From Isometric Embeddings to C 1 Fractals Francis Lazarus CNRS, - PowerPoint PPT Presentation

John F . Nash Nicolaas Kuiper Nash-Kuiper theorem, 1954-55 If h : ( M n , g ) → E k with k > n is a short embedding ( h ∗ �· , ·� E k < g ), then ∀ ε > 0 there exists a C 1 isometric f : ( M n , g ) → E k s.t. � f − h � C 0 < ε

Nash-Kuiper theorem, 1954-55 If h : ( M n , g ) → E k with k > n is a short embedding ( h ∗ �· , ·� E k < g ), then ∀ ε > 0 there exists a C 1 isometric f : ( M n , g ) → E k s.t. � f − h � C 0 < ε × ( ε/ 2 ) − →

Nash-Kuiper theorem, 1954-55 If h : ( M n , g ) → E k with k > n is a short embedding ( h ∗ �· , ·� E k < g ), then ∀ ε > 0 there exists a C 1 isometric f : ( M n , g ) → E k s.t. � f − h � C 0 < ε − →

Nash’s Method in a Nutshell Let h s.t. g − h ∗ �· , ·� E k is a metric, i.e. h is short.

Nash’s Method in a Nutshell Let h s.t. g − h ∗ �· , ·� E k is a metric, i.e. h is short. Choose a locally finite cover of Sym + n by simplices. g i = ℓ 2 i , 1 + ℓ 2 i , 2 + ℓ 2 i , 3 g − h ∗ �· , ·� E 3 � � � g ( p ) − h ∗ �· , ·� E k ( p ) = a i , j ( p ) ℓ 2 ϕ σ ( p ) α i ( p ) g i = i , j σ i ∈ σ i , j

Nash’s Method in a Nutshell g ( p ) − h ∗ �· , ·� E k ( p ) = � i , j a i , j ( p ) ℓ 2 i , j Step i , j : Replace h by √ a i , j � � h i , j = h + cos ( N i , j ℓ i , j ) u + sin ( N i , j ℓ i , j ) v N i , j u v ∗ �· , ·� E k − h ∗ �· , ·� E k = a i , j ( p ) ℓ 2 h i , j i , j + O ( 1 / N i , j )

Nash’s Method in a Nutshell g ( p ) − h ∗ �· , ·� E k ( p ) = � i , j a i , j ( p ) ℓ 2 i , j Step i , j : Replace h by √ a i , j � � h i , j = h + cos ( N i , j ℓ i , j ) u + sin ( N i , j ℓ i , j ) v N i , j u v ∗ �· , ·� E k − h ∗ �· , ·� E k = a i , j ( p ) ℓ 2 h i , j i , j + O ( 1 / N i , j ) Stage = all steps i , j � h 1 .

Nash’s Method in a Nutshell g ( p ) − h ∗ �· , ·� E k ( p ) = � i , j a i , j ( p ) ℓ 2 i , j Step i , j : Replace h by √ a i , j � � h i , j = h + cos ( N i , j ℓ i , j ) u + sin ( N i , j ℓ i , j ) v N i , j u v ∗ �· , ·� E k − h ∗ �· , ·� E k = a i , j ( p ) ℓ 2 h i , j i , j + O ( 1 / N i , j ) Stage = all steps i , j � h 1 . Repeating the stages we get h 1 , h 2 , . . . . Choosing the N i , j large C 1 → h ∞ with h ∞ a C 1 isometric embedding. enough h k

Nash’s Method Revised by Kuiper g ( p ) − h ∗ �· , ·� E k ( p ) = � i , j a i , j ( p ) ℓ 2 i , j Step i , j : In a suitable chart, replace h by √ a i , j h i , j = h − a i , j sin ( N i , j ℓ i , j − a i , j sin ( 2 N i , j ℓ i , j ) ∂ f ∂ x + 4 sin ( 2 N i , j ℓ i , j )) w 4 N i , j N i , j w ∗ �· , ·� E k − h ∗ �· , ·� E k = a i , j ( p ) ℓ 2 h i , j i , j + small terms Stage = all steps i , j � h 1 . Repeating the stages we get h 1 , h 2 , . . . . Choosing the N i , j large C 1 → h ∞ with h ∞ a C 1 isometric embedding. enough h k

Can you guess the shape of h ∞ ?

Sphere Rigidity: A Paradox 1 Historical Background 2 3 Nash’s Isometric Embedding Gromov’s Point of View 4 Application to Isometric Embeddings 5 The C 1 Fractal Structure 6

From PDEs to Differential Relations Misha Gromov f : ( S , g ) → E 3 is an isometry if � ∂ f ∂ x , ∂ f � ∂ f ∂ x , ∂ f � ∂ f ∂ y , ∂ f ∂ x � E 3 = E , ∂ y � E 3 = F , ∂ y � E 3 = G

From PDEs to Differential Relations Misha Gromov f : ( S , g ) → E 3 is an isometry if � ∂ f ∂ x , ∂ f � ∂ f ∂ x , ∂ f � ∂ f ∂ y , ∂ f ∂ x � E 3 = E , ∂ y � E 3 = F , ∂ y � E 3 = G ⇔ j 1 f : p = ( x , y ) �→ ( p , f ( p ) , ∂ f ∂ x ( p ) , ∂ f ∂ y ( p )) satisfies R ( j 1 f ) = ( 0 , 0 , 0 ) , where R ( p , f , u , v ) = ( � u , u � E 3 − E ( p ) , � u , v � E 3 − F ( p ) , � v , v � E 3 − G ( p ))

From PDEs to Differential Relations Idea: Decouple the derivatives from the map to solve R ( p , f ( p ) , u ( p ) , v ( p )) = 0 with p ∈ S , f ( p ) ∈ E 3 , u ( p ) ∈ T f ( p ) E 3 , v ( p ) ∈ T f ( p ) E 3 .

From PDEs to Differential Relations Idea: Decouple the derivatives from the map to solve R ( p , f ( p ) , u ( p ) , v ( p )) = 0 with p ∈ S , f ( p ) ∈ E 3 , u ( p ) ∈ T f ( p ) E 3 , v ( p ) ∈ T f ( p ) E 3 . A solution to R = 0 is said formal. A solution of the form j 1 f is a true (or holonomic) solution.

From PDEs to Differential Relations Idea: Decouple the derivatives from the map to solve R ( p , f ( p ) , u ( p ) , v ( p )) = 0 with p ∈ S , f ( p ) ∈ E 3 , u ( p ) ∈ T f ( p ) E 3 , v ( p ) ∈ T f ( p ) E 3 . A solution to R = 0 is said formal. A solution of the form j 1 f is a true (or holonomic) solution. R = 0, or more generally a differential relation, satisfies the h-principle if every formal solution is homotopic (through formal solutions) to a true solution.

The h-Principle R = 0, or more generally a differential relation, satisfies the h-principle if every formal solution is homotopic (through formal solutions) to a true solution. Existence of formal solutions are of a topological nature. A (counter-)example: There is no immersion S 2 → R 2 . I := { ( p , f , L ) | p ∈ S 2 , f ∈ R 2 , L ∈ L ( T p S 2 , T f R 2 ) : rank L = 2 }

The h-Principle R = 0, or more generally a differential relation, satisfies the h-principle if every formal solution is homotopic (through formal solutions) to a true solution. Existence of formal solutions are of a topological nature. A (counter-)example: There is no immersion S 2 → R 2 . I := { ( p , f , L ) | p ∈ S 2 , f ∈ R 2 , L ∈ L ( T p S 2 , T f R 2 ) : rank L = 2 }

The h-Principle R = 0, or more generally a differential relation, satisfies the h-principle if every formal solution is homotopic (through formal solutions) to a true solution. Existence of formal solutions are of a topological nature. A (counter-)example: There is no immersion S 2 → R 2 . p f ( p ) I := { ( p , f , L ) | p ∈ S 2 , f ∈ R 2 , L ∈ L ( T p S 2 , T f R 2 ) : rank L = 2 }

1-dimensional Convex Integration Convex integration is a tool invented by Gromov to prove the h-principle for many differential relations. R p A simple observation

1-dimensional Convex Integration Convex integration is a tool invented by Gromov to prove the h-principle for many differential relations. p 3 R p 2 � p = α i p i p p 1 A simple observation

1-dimensional Convex Integration Convex integration is a tool invented by Gromov to prove the h-principle for many differential relations. R γ � p = γ, γ ⊂ R p I A simple observation

1-dimensional Convex Integration Convex integration is a tool invented by Gromov to prove the h-principle for many differential relations. R γ � p = S 1 γ, γ ⊂ R p A simple observation

1-dimensional Convex Integration Lemma (Gromov, 1973) Let f 0 : I → R 3 . For all x ∈ I , suppose R x ⊂ R 3 is open and f ′ 0 ( x ) ∈ IntConv ( R x ) . Then, ∀ ε > 0, there exists a true solution f of R = ∪ x R x s.t. � f − f 0 � C 0 < ε R I f ′ 0

1-dimensional Convex Integration Step 1 Build a continuous family of loops I × S 1 γ : → R ( x , s ) �→ γ x ( s ) such that � f ′ ∀ x ∈ I , 0 ( x ) = S 1 γ x γ x R f ′ 0 ( x ) x

1-dimensional Convex Integration Step 2 Put � x f ( x ) := f 0 ( 0 ) + γ s ( { Ns } ) ds 0 where N ∈ N ∗ et { Ns } is the fractional part of Ns . γ x ( { Nx } ) x

1-dimensional Convex Integration Step 2 Put � x f ( x ) := f 0 ( 0 ) + γ s ( { Ns } ) ds 0 where N ∈ N ∗ et { Ns } is the fractional part of Ns . γ x ( { Nx } ) x f ′ ( x ) = γ x ( { Nx } ) ∈ R x and ⌊ Nx ⌋ ⌊ Nx ⌋ i + 1 � 1 0 ( i N � � N f ′ f ( x ) ≈ f 0 ( 0 )+ γ s ( { Ns } ) ds ≈ f 0 ( 0 )+ N ) ≈ f 0 ( x ) i i = 0 i = 0 N

1-dimensional Convex Integration Step 2 Put � x f ( x ) := f 0 ( 0 ) + γ s ( { Ns } ) ds 0 where N ∈ N ∗ et { Ns } is the fractional part of Ns . γ x ( { Nx } ) x f ′ ( x ) = γ x ( { Nx } ) ∈ R x and ⌊ Nx ⌋ ⌊ Nx ⌋ i + 1 � 1 0 ( i N � � N f ′ f ( x ) ≈ f 0 ( 0 )+ γ s ( { Ns } ) ds ≈ f 0 ( 0 )+ N ) ≈ f 0 ( x ) i i = 0 i = 0 N

The h-Principle for Ample Relations Theorem (Gromov, 1973) Let R ⊂ J 1 ( M , N ) be an open and ample differential relation. Then the inclusion of true solutions into the space of formal solutions is a weak homotopy equivalence. ample non-ample The relation of immersions The differential relation of immersions from M m to N n satisfies the h-principle for n > m . In particular, S 2 can be everted in R 3 .

Sphere Rigidity: A Paradox 1 Historical Background 2 3 Nash’s Isometric Embedding Gromov’s Point of View 4 Application to Isometric Embeddings 5 The C 1 Fractal Structure 6

Back to Isometries The relation R iso := { R = ( 0 , 0 , 0 ) } of isometries S 2 → E 3 is neither open nor ample and S 2 is 2-dimensional . R ( p , q , u , v ) = ( � u , u � E 3 − E ( p ) , � u , v � E 3 − F ( p ) , � v , v � E 3 − G ( p ))

Back to Isometries The relation R iso := { R = ( 0 , 0 , 0 ) } of isometries S 2 → E 3 is neither open nor ample and S 2 is 2-dimensional . R ( p , q , u , v ) = ( � u , u � E 3 − E ( p ) , � u , v � E 3 − F ( p ) , � v , v � E 3 − G ( p )) Find f 0 s.t. f ′ 0 ∈ IntConv ( R iso ) , i.e. f 0 is short. 1

Back to Isometries The relation R iso := { R = ( 0 , 0 , 0 ) } of isometries S 2 → E 3 is neither open nor ample and S 2 is 2-dimensional . R ( p , q , u , v ) = ( � u , u � E 3 − E ( p ) , � u , v � E 3 − F ( p ) , � v , v � E 3 − G ( p )) Find f 0 s.t. f ′ 0 ∈ IntConv ( R iso ) , i.e. f 0 is short. 1 Thicken R iso to get an open relation. 2

Back to Isometries The relation R iso := { R = ( 0 , 0 , 0 ) } of isometries S 2 → E 3 is neither open nor ample and S 2 is 2-dimensional . R ( p , q , u , v ) = ( � u , u � E 3 − E ( p ) , � u , v � E 3 − F ( p ) , � v , v � E 3 − G ( p )) Find f 0 s.t. f ′ 0 ∈ IntConv ( R iso ) , i.e. f 0 is short. 1 Thicken R iso to get an open relation. 2 Use a single coordinate chart. 3

Back to Isometries The relation R iso := { R = ( 0 , 0 , 0 ) } of isometries S 2 → E 3 is neither open nor ample and S 2 is 2-dimensional . R ( p , q , u , v ) = ( � u , u � E 3 − E ( p ) , � u , v � E 3 − F ( p ) , � v , v � E 3 − G ( p )) Find f 0 s.t. f ′ 0 ∈ IntConv ( R iso ) , i.e. f 0 is short. 1 Thicken R iso to get an open relation. 2 Use a single coordinate chart. 3 Deal with boundary conditions. 4

Back to Isometries The relation R iso := { R = ( 0 , 0 , 0 ) } of isometries S 2 → E 3 is neither open nor ample and S 2 is 2-dimensional . R ( p , q , u , v ) = ( � u , u � E 3 − E ( p ) , � u , v � E 3 − F ( p ) , � v , v � E 3 − G ( p )) Find f 0 s.t. f ′ 0 ∈ IntConv ( R iso ) , i.e. f 0 is short. 1 Thicken R iso to get an open relation. 2 Use a single coordinate chart. 3 Deal with boundary conditions. 4 Extend 1D-CI to 2D. 5

Back to Isometries Find f 0 s.t. f ′ 0 ∈ IntConv ( R iso ) , i.e. f 0 is short. Use a single coordinate chart.

Back to Isometries Extend 1D-CI to 2D. 3 g − f ∗ � 0 �· , ·� E 3 = ρ i ℓ i ⊗ ℓ i i = 1 g i = f ∗ i − 1 �· , ·� E 3 + ρ i ℓ i ⊗ ℓ i

Back to Isometries Extend 1D-CI to 2D. γ s ( Ns ) = r ( s ) e i α s cos ( 2 π Ns ) � x f ( x ) := f 0 ( 0 ) + γ s ( { Ns } ) ds 0

Back to Isometries Extend 1D-CI to 2D.

Back to Isometries Extend 1D-CI to 2D. Corrugations

Back to Isometries Thicken R iso to get an open relation. f ′ 0 R iso

Back to Isometries Thicken R iso to get an open relation. R iso f ′

Back to Isometries Thicken R iso to get an open relation. R iso R 1 f ′ 1

Back to Isometries Thicken R iso to get an open relation. R ∞ = R iso R 1 f ′ 1

Back to Isometries Thicken R iso to get an open relation. R ∞ = R iso f ′ R 2 2

Back to Isometries Thicken R iso to get an open relation. R ∞ = R iso R 3 f ′ 3

Back to Isometries Deal with boundary conditions. g g 2 g 1 , 1 g 1 f ∗ 0 � , � E 3 g 1 , 3 g 1 , 2 D D 0 , 3 D 1 , 1 D 1 , 2 D 1 , 3 Compute embeddings f i , j so that f ∗ i , j � , � E 3 ≈ g i , j .

Back to Isometries Deal with boundary conditions. g g 2 g 1 , 1 g 1 f ∗ 0 � , � E 3 g 1 , 3 g 1 , 2 D D 0 , 3 D 1 , 1 D 1 , 2 D 1 , 3 Compute embeddings f i , j so that f ∗ i , j � , � E 3 ≈ g i , j . = ⇒ sequence f 0 , f 1 , 1 , f 1 , 2 , f 1 , 3 , f 2 , 1 , . . . C 1 converging to an isometry.

Sphere Rigidity: A Paradox 1 Historical Background 2 3 Nash’s Isometric Embedding Gromov’s Point of View 4 Application to Isometric Embeddings 5 The C 1 Fractal Structure 6

The C 1 Fractal Structure The IC process applied on a circle of radius < 1 .

The C 1 Fractal Structure The IC process applied on a circle of radius < 1 .

The C 1 Fractal Structure The IC process applied on a circle of radius < 1 .

The C 1 Fractal Structure The IC process applied on a circle of radius < 1 .

The C 1 Fractal Structure The IC process applied on a circle of radius < 1 . � ∞ � � ∀ x ∈ S 1 , e i α k ( x ) cos 2 π N k x n 0 ( x ) = e iA ∞ ( x ) n 0 ( x ) n ∞ ( x ) = k = 1 where n 0 is ⊥ to f 0 and A ∞ ( x ) = � ∞ k = 1 α k ( x ) cos 2 π N k x . k = 0 a k cos ( 2 π b k x )) ≤ ln ( a ) / ln ( b ) + 2 dim H graph ( � ∞

The C 1 Fractal Structure The IC process applied on a circle of radius < 1 . � ∞ � � ∀ x ∈ S 1 , e i α k ( x ) cos 2 π N k x n 0 ( x ) = e iA ∞ ( x ) n 0 ( x ) n ∞ ( x ) = k = 1 where n 0 is ⊥ to f 0 and A ∞ ( x ) = � ∞ k = 1 α k ( x ) cos 2 π N k x . � ∞ � � t 0 � t ∞ � � � = C k n ∞ n 0 k = 0 � cos θ k � sin θ k where C k = and θ k ( x ) = α k ( x ) cos 2 π N k x − sin θ k cos θ k

The C 1 Fractal Structure Let k , j + 1 v k , j + 1 n k , j + 1 ) t = C k , j + 1 · ( v ⊥ ( v ⊥ k , j v k , j n k , j ) t , C k , j + 1 ∈ SO ( 3 ) The corrugation matrix is:   ∞ 3 3 � � � R ( k , i ) = C ℓ, j C k , j   ℓ = k + 1 j = 1 j = i