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¨

• lder maps to the Heisenberg group

◮ H¨

• lder 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 : 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.

The Heisenberg group

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

• x, y, z ∈ R

   .

The Heisenberg group

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

• x, y, z ∈ R

   . This contains a lattice HZ = X, Y , Z | [X, Y ] = Z, all other pairs commute.

A lattice in H3

A lattice in H3

z = xyx−1y−1

A lattice in H3

z = xyx−1y−1 z4 = x2y2x−2y−2

A lattice in H3

z = xyx−1y−1 z4 = x2y2x−2y−2 zn2 = xnynx−ny−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} ◮ st(x, y, z) = (tx, ty, t2z) 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} ◮ st(x, y, z) = (tx, ty, t2z) scales the metric by t ◮ The ball of radius ǫ is roughly an ǫ × ǫ × ǫ2 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} ◮ st(x, y, z) = (tx, ty, t2z) scales the metric by t ◮ The ball of radius ǫ is roughly an ǫ × ǫ × ǫ2 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 C2 surface can be horizontal – most curves in a C2 surface have Hausdorff dimension 2.

A surface in H

◮ 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.

A surface in H

◮ 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. ◮ What’s the shape of a surface in H?

What’s the shape of a surface in H?

Question (Gromov)

Let 0 < α ≤ 1. What do α–H¨

• lder maps from D2 or D3 to H look

like?

What’s the shape of a surface in H?

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.

What’s the shape of a surface in H?

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.

What’s the shape of a surface in H?

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.

◮ It follows that 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.

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 : 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 of 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). ◮ 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 of 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). ◮ A smooth signed-area preserving map must preserve

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

◮ When α < 1

2 and γ : S1 → D2 is α–H¨

• lder, γ can be

space-filling, so σ(γ) may be undefined.

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 : D2 → D2 is signed-area preserving if for every Lipschitz closed curve γ, σ(γ) = σ(f ◦ γ). ◮ A smooth signed-area preserving map must preserve

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

◮ When α < 1

2 and γ : S1 → D2 is α–H¨

• lder, γ can be

space-filling, so σ(γ) may be undefined. ◮ (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.)

Self-similar signed-area preserving maps

Theorem (Guth–Y.)

When 1

2 < α < 2 3, the α–H¨

• lder signed-area preserving maps from

D2 to R2 are dense in C0(D2, R2).

Self-similar signed-area preserving maps

Theorem (Guth–Y.)

When 1

2 < α < 2 3, the α–H¨

• lder signed-area preserving maps from

D2 to R2 are dense in C0(D2, R2).

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.

Open questions

◮ What else can this be used for?