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¨

• lder signed-area preserving maps

◮ H¨

• lder 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 : Sn+1 → Sn 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 : Sn+1 → Sn such that f is not null-homotopic. This follows from:

Extension Lemma (Wenger–Y.)

Let α : Sm−2 → Sn−2 be a map with m > n. The suspension Σα : Sm−1 → Sn−1 extends to a corank–1 map β : Dm → Dn.

Suspensions

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

Suspensions

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

Proof of Theorem given Extension Lemma

Theorem

Let n ≥ 4. There is a corank–1 map f : Sn+1 → Sn such that f is not null-homotopic.

Proof of Theorem given Extension Lemma

Theorem

Let n ≥ 4. There is a corank–1 map f : Sn+1 → Sn such that f is not null-homotopic.

Proof.

◮ Let h : S3 → S2 be the Hopf fibration. Then Σkh is homotopically nontrivial for every k.

Proof of Theorem given Extension Lemma

Theorem

Let n ≥ 4. There is a corank–1 map f : Sn+1 → Sn such that f is not null-homotopic.

Proof.

◮ Let h : S3 → S2 be the Hopf fibration. Then Σkh is homotopically nontrivial for every k. ◮ Let k > 0. By the Extension Lemma, there is a corank–1 extension β : D4+k → D3+k of Σkh.

Proof of Theorem given Extension Lemma

Theorem

Let n ≥ 4. There is a corank–1 map f : Sn+1 → Sn such that f is not null-homotopic.

Proof.

◮ Let h : S3 → S2 be the Hopf fibration. Then Σkh is homotopically nontrivial for every k. ◮ Let k > 0. By the Extension Lemma, there is a corank–1 extension β : D4+k → D3+k of Σkh. ◮ Let f : S4+k → S3+k be two copies of β glued along the

• equator. This map has corank 1 and f ∼ Σ(Σkh) = Σk+1h.

Proof of Theorem given Extension Lemma

Theorem

Let n ≥ 4. There is a corank–1 map f : Sn+1 → Sn such that f is not null-homotopic.

Proof.

◮ Let h : S3 → S2 be the Hopf fibration. Then Σkh is homotopically nontrivial for every k. ◮ Let k > 0. By the Extension Lemma, there is a corank–1 extension β : D4+k → D3+k of Σkh. ◮ Let f : S4+k → S3+k be two copies of β glued along the

• equator. This map has corank 1 and f ∼ Σ(Σkh) = Σk+1h.

It remains to prove the Extension Lemma.

Higher dimensions (in progress)

Theorem

Let h : S3 → S2 be the Hopf fibration. Then Σ2h : S5 → S4 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 Σ2kh.

Higher dimensions (in progress)

Theorem

Let h : S3 → S2 be the Hopf fibration. Then Σ2h : S5 → S4 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 Σ2kh. This is sharp; Σ2kh is not homotopic to a Lipschitz map with corank k + 1 and Σ2k−1h 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

• f 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

• f the winding number of γ). A map f : D2 → D2 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

• f the winding number of γ). A map f : D2 → D2 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

• rientation; 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

• f the winding number of γ). A map f : D2 → D2 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

• rientation; in fact, the Jacobian must equal 1.

◮ Likewise, a Lipschitz signed-area preserving map must have Jacobian 1 almost everywhere.

H¨

• lder signed-area preserving maps

◮ Let 0 < α ≤ 1. A map f : X → Y is α–H¨

• lder if there is

some L > 0 such that for all x1, x2 ∈ X, dY (f (x1), f (x2)) ≤ LdX(x1, x2)α.

H¨

• lder signed-area preserving maps

◮ Let 0 < α ≤ 1. A map f : X → Y is α–H¨

• lder if there is

some L > 0 such that for all x1, x2 ∈ X, dY (f (x1), f (x2)) ≤ LdX(x1, x2)α. ◮ If f is α–H¨

• lder, then dimHaus f (X) ≤ α−1 dimHaus X.

H¨

• lder signed-area preserving maps

◮ Let 0 < α ≤ 1. A map f : X → Y is α–H¨

• lder if there is

some L > 0 such that for all x1, x2 ∈ X, dY (f (x1), f (x2)) ≤ LdX(x1, x2)α. ◮ If f is α–H¨

• lder, then dimHaus f (X) ≤ α−1 dimHaus X.

◮ (Olbermann, Z¨ ust) If γ : S1 → R2 is an α–H¨

• lder curve with

α > 1

2, then σ(α) is well-defined.

H¨

• lder signed-area preserving maps

◮ Let 0 < α ≤ 1. A map f : X → Y is α–H¨

• lder if there is

some L > 0 such that for all x1, x2 ∈ X, dY (f (x1), f (x2)) ≤ LdX(x1, x2)α. ◮ If f is α–H¨

• lder, then dimHaus f (X) ≤ α−1 dimHaus X.

◮ (Olbermann, Z¨ ust) If γ : S1 → R2 is an α–H¨

• lder curve with

α > 1

2, then σ(α) is well-defined.

◮ (De Lellis–Hirsch–Inauen) When α > 2

3, an α–H¨

• lder

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¨

• lder signed-area preserving map

from D2 to R2 approximating any continuous map.

Wild signed-area preserving maps

Theorem (Guth–Y.)

When 1

2 < α < 2 3, there is an α–H¨

• lder signed-area preserving map

from D2 to R2 approximating any continuous map.

Extension Lemma

Let 1

2 < α < 2 3, let γ : S1 → R2 be a curve such that

σ(γ) = area D2. Then γ extends to an α–H¨

• lder signed-area

preserving map β : D2 → R2.

H¨

• lder maps to the Heisenberg group

Question (Gromov)

Let 0 < α ≤ 1. What do α–H¨

• lder maps from D2 to the

Heisenberg group H look like?

H¨

• lder maps to the Heisenberg group

Question (Gromov)

Let 0 < α ≤ 1. What do α–H¨

• lder maps from D2 to the

Heisenberg group H look like?

Answer

◮ Any smooth map to H is 1

2–H¨

• lder, so when α ≤ 1

2, there are

plenty of maps.

H¨

• lder maps to the Heisenberg group

Question (Gromov)

Let 0 < α ≤ 1. What do α–H¨

• lder maps from D2 to the

Heisenberg group H look like?

Answer

◮ Any smooth map to H is 1

2–H¨

• lder, so when α ≤ 1

2, there are

plenty of maps. ◮ (Gromov) Any embedding of D2 in H has Hausdorff dimension at least 3, so for α > 2

3, there is no α–H¨

• lder

embedding of D2 in H.

H¨

• lder maps to the Heisenberg group

Question (Gromov)

Let 0 < α ≤ 1. What do α–H¨

• lder maps from D2 to the

Heisenberg group H look like?

Answer

◮ Any smooth map to H is 1

2–H¨

• lder, so when α ≤ 1

2, there are

plenty of maps. ◮ (Gromov) Any embedding of D2 in H has Hausdorff dimension at least 3, so for α > 2

3, there is no α–H¨

• lder

embedding of D2 in H. ◮ (Z¨ ust) For α > 2

3, α–H¨

• lder maps from D2 to H factor

through a tree.

H¨

• lder maps to the Heisenberg group

Question (Gromov)

Let 0 < α ≤ 1. What do α–H¨

• lder maps from D2 to the

Heisenberg group H look like?

Answer

◮ Any smooth map to H is 1

2–H¨

• lder, so when α ≤ 1

2, there are

plenty of maps. ◮ (Gromov) Any embedding of D2 in H has Hausdorff dimension at least 3, so for α > 2

3, there is no α–H¨

• lder

embedding of D2 in H. ◮ (Z¨ ust) For α > 2

3, α–H¨

• lder maps from D2 to H factor

through a tree. ◮ (Wenger–Y.) When 1

2 < α < 2 3, there is an α–H¨

• lder map

approximating any continuous map from Dn to H!

Open questions

◮ What else can this be used for?

A quick trip through the Heisenberg group

◮ Let H be the 3–dimensional nilpotent Lie group H =      1 x z 1 y 1  

• x, y, z ∈ R

   .

A quick trip through the Heisenberg group

◮ Let H be the 3–dimensional nilpotent Lie group H =      1 x z 1 y 1  

• x, y, z ∈ R

   . ◮ H is equipped with a natural sub-Riemannian metric — like a Riemannian metric except some directions have infinite length.

A quick trip through the Heisenberg group

◮ Let H be the 3–dimensional nilpotent Lie group H =      1 x z 1 y 1  

• x, y, z ∈ R

   . ◮ H is equipped with a natural sub-Riemannian metric — like a Riemannian metric except some directions have infinite length. ◮ This causes H to have Hausdorff dimension 4 but topological dimension 3.

The sub-Riemannian metric

◮ There is a distribution of horizontal planes spanned by red and blue edges.

The 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}

The 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} ◮ Non-horizontal curves have Hausdorff dimension 2.

The sub-Riemannian metric

◮ No C2 surface can be horizontal – most curves in a C2 surface have Hausdorff dimension 2.

The sub-Riemannian metric

◮ No C2 surface can be horizontal – most curves in a C2 surface have Hausdorff dimension 2. ◮ (Gromov) In fact, any surface in H has Hausdorff dimension at least 3.

Gromov’s question

Question (Gromov)

Let 0 < α ≤ 1. What do α–H¨

• lder maps from D2 or D3 to H look

like?

Gromov’s question

Question (Gromov)

Let 0 < α ≤ 1. What do α–H¨

• lder maps from D2 or D3 to H look

like? Recall: ◮ Let 0 < α ≤ 1. A map f : X → Y is α–H¨

• lder if there is

some L > 0 such that for all x1, x2 ∈ X, dY (f (x1), f (x2)) ≤ LdX(x1, x2)α.

Gromov’s question

Question (Gromov)

Let 0 < α ≤ 1. What do α–H¨

• lder maps from D2 or D3 to H look

like? Recall: ◮ Let 0 < α ≤ 1. A map f : X → Y is α–H¨

• lder if there is

some L > 0 such that for all x1, x2 ∈ X, dY (f (x1), f (x2)) ≤ LdX(x1, x2)α. ◮ If f is α–H¨

• lder, then dimHaus f (X) ≤ α−1 dimHaus X.

Gromov’s question

Question (Gromov)

Let 0 < α ≤ 1. What do α–H¨

• lder maps from D2 or D3 to H look

like? Recall: ◮ Let 0 < α ≤ 1. A map f : X → Y is α–H¨

• lder if there is

some L > 0 such that for all x1, x2 ∈ X, dY (f (x1), f (x2)) ≤ LdX(x1, x2)α. ◮ If f is α–H¨

• lder, then dimHaus f (X) ≤ α−1 dimHaus X.

◮ Any embedding of D2 in H has Hausdorff dimension at least 3, so for α > 2

3, there is no α–H¨

• lder embedding of D2 in H.

H¨

• lder maps to H

In fact, when α > 2

3, H¨

• lder maps factor through trees.

Theorem (Z¨ ust)

If f : Dn → H is an α–H¨

• lder map, α > 2

3, then there is an R–tree

T such that f = g ◦ h, where h : M → T and g : T → N are Lipschitz maps.

H¨

• lder maps to H

In fact, when α > 2

3, H¨

• lder maps factor through trees.

Theorem (Z¨ ust)

If f : Dn → H is an α–H¨

• lder map, α > 2

3, then there is an R–tree

T such that f = g ◦ h, where h : M → T and g : T → N are Lipschitz maps. On the other hand, any smooth map is 1

2–H¨

• lder, so when α ≤ 1

2,

there are plenty of maps.

H¨

• lder maps to H

In fact, when α > 2

3, H¨

• lder maps factor through trees.

Theorem (Z¨ ust)

If f : Dn → H is an α–H¨

• lder map, α > 2

3, then there is an R–tree

T such that f = g ◦ h, where h : M → T and g : T → N are Lipschitz maps. On the other hand, any smooth map is 1

2–H¨

• lder, so when α ≤ 1

2,

there are plenty of maps. What happens when 1

2 < α < 2 3?

Self-similar H¨

• lder maps to H

Theorem (Wenger–Y.)

When 1

2 < α < 2 3, the set of α–H¨

• lder maps is dense in C0(Dn, H).

Self-similar H¨

• lder maps to H

Theorem (Wenger–Y.)

When 1

2 < α < 2 3, the set of α–H¨

• lder maps is dense in C0(Dn, H).

Extension Lemma

Let γ : S1 → H be a Lipschitz closed curve in H and let

1 2 < α < 2

• 3. Then γ extends to a map β : D2 → H which is

α–H¨

• lder.

Self-similar H¨

• lder maps to H

Let H be the 3-D Heisenberg group.

Theorem (Wenger–Y.)

When 1

2 < α < 2 3, the set of α–H¨

• lder maps is dense in C0(Dn, H).

Self-similar H¨

• lder maps to H

Let H be the 3-D Heisenberg group.

Theorem (Wenger–Y.)

When 1

2 < α < 2 3, the set of α–H¨

• lder maps is dense in C0(Dn, H).

Extension Lemma

Let γ : S1 → H be a Lipschitz closed curve in H and let

1 2 < α < 2

• 3. Then γ extends to a map β : D2 → H which is

α–H¨

• lder.