ∗ h., .i E

Definition.– A map f : (M ⁿ , g) −→ ^C

E ^q is an isometry if f ^∗ h., .i = g.

In a local coordinate system x = (x ₁ , ..., x _n ) : 1 ≤ i ≤ j ≤ n, h ∂f

∂x _i (x ), ∂f

∂x _j (x )i = g _ij (x ) The Janet dimension :

s _n = n(n + 1)

Definition.– A map f : (M ⁿ , g) −→ ^C

E ^q is an isometry if f ^∗ h., .i = g.

In a local coordinate system x = (x ₁ , ..., x _n ) : 1 ≤ i ≤ j ≤ n, h ∂f

∂x _i (x ), ∂f

∂x _j (x )i = g _ij (x )

s _n = n(n + 1)

Definition.– A map f : (M ⁿ , g) −→ ^C

E ^q is an isometry if f ^∗ h., .i = g.

In a local coordinate system x = (x ₁ , ..., x _n ) : 1 ≤ i ≤ j ≤ n, h ∂f

∂x _i (x ), ∂f

∂x _j (x )i = g _ij (x ) The Janet dimension :

s _n = n(n + 1)

Definition.– A map f : (M ⁿ , g) −→ ^C

E ^q is called strictly short if f ^∗ h., .i < g.

Theorem (1954-55-86).– Let M ⁿ be a compact Riemannian manifold and f ₀ : (M ⁿ , g) ^C

−→ E ^q be a strictly short

embedding. Then, for every > 0, there exists a C ¹

isometric embedding f : (M ⁿ , g) −→ E ^q such that

kf − f ₀ k _C

Definition.– A map f : (M ⁿ , g) −→ ^C

E ^q is called strictly short if f ^∗ h., .i < g.

Theorem (1954-55-86).– Let M ⁿ be a compact Riemannian manifold and f ₀ : (M ⁿ , g) −→ ^C

E ^q be a strictly short

embedding. Then, for every > 0, there exists a C ¹

isometric embedding f : (M ⁿ , g) −→ E ^q such that

kf − f ₀ k _C

C ¹ -isometric embedding of the unit sphere of E ³ in a ball of

Sphere Eversion.– The sphere S ² ⊂ E ³ can be turned

inside out by a regular homotopy of isometric C ¹

Definition.– Any quotient E ² /Λ of the Euclidean 2-space by a lattice Λ ⊂ E ² is called a flat torus

Flat Tori.– Any flat torus E ² /Λ admits a C ¹ isometric

embedding in E ³ .

Definition.– Any quotient E ² /Λ of the Euclidean 2-space by a lattice Λ ⊂ E ² is called a flat torus

Flat Tori.– Any flat torus E ² /Λ admits a C ¹ isometric

embedding in E ³ .

Goal of this talk.– To recover the Nash-Kuiper result on C ¹ isometric embeddings from the machinery of the Gromov Integration Theory.

Our main ingredient.– The Gromov Theorem : Let R ⊂ J ¹ (M, N) be an open and ample differential relation. Then R satisfies the parametric h-principle i. e.

• For simplicity M ⁿ = [0, 1] ⁿ .

∆ := g − f ₀ ^∗ h., .i _E

lies inside the positive cone M of inner products of E ⁿ .

• There exist S ≥ ⁿ⁽ⁿ⁺¹⁾ ₂ linear forms ` ₁ , . . . ` _S of E ⁿ such that

g − f ₀ ^∗ h., .i _E

ρ _j ` _j ⊗ ` _j

where ρ _j > 0.

` ₁ , ..., ` _S .

along the n directions of the coordinates in [0, 1] ⁿ .

• This will produce S intermediary maps f ₁ , ..., f _S such that

g − f ₁ ^∗ h., .i _E

≈ρ ₂ ` ² ₂ +ρ ₃ ` ² ₃ +...+ρ _S ` ² _S g − f ₂ ^∗ h., .i _E

≈ ρ 3 ` ² ₃ +...+ρ _S ` ² _S

g − f _S−1 ^∗ h., .i _E

≈ ρ _S ` ² _S g − f _S ^∗ h., .i _E

• The map f := f _S is then a solution of R e = Op(R).

f _j ^∗ h., .i _E

− f _j−1 ^∗ h., .i _E

=(g − f _j−1 ^∗ h., .i _E

) − (g − f _j ^∗ h., .i _E

≈ ρ _j ` _j ⊗ ` _j .

Hence, the fundamental problem is the following : Fundamental Problem.– Given a positive function ρ, a linear form ` 6= 0 and an embedding f ₀ how to build an other embedding f such that

f ^∗ h., .i _E

where µ := f ₀ ^∗ h., .i _E

f ₀ : [0, 1] −→ E ^q be an embedding, ρ a positive function,

` 6= 0 a linear form on R , how to build an other embedding f : [0, 1] −→ E ^q such that

∀u ∈ [0, 1], kf ⁰ (u)k ² ≈ kf ₀ ⁰ (u)k ² + ρ(u)` ² (∂ _u ) ?

kf ₀ ⁰ (u)k ² + ρ(u)` ² (∂ u ).

∀s ∈ R / Z , h _u (s) := r (u)e ^iα(u) ^cos(2πs)

r (u)e ^iα(u) ^cos(2πs) ds = f ₀ ⁰ (u).

∀s ∈ R / Z , h _u (s) := r (u)e ^iα(u) ^cos(2πs) and α(u) > 0 is such that

r (u)e ^iα(u) ^cos(2πs) ds = f ₀ ⁰ (u).

• The convex integration formula : f (t) := f ₀ (0) +

r (u)e ^iα(u) ^{cos 2π} N u du.

where e ^iθ := cos θ t + sin θ n with t := ^f

k and n is a unit normal to the curve f ₀ .

fundamental problem. Its speed kf ⁰ k is equal to the given function r = (kf ₀ ⁰ k ² + ρ` ² (∂ c ))