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) April 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 maps to the Heisenberg group ◮ H¨ older signed-area preserving maps

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.

The Heisenberg group Let H be the 3–dimensional nilpotent Lie group  �    1 x z �   � H = 0 1 y x , y , z ∈ R  .   � � 0 0 1  �

The Heisenberg group Let H be the 3–dimensional nilpotent Lie group  �    1 x z �   � H = 0 1 y x , y , z ∈ R  .   � � 0 0 1  � This contains a lattice H Z = � X , Y , Z | [ X , Y ] = Z , all other pairs commute � .

A lattice in H 3

A lattice in H 3 z = xyx − 1 y − 1

A lattice in H 3 z = xyx − 1 y − 1 z 4 = x 2 y 2 x − 2 y − 2

A lattice in H 3 z = xyx − 1 y − 1 z 4 = x 2 y 2 x − 2 y − 2 z n 2 = x n y n x − n y − n

From Cayley graph to sub-riemannian metric ◮ There is a distribution of horizontal planes spanned by red and blue edges.

From Cayley graph to sub-riemannian metric ◮ There is a distribution of horizontal planes spanned by red and blue edges. ◮ d ( u , v ) = inf { ℓ ( γ ) | γ is a horizontal curve from u to v }

From Cayley graph to sub-riemannian metric ◮ There is a distribution of horizontal planes spanned by red and blue edges. ◮ d ( u , v ) = inf { ℓ ( γ ) | γ is a horizontal curve from u to v } ◮ s t ( x , y , z ) = ( tx , ty , t 2 z ) scales the metric by t

From Cayley graph to sub-riemannian metric ◮ There is a distribution of horizontal planes spanned by red and blue edges. ◮ d ( u , v ) = inf { ℓ ( γ ) | γ is a horizontal curve from u to v } ◮ s t ( x , y , z ) = ( tx , ty , t 2 z ) scales the metric by t ◮ The ball of radius ǫ is roughly an ǫ × ǫ × ǫ 2 box.

From Cayley graph to sub-riemannian metric ◮ There is a distribution of horizontal planes spanned by red and blue edges. ◮ d ( u , v ) = inf { ℓ ( γ ) | γ is a horizontal curve from u to v } ◮ s t ( x , y , z ) = ( tx , ty , t 2 z ) scales the metric by t ◮ The ball of radius ǫ is roughly an ǫ × ǫ × ǫ 2 box. ◮ Non-horizontal curves have Hausdorff dimension 2.

A geodesic in H ◮ Every horizontal curve is the lift of a curve in the plane.

A geodesic in H ◮ Every horizontal curve is the lift of a curve in the plane. ◮ The length of the lift is the length of the original curve.

A geodesic in H ◮ Every horizontal curve is the lift of a curve in the plane. ◮ The length of the lift is the length of the original curve. ◮ The change in height along the lift of a closed curve is the signed area of the curve.

A geodesic in H ◮ Every horizontal curve is the lift of a curve in the plane. ◮ The length of the lift is the length of the original curve. ◮ The change in height along the lift of a closed curve is the signed area of the curve. ◮ By the isoperimetric inequality, geodesics are lifts of circular arcs.

A surface in H ◮ No C 2 surface can be horizontal – most curves in a C 2 surface have Hausdorff dimension 2.

A surface in H ◮ No C 2 surface can be horizontal – most curves in a C 2 surface have Hausdorff dimension 2. ◮ (Gromov) In fact, any surface in H has Hausdorff dimension at least 3.

A surface in H ◮ No C 2 surface can be horizontal – most curves in a C 2 surface have Hausdorff dimension 2. ◮ (Gromov) In fact, any surface in H has Hausdorff dimension at least 3. ◮ What’s the shape of a surface in H ?

What’s the shape of a surface in H ? Question (Gromov) older maps from D 2 or D 3 to H look Let 0 < α ≤ 1 . What do α –H¨ like?