Flat 2-Tori in E 3

(1)

E³

V.Borrelli

Flat 2-Tori in E ³

(2)

Flat 2-Tori in E³ V.Borrelli

Implementing the Convex Integration Process

C¹Fractals Pictures !

The Hevea Project

Francis Lazarus Boris Thibert Saïd Jabrane

Gipsa-Lab, Grenoble LJK, Grenoble ICJ, Lyon

(3)

E³

V.Borrelli

The initial map

(4)

Step 1 : decomposition of ∆

(5)

E³ V.Borrelli

The choice of the ` _j

(6)

The choice of the ` _j

` ₂ (.) = h 1

5 (e ₁ + 2e ₂ ), .i

E

²

(7)

E³ V.Borrelli

The choice of the ` _j

(8)

Step 2 : Three convex integrations

• From f ₀ and by applying three convex integrations along the directions given by ` ₁ , ` ₂ and ` ₂ , we build iteratively three maps (a salvo) :

f _1,1 , f _1,2 , f _1,3

such that

g − f _1,1 ^∗ h., .i _E

^q

=ρ ₂ ` ² ₂ +ρ ₃ ` ² ₃ + O( _N ¹

1,1

) g − f _1,2 ^∗ h., .i _E

^q

= ρ ₃ ` ² ₃ + O( _N ¹

1,1

) + O( _N ¹

1,2

) g − f _1,3 ^∗ h., .i _E

^q

= 0 + O( _N ¹

1,1

) + O( _N ¹

1,2

) + O( _N ¹

1,3

)

• We choose N _1,1 , N _1,2 and N _1,3 such that :

kh., .i

E

²

− f _1,3 ^∗ h., .i

E

³

k _C

0

≤ 1 2 kh., .i

E

²

− f ₀ ^∗ h., .i

E

³

k _C

0

.

(9)

E³

V.Borrelli

Torus of revolution

(10)

Convex Integration in the First

Direction

(11)

E³

V.Borrelli

Convex Integration in the

Second Direction

(12)

Convex Integration in the Third

Direction

(13)

E³ V.Borrelli

Three directions

(14)

Three directions

The flow in direction 2

(15)

E³ V.Borrelli

Three directions

(16)

A zoom

The vertical scale is exaggerated for emphasis.

(17)

Flat 2-Tori in E³ V.Borrelli Implementing

Step 3 : Iterating the process

• After the salvo n˚ k, the isometric default is divided by 2 ^k

kh., .i

E

²

− f _k,3 ^∗ h., .i

E

³

k _C

0

≤ 1 2 ^k kh., .i

E

²

− f ₀ ^∗ h., .i

E

³

k _C

0

.

• The limit

f _∞ := lim

k−→+∞ f _k,3

is a C ¹ isometric immersion of the flat torus.

(18)

Step 3 : Iterating the process

• We continue the process to create infinite sequence of maps

f ₀ , f _1,1 , f _1,2 , f _1,3 , ... f _k,1 , f _k,2 , f _k,3 , ...

• After the salvo n˚ k, the isometric default is divided by 2 ^k

kh., .i

E

²

− f _k,3 ^∗ h., .i

E

³

k _C

0

≤ 1 2 ^k kh., .i

E

²

− f ₀ ^∗ h., .i

E

³

k _C

0

.

• The limit

f _∞ := lim

k−→+∞ f _k,3

is a C ¹ isometric immersion of the flat torus.

(19)

E³ V.Borrelli

Step 3 : Iterating the process

• We continue the process to create infinite sequence of maps

f ₀ , f _1,1 , f _1,2 , f _1,3 , ... f _k,1 , f _k,2 , f _k,3 , ...

• After the salvo n˚ k, the isometric default is divided by 2 ^k

kh., .i

E

²

− f _k,3 ^∗ h., .i

E

³

k _C

0

≤ 1 2 ^k kh., .i

E

²

− f ₀ ^∗ h., .i

E

³

k _C

0

.

(20)

Another CI in the first direction

(21)

E³

V.Borrelli

And so on... up to the Flat Torus

(22)

Implementing the Convex Integration Process C¹Fractals Pictures !

Nash-Kuiper in a 1D setting !

(23)

E³

V.Borrelli

Nash-Kuiper in a 1D setting !

(24)

Nash-Kuiper in a 1D setting !

(25)

E³

V.Borrelli

Nash-Kuiper in a 1D setting !

(26)

Spectrum of the Gauss map

(27)

E³

V.Borrelli

Moscow University

(28)

Riesz Products

Frigyes Riesz

• These spectra arise from the study of Riesz products :

u 7−→

∞

Y

j=0

(1 + α _j cos(2πb ^j u)).

In that product, b ≥ 3 is an integer and (α _j ) j∈ N is a sequence of real numbers such that, for every j ∈ N ,

|α _j | ≤ 1.

• Here, products of corrugation matrices take the place of

Riesz products.

(29)

E³ V.Borrelli

Riesz Products

Frigyes Riesz

• These spectra arise from the study of Riesz products :

u 7−→

∞

Y

j=0

(1 + α _j cos(2πb ^j u)).

In that product, b ≥ 3 is an integer and (α _j ) j∈ N is a

sequence of real numbers such that, for every j ∈ N ,

(30)

Corrugation matrices

• Let C _k : S ¹ −→ O(2) be the matrix valued map such that

∀u ∈ S ¹ ,

t _k (u) n _k (u)

= C _k (u) ·

t _k ₋₁ (u) n k−1 (u)

We call C _k a corrugation matrix.

(31)

Structure of the Gauss Map

• Corrugation matrices encode the data of the successive convex integrations :

cos θ (u) sin θ (u)

• The Gauss map n ∞ of the limit embedding is given by a Riesz-like product :

t _∞ n ∞

=

∞

Y

k =1

C _k

!

· t ₀

n ₀

(32)

Structure of the Gauss Map

• Corrugation matrices encode the data of the successive convex integrations :

C _k (u) :=

cos θ _k (u) sin θ _k (u)

− sin θ k (u) cos θ k (u)

with

θ k (u) = arg h _u ({N _k u}) = α k cos(2πN _k u).

• The Gauss map n ∞ of the limit embedding is given by a Riesz-like product :

t _∞ n ∞

=

∞

Y

k=1

C _k

!

· t ₀

n ₀

(33)

E³ V.Borrelli

From the circle to the torus

We denote by C _k,j the SO(3) matrix such that :

(34)

From the circle to the torus

Let f ∞ : T ² −→ E ³ be the limit of the maps : f ₀ , f _1,1 , f _1,2 , f _1,3 , f _2,1 , f _2,2 , f _2,3 , ...

We have



 v _∞ ^⊥ v ∞

n ∞



 =

∞

Y

k=1





3 Y

j=1

C _k,j



 ·



 v ₀ ^⊥

v ₀ n ₀





Beware ! – Unlike the 1-dimensional case, the analytic

expressions of the C _k,j ’s are simply ugly... but fortunately,

their asymptotic expressions are nice.

(35)

From the circle to the torus

Let f ∞ : T ² −→ E ³ be the limit of the maps : f ₀ , f _1,1 , f _1,2 , f _1,3 , f _2,1 , f _2,2 , f _2,3 , ...

We have

 v _∞ ^⊥ v



=

∞

Y

 3

Y C



·  v ₀ ^⊥ v



but fortunately,

their asymptotic expressions are nice.

(36)

From the circle to the torus

Let f ∞ : T ² −→ E ³ be the limit of the maps : f ₀ , f _1,1 , f _1,2 , f _1,3 , f _2,1 , f _2,2 , f _2,3 , ...

We have



 v _∞ ^⊥ v ∞

n ∞



 =

∞

Y

k=1





3 Y

j=1

C _k,j



 ·



 v ₀ ^⊥

v ₀ n ₀





Beware ! – Unlike the 1-dimensional case, the analytic

expressions of the C _k,j ’s are simply ugly... but fortunately,

their asymptotic expressions are nice.

(37)

Implementing the Convex Integration Process C¹Fractals

Loss of derivatives

• In the one dimensional setting we have :

t

• In particular

∂f

∂t (t) = r (t)e ^iα(t) ^{cos 2πNt} ,

therefore, if f ₀ is C ^k then f is C ^k also.

(38)

Loss of derivatives

• In the one dimensional setting we have :

f (t) := f ₀ (0) + Z t

0 r (u)e ^iα(u) ^{cos 2πNu} du.

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

0 0

kf

₀⁰

k .

• In particular

∂f

∂t (t) = r (t)e ^iα(t) ^{cos 2πNt} ,

therefore, if f ₀ is C ^k then f is C ^k also.

(39)

Loss of derivatives

• In the two dimensional setting we have :

f (t, s) := f ₀ (0, s) + Z t

0 r (u, s)e ^iα(u,s) ^{cos 2πNu} du + gluing term

• The integral over the variable t can not recover the loss of

derivative due to the presence of the partial derivative ∂ s f in

the definition of n. Therefore if f ₀ is C ^k then, generically, f is

C ^k ⁻¹ only.

(40)

Loss of derivatives

• In the two dimensional setting we have :

f (t, s) := f ₀ (0, s) + Z t

0 r (u, s)e ^iα(u,s) ^{cos 2πNu} du + gluing term

where e ^iθ := cos θ t + sin θ n with

t := ∂ _t f ₀

k∂ _t f ₀ k and n := ∂ _t f ₀ ∧ ∂ _s f ₀ k∂ _t f ₀ ∧ ∂ s f ₀ k

• The integral over the variable t can not recover the loss of

derivative due to the presence of the partial derivative ∂ s f in

the definition of n. Therefore if f ₀ is C ^k then, generically, f is

C ^k−1 only.

(41)

E³ V.Borrelli

Corrugation Theorem

Corrugation Theorem (∼, Jabrane, Lazarus, Thibert).–

For every p ∈ T ² , we have

C _k,j+1 (p) = L _k,j+1 (p) · R _k,j (p) where

L _k,j+1 :=





cos θ _k,j+1 0 sin θ _k,j+1

0 1 0

− sin θ _k,j+1 0 cos θ _k,j+1



+ O 1

N _k,j+1

with θ _k,j+1 (p) := α _k (p) cos(2πN _k X _j+1 ) and

 − sin β j − cos β j 0 

(42)

It is time for pictures !

The new CSCS (Swiss National Supercomputing Centre) building

in Lugano-Cornaredo.

(43)

E³

V.Borrelli

A wide-angle view

(44)

As an arch

(45)

E³

V.Borrelli

Tangential view

(46)

The bobsleigh track

(47)

E³

V.Borrelli

The tokamak

(48)

"Cinq colonnes à la une"

(49)

E³

V.Borrelli

Guts and tripe

(50)

In the belly of the beast

(51)

E³

V.Borrelli

The eye

(52)

Flexible electrical conduits

(53)

E³

V.Borrelli

The truck suspension

(54)

Velodrome

(55)

E³

V.Borrelli

The tire

(56)

The landing

(57)

E³

V.Borrelli

The rope

(58)

UFO

YouTube : Flat Torus

(59)

E³ V.Borrelli

Zoom !

(60)

Zoom !

First integration : 8 oscillations

(61)

E³ V.Borrelli

Zoom !

(62)

Zoom !

Zoom in on the second integration

(63)

E³ V.Borrelli

Zoom !

(64)

Zoom !

Third integration : 4096 oscillations

(65)

E³ V.Borrelli

Zoom !

(66)

Zoom !

Zoom in on the third integration

(67)

E³ V.Borrelli

Zoom !

(68)

Zoom !

The fourth integration : 524 288 oscillations

(69)

E³ V.Borrelli

Zoom !

(70)

Zoom !

The fourth integration : 524 288 oscillations

(71)

E³ V.Borrelli

Zoom !

(72)

Zoom !

The fifth integration : 2 097 152 oscillations

(73)

E³ V.Borrelli

Zoom !

(74)

Zoom !

The sixth integration : 16 777 216 oscillations

(75)

E³ V.Borrelli

Zoom !

(76)

Zoom !

Zoom in on the sixth integration

(77)

E³ V.Borrelli

Zoom !

(78)

"Dites 33 !"

(79)

E³

V.Borrelli

"Dites 33 !"

(80)

"Dites 33 !"

(81)

E³

V.Borrelli

"Dites 33 !"

(82)

"Dites 33 !"

(83)

E³

V.Borrelli

"Dites 33 !"

(84)

"Dites 33 !"

(85)

E³

V.Borrelli

"Dites 33 !"

(86)

"Dites 33 !"

(87)

E³

V.Borrelli

"Dites 33 !"

(88)

Artefacts

(89)

E³

V.Borrelli

Artefacts

(90)

Artefacts

(91)

E³ V.Borrelli

Corrugations in real life ?

(92)

Corrugations in real life ?

A whelk

(93)

E³ V.Borrelli

Corrugations in real life ?

(94)

Corrugations in real life ?

(95)

E³

V.Borrelli

Flat 2-Tori in E 3

Flat 2-Tori in E 3

The Hevea Project

Francis Lazarus Boris Thibert Saïd Jabrane

Gipsa-Lab, Grenoble LJK, Grenoble ICJ, Lyon

The initial map

Step 1 : decomposition of ∆

The choice of the ` j

The choice of the ` j

` 2 (.) = h 1

5 (e 1 + 2e 2 ), .i

E

The choice of the ` j

Step 2 : Three convex integrations

• From f 0 and by applying three convex integrations along the directions given by ` 1 , ` 2 and ` 2 , we build iteratively three maps (a salvo) :

f 1,1 , f 1,2 , f 1,3

such that

g − f 1,1 ∗ h., .i E

=ρ 2 ` 2 2 +ρ 3 ` 2 3 + O( N 1

) g − f 1,2 ∗ h., .i E

= ρ 3 ` 2 3 + O( N 1

) + O( N 1

) g − f 1,3 ∗ h., .i E

= 0 + O( N 1

) + O( N 1

) + O( N 1

)

• We choose N 1,1 , N 1,2 and N 1,3 such that :

kh., .i

E

− f 1,3 ∗ h., .i

E

k C

≤ 1 2 kh., .i

E

− f 0 ∗ h., .i

E

k C

.

Torus of revolution

Convex Integration in the First

Direction

Convex Integration in the

Second Direction

Convex Integration in the Third

Direction

Three directions

Three directions

The flow in direction 2

Three directions

A zoom

The vertical scale is exaggerated for emphasis.

Step 3 : Iterating the process

• After the salvo n˚ k, the isometric default is divided by 2 k

kh., .i

E

− f k,3 ∗ h., .i

E

k C

≤ 1 2 k kh., .i

E

− f 0 ∗ h., .i

E

k C

.

• The limit

f ∞ := lim

k−→+∞ f k,3

is a C 1 isometric immersion of the flat torus.

Step 3 : Iterating the process

• We continue the process to create infinite sequence of maps

f 0 , f 1,1 , f 1,2 , f 1,3 , ... f k,1 , f k,2 , f k,3 , ...

• After the salvo n˚ k, the isometric default is divided by 2 k

kh., .i

E

− f k,3 ∗ h., .i

E

k C

≤ 1 2 k kh., .i

E

Flat 2-Tori in E ³

The choice of the ` _j

The choice of the ` _j

` ₂ (.) = h 1

5 (e ₁ + 2e ₂ ), .i

The choice of the ` _j

• From f ₀ and by applying three convex integrations along the directions given by ` ₁ , ` ₂ and ` ₂ , we build iteratively three maps (a salvo) :

f _1,1 , f _1,2 , f _1,3

g − f _1,1 ^∗ h., .i _E

=ρ ₂ ` ² ₂ +ρ ₃ ` ² ₃ + O( _N ¹

) g − f _1,2 ^∗ h., .i _E

= ρ ₃ ` ² ₃ + O( _N ¹

) + O( _N ¹

) g − f _1,3 ^∗ h., .i _E

= 0 + O( _N ¹

) + O( _N ¹

) + O( _N ¹

• We choose N _1,1 , N _1,2 and N _1,3 such that :

− f _1,3 ^∗ h., .i

k _C

− f ₀ ^∗ h., .i

k _C

• After the salvo n˚ k, the isometric default is divided by 2 ^k

− f _k,3 ^∗ h., .i

k _C

≤ 1 2 ^k kh., .i

− f ₀ ^∗ h., .i

k _C

f _∞ := lim

k−→+∞ f _k,3

is a C ¹ isometric immersion of the flat torus.

f ₀ , f _1,1 , f _1,2 , f _1,3 , ... f _k,1 , f _k,2 , f _k,3 , ...

• After the salvo n˚ k, the isometric default is divided by 2 ^k

− f _k,3 ^∗ h., .i

k _C

≤ 1 2 ^k kh., .i

− f ₀ ^∗ h., .i

k _C

f _∞ := lim

k−→+∞ f _k,3

is a C ¹ isometric immersion of the flat torus.

f ₀ , f _1,1 , f _1,2 , f _1,3 , ... f _k,1 , f _k,2 , f _k,3 , ...

• After the salvo n˚ k, the isometric default is divided by 2 ^k

− f _k,3 ^∗ h., .i

k _C

≤ 1 2 ^k kh., .i

− f ₀ ^∗ h., .i

k _C

(1 + α _j cos(2πb ^j u)).

In that product, b ≥ 3 is an integer and (α _j ) j∈ N is a sequence of real numbers such that, for every j ∈ N ,

|α _j | ≤ 1.

(1 + α _j cos(2πb ^j u)).

In that product, b ≥ 3 is an integer and (α _j ) j∈ N is a

• Let C _k : S ¹ −→ O(2) be the matrix valued map such that

∀u ∈ S ¹ ,

t _k (u) n _k (u)

= C _k (u) ·

t _k ₋₁ (u) n k−1 (u)

We call C _k a corrugation matrix.

t _∞ n ∞