Self-similar solutions to extension and approximation problems - PowerPoint PPT Presentation

Self-similar solutions to extension and approximation problems Robert Young New York University (joint with Larry Guth and Stefan Wenger) June 2019 Parts of this work were supported by NSF grant DMS 1612061, the Sloan Foundation, and the Natural Sciences and Engineering Research Council of Canada

Outline ◮ Kaufman’s construction: rank–1 maps from the cube to the square ◮ Topologically nontrivial low-rank maps ◮ H¨ older signed-area preserving maps ◮ H¨ older maps to the Heisenberg group

Kaufman’s construction Theorem (Kaufman) There is a Lipschitz map f : [0 , 1] 3 → [0 , 1] 2 which is surjective and satisfies rank Df ≤ 1 almost everywhere.

Kaufman’s construction Theorem (Kaufman) There is a Lipschitz map f : [0 , 1] 3 → [0 , 1] 2 which is surjective and satisfies rank Df ≤ 1 almost everywhere. By Sard’s Theorem, if f is smooth and rank Df ≤ 1 everywhere, then f ([0 , 1] 3 ) has measure zero, so there is no smooth map satisfying the theorem.

Kaufman’s construction Theorem (Kaufman) There is a Lipschitz map f : [0 , 1] 3 → [0 , 1] 2 which is surjective and satisfies rank Df ≤ 1 almost everywhere. By Sard’s Theorem, if f is smooth and rank Df ≤ 1 everywhere, then f ([0 , 1] 3 ) has measure zero, so there is no smooth map satisfying the theorem. But there is a self-similar map!

Rank–1 maps are topologically trivial Theorem (Wenger–Y.) Let M be a simply-connected manifold and let f : M → N be a Lipschitz map such that rank Df ≤ 1 almost everywhere. Then there is an R –tree T such that f = g ◦ h, where h : M → T and g : T → N are Lipschitz maps.

Topologically nontrivial rank–( n − 1) maps We say a Lipschitz map to an n –manifold with rank Df ≤ n − 1 almost everywhere is corank– 1.

Topologically nontrivial rank–( n − 1) maps We say a Lipschitz map to an n –manifold with rank Df ≤ n − 1 almost everywhere is corank– 1. Theorem (Wenger–Y.) Let n ≥ 4 . There is a corank– 1 map f : S n +1 → S n such that f is not null-homotopic.

Topologically nontrivial rank–( n − 1) maps We say a Lipschitz map to an n –manifold with rank Df ≤ n − 1 almost everywhere is corank– 1. Theorem (Wenger–Y.) Let n ≥ 4 . There is a corank– 1 map f : S n +1 → S n such that f is not null-homotopic. This follows from: Extension Lemma (Wenger–Y.) Let α : S m − 2 → S n − 2 be a map with m > n. The suspension Σ α : S m − 1 → S n − 1 extends to a corank–1 map β : D m → D n .

Suspensions Let X be a topological space. The suspension Σ X is the space Σ X = X × [0 , 1] / ∼ , where ∼ identifies all the points in X × 0 and identifies all the points in X × 1.

Suspensions Let X be a topological space. The suspension Σ X is the space Σ X = X × [0 , 1] / ∼ , where ∼ identifies all the points in X × 0 and identifies all the points in X × 1. In particular, Σ S m = S m +1 for all m .

Suspensions Let X be a topological space. The suspension Σ X is the space Σ X = X × [0 , 1] / ∼ , where ∼ identifies all the points in X × 0 and identifies all the points in X × 1. In particular, Σ S m = S m +1 for all m . For f : S m → S n , let Σ f : S m +1 → S n +1 , Σ f ( x , t ) = ( f ( x ) , t ) .

Proof of Theorem given Extension Lemma Theorem Let n ≥ 4 . There is a corank– 1 map f : S n +1 → S n such that f is not null-homotopic.

Proof of Theorem given Extension Lemma Theorem Let n ≥ 4 . There is a corank– 1 map f : S n +1 → S n such that f is not null-homotopic. Proof. ◮ Let h : S 3 → S 2 be the Hopf fibration. Then Σ k h is homotopically nontrivial for every k .

Proof of Theorem given Extension Lemma Theorem Let n ≥ 4 . There is a corank– 1 map f : S n +1 → S n such that f is not null-homotopic. Proof. ◮ Let h : S 3 → S 2 be the Hopf fibration. Then Σ k h is homotopically nontrivial for every k . ◮ Let k > 0. By the Extension Lemma, there is a corank–1 extension β : D 4+ k → D 3+ k of Σ k h .

Proof of Theorem given Extension Lemma Theorem Let n ≥ 4 . There is a corank– 1 map f : S n +1 → S n such that f is not null-homotopic. Proof. ◮ Let h : S 3 → S 2 be the Hopf fibration. Then Σ k h is homotopically nontrivial for every k . ◮ Let k > 0. By the Extension Lemma, there is a corank–1 extension β : D 4+ k → D 3+ k of Σ k h . ◮ Let f : S 4+ k → S 3+ k be two copies of β glued along the equator. This map has corank 1 and f ∼ Σ(Σ k h ) = Σ k +1 h .

Proof of Theorem given Extension Lemma Theorem Let n ≥ 4 . There is a corank– 1 map f : S n +1 → S n such that f is not null-homotopic. Proof. ◮ Let h : S 3 → S 2 be the Hopf fibration. Then Σ k h is homotopically nontrivial for every k . ◮ Let k > 0. By the Extension Lemma, there is a corank–1 extension β : D 4+ k → D 3+ k of Σ k h . ◮ Let f : S 4+ k → S 3+ k be two copies of β glued along the equator. This map has corank 1 and f ∼ Σ(Σ k h ) = Σ k +1 h . It remains to prove the Extension Lemma.

Higher dimensions (in progress) Theorem Let h : S 3 → S 2 be the Hopf fibration. Then Σ 2 h : S 5 → S 4 is homotopic to a corank– 1 map. Conjecture/Theorem (Guth–Y., in progress) Let k ≥ 1 . Then there is a corank–k map homotopic to Σ 2 k h.

Higher dimensions (in progress) Theorem Let h : S 3 → S 2 be the Hopf fibration. Then Σ 2 h : S 5 → S 4 is homotopic to a corank– 1 map. Conjecture/Theorem (Guth–Y., in progress) Let k ≥ 1 . Then there is a corank–k map homotopic to Σ 2 k h. This is sharp; Σ 2 k h is not homotopic to a Lipschitz map with corank k + 1 and Σ 2 k − 1 h is not homotopic to a Lipschitz map with corank k.

Signed-area preserving maps For a closed curve γ , let σ ( γ ) be the signed area of γ (the integral of the winding number of γ ).

Signed-area preserving maps For a closed curve γ , let σ ( γ ) be the signed area of γ (the integral of the winding number of γ ). A map f : D 2 → D 2 is signed-area preserving if for every Lipschitz closed curve γ , σ ( γ ) = σ ( f ◦ γ ).

Signed-area preserving maps For a closed curve γ , let σ ( γ ) be the signed area of γ (the integral of the winding number of γ ). A map f : D 2 → D 2 is signed-area preserving if for every Lipschitz closed curve γ , σ ( γ ) = σ ( f ◦ γ ). Then: ◮ A smooth signed-area preserving map must preserve orientation; in fact, the Jacobian must equal 1.

Signed-area preserving maps For a closed curve γ , let σ ( γ ) be the signed area of γ (the integral of the winding number of γ ). A map f : D 2 → D 2 is signed-area preserving if for every Lipschitz closed curve γ , σ ( γ ) = σ ( f ◦ γ ). Then: ◮ A smooth signed-area preserving map must preserve orientation; in fact, the Jacobian must equal 1. ◮ Likewise, a Lipschitz signed-area preserving map must have Jacobian 1 almost everywhere.

H¨ older signed-area preserving maps ◮ Let 0 < α ≤ 1. A map f : X → Y is α –H¨ older if there is some L > 0 such that for all x 1 , x 2 ∈ X , d Y ( f ( x 1 ) , f ( x 2 )) ≤ Ld X ( x 1 , x 2 ) α .

H¨ older signed-area preserving maps ◮ Let 0 < α ≤ 1. A map f : X → Y is α –H¨ older if there is some L > 0 such that for all x 1 , x 2 ∈ X , d Y ( f ( x 1 ) , f ( x 2 )) ≤ Ld X ( x 1 , x 2 ) α . older, then dim Haus f ( X ) ≤ α − 1 dim Haus X . ◮ If f is α –H¨

H¨ older signed-area preserving maps ◮ Let 0 < α ≤ 1. A map f : X → Y is α –H¨ older if there is some L > 0 such that for all x 1 , x 2 ∈ X , d Y ( f ( x 1 ) , f ( x 2 )) ≤ Ld X ( x 1 , x 2 ) α . older, then dim Haus f ( X ) ≤ α − 1 dim Haus X . ◮ If f is α –H¨ ust) If γ : S 1 → R 2 is an α –H¨ ◮ (Olbermann, Z¨ older curve with α > 1 2 , then σ ( α ) is well-defined.

H¨ older signed-area preserving maps ◮ Let 0 < α ≤ 1. A map f : X → Y is α –H¨ older if there is some L > 0 such that for all x 1 , x 2 ∈ X , d Y ( f ( x 1 ) , f ( x 2 )) ≤ Ld X ( x 1 , x 2 ) α . older, then dim Haus f ( X ) ≤ α − 1 dim Haus X . ◮ If f is α –H¨ ust) If γ : S 1 → R 2 is an α –H¨ ◮ (Olbermann, Z¨ older curve with α > 1 2 , then σ ( α ) is well-defined. ◮ (De Lellis–Hirsch–Inauen) When α > 2 3 , an α –H¨ older signed-area preserving map must preserve orientation. (The image of a positively-oriented simple closed curve has nonnegative winding number around any point.)

Wild signed-area preserving maps Theorem (Guth–Y.) When 1 2 < α < 2 3 , there is an α –H¨ older signed-area preserving map from D 2 to R 2 approximating any continuous map.

Wild signed-area preserving maps Theorem (Guth–Y.) When 1 2 < α < 2 3 , there is an α –H¨ older signed-area preserving map from D 2 to R 2 approximating any continuous map. Extension Lemma 3 , let γ : S 1 → R 2 be a curve such that Let 1 2 < α < 2 σ ( γ ) = area D 2 . Then γ extends to an α –H¨ older signed-area preserving map β : D 2 → R 2 .

H¨ older maps to the Heisenberg group Question (Gromov) older maps from D 2 to the Let 0 < α ≤ 1 . What do α –H¨ Heisenberg group H look like?

H¨ older maps to the Heisenberg group Question (Gromov) older maps from D 2 to the Let 0 < α ≤ 1 . What do α –H¨ Heisenberg group H look like? Answer ◮ Any smooth map to H is 1 older, so when α ≤ 1 2 –H¨ 2 , there are plenty of maps.