Slits, spirals and Loewner hulls Joan Lind University of Tennessee - - PowerPoint PPT Presentation

Slits, spirals and Loewner hulls

Joan Lind University of Tennessee

The story begins

Bieberbach conjecture (1916): For f (z) = z + a2z2 + a3z3 + · · · conformal on D, then |an| ≤ n.

The story begins

Bieberbach conjecture (1916): For f (z) = z + a2z2 + a3z3 + · · · conformal on D, then |an| ≤ n. Charles Loewner introduced the Loewner equation in 1923 to prove the n = 3 case.

The story begins

Bieberbach conjecture (1916): For f (z) = z + a2z2 + a3z3 + · · · conformal on D, then |an| ≤ n. Charles Loewner introduced the Loewner equation in 1923 to prove the n = 3 case. Louis des Branges again used the Loewner equation when he proved the conjecture in 1985.

The story continues

Simple Random Walk (SRW) in 2 dimensions – has a conformally invariant scaling limit: 2-d Brownian motion

The story continues

Schramm’s question: Do other 2-dimensional random walks (such as SAW, LERW, etc.) have conformally invariant scaling limits?

The story continues

In 2000, Oded Schramm introduced SLEκ, a family of random processes that contain the possible conformally invariant scaling limits.

The story continues

In 2000, Oded Schramm introduced SLEκ, a family of random processes that contain the possible conformally invariant scaling limits. Through the Loewner equation, SLEκ correspond to √κBt.

The Loewner equation

L Equ

← !

growing families of 2-d sets real-valued functions

From curves to functions

γ(t)

From curves to functions

γ(t)

From curves to functions

γ(t) gt λ(t)

From curves to functions

γ(t) gt λ(t)

gt(z) = z + ct z + O 1 z2

• for z near infinity

From curves to functions

γ(t) gt λ(t)

gt(z) = z + 2t z + O 1 z2

• for z near infinity

From curves to functions

γ(t) gt λ(t)

∂ ∂t gt(z) = 2 gt(z) − λ(t)

From functions to growing families of sets

For a continuous, real-valued function λ(t) and z ∈ H, consider ∂ ∂t gt(z) = 2 gt(z) − λ(t), g0(z) = z

From functions to growing families of sets

For a continuous, real-valued function λ(t) and z ∈ H, consider ∂ ∂t gt(z) = 2 gt(z) − λ(t), g0(z) = z Loewner hulls: Kt = {z ∈ H : gs(z) = λ(s) for some s ≤ t}.

From functions to growing families of sets

For a continuous, real-valued function λ(t) and z ∈ H, consider ∂ ∂t gt(z) = 2 gt(z) − λ(t), g0(z) = z Loewner hulls: Kt = {z ∈ H : gs(z) = λ(s) for some s ≤ t}. Theorem: gt is a conformal map from H \ Kt onto H.

Loewner flow

∂ ∂t gt(z) = 2 Re gt(z) − λ(t) |gt(z) − λ(t)|2 − 2i Im gt(z) |gt(z) − λ(t)|2

0.2 0.4 0.6 0.8 1 –1 –0.5 0.5 1

Example

Loewner hull generated by λ(t) ≡ 0.

Loewner equation

L Equ

← !

growing families of 2-d sets real-valued functions

The geometry of SLE curves

L Equ SLEκ ← → √κBt

The geometry of SLE curves

L Equ SLEκ ← → √κBt 0 ≤ κ ≤ 4 4 < κ < 8 8 ≤ κ

The geometry of SLE curves

SLE2 SLE6

Simple curve Loewner hulls

Question: When does the deterministic Loewner equation generate a simple curve?

Simple curve Loewner hulls

Question: When does the deterministic Loewner equation generate a simple curve? Lip(1/2) functions: |λ(t) − λ(s)| ≤ M |t − s|1/2 for all t, s is the domain of λ. The smallest such M is ||λ||1/2.

Simple curve Loewner hulls

Question: When does the deterministic Loewner equation generate a simple curve? Lip(1/2) functions: |λ(t) − λ(s)| ≤ M |t − s|1/2 for all t, s is the domain of λ. The smallest such M is ||λ||1/2. Answer: (Marshall, Rohde) There exists C0 > 0 so that for λ ∈ Lip(1/2) with ||λ||1/2 < C0, then the Loewner hull is a quasislit γ.

Simple curve Loewner hulls

Question: Do all Lip(1/2) driving functions generate simple curves?

Simple curve Loewner hulls

Question: Do all Lip(1/2) driving functions generate simple curves? Answer: (Marshall, Rohde) No. There is a non-simple example (a curve that spirals around a disc) that is generated by a Lip(1/2) driving function.

Simple curve Loewner hulls

Question: What is the optimal value of C0 for the Marshall-Rohde theorem?

Simple curve Loewner hulls

Question: What is the optimal value of C0 for the Marshall-Rohde theorem? Answer: (L) C0 = 4.

Simple curve Loewner hulls

Question: What is the optimal value of C0 for the Marshall-Rohde theorem? Answer: (L) C0 = 4. Key examples: λ(t) = −c√1 − t c = 3 c = 5

Loewner deformations

Omer Angel’s question: For r < 1, what can you say about Loewner hull driven by rλ?

Loewner deformations

Omer Angel’s question: For r < 1, what can you say about Loewner hull driven by rλ? Is it at least as nice as the hull driven by λ?

Loewner deformations

Omer Angel’s question: For r < 1, what can you say about Loewner hull driven by rλ? Is it at least as nice as the hull driven by λ? This is true for SLE. 0 ≤ κ ≤ 4 4 < κ < 8 8 ≤ κ

Loewner deformations

Omer Angel’s question: For r < 1, what can you say about Loewner hull driven by rλ? Is it at least as nice as the hull driven by λ? This is true for λ(t) = −c√1 − t. c = 3 c = 5

Loewner deformations

Omer Angel’s question: For r < 1, what can you say about Loewner hull driven by rλ? Is it at least as nice as the hull driven by λ? Is it always true?

Loewner deformations

Omer Angel’s question: For r < 1, what can you say about Loewner hull driven by rλ? Is it at least as nice as the hull driven by λ? Is it always true? Answer: N0.

Loewner deformations

Theorem (L, Marshall, Rohde)

Let λ be the driving function for the star spiral. When 0 < r < 1, then rλ generates a simple curve, and when r > 1, then rλ generates a “bubble” curve.

Spirals and collisions

Theorem (L, Marshall, Rohde)

If γ is a suffiently nice infinite spiral of half-plane capacity T, or if γ has a tangential self-intersection, then its driving term λ satisfies lim

t→T

|λ(T) − λ(t)| √ T − t = 4.

Spirals and collisions

Theorem (L, Marshall, Rohde)

If γ is a suffiently nice infinite spiral of half-plane capacity T, or if γ has a tangential self-intersection, then its driving term λ satisfies lim

t→T

|λ(T) − λ(t)| √ T − t = 4.

Theorem (L, Marshall, Rohde)

If λ : [0, T] → R is sufficiently regular on [0, T) and if lim

t→T

|λ(T) − λ(t)| √ T − t = κ > 4, then γ(T) = lim

t→T γ(t) exists, is real, and γ intersects R in the

same angle as the trace for κ√1 − t.

Spirals and collisions

Scaling Property:

◮ If λ generates hull Kt, then cλ(t/c2) generates cKt/c2.

Spirals and collisions

Scaling Property:

◮ If λ generates hull Kt, then cλ(t/c2) generates cKt/c2.

Example: λ(t) = c√t

Spirals and collisions

Concatenation Property:

◮ If λ generates hull Kt for t ∈ [0, T], then λ restricted to

[t0, T] generates hull gt0(KT \ Kt0).

Spirals and collisions

Concatenation Property:

◮ If λ generates hull Kt for t ∈ [0, T], then λ restricted to

[t0, T] generates hull gt0(KT \ Kt0). Kt0 λ(0) gt0 gt0(KT \ Kt0) λ(t0)

Spirals and collisions

Start with λ = −c√1 − t. For t0 ∈ (0, 1), map down by gt0. Rescale by 1/√1 − t0.

Spirals and collisions

Start with λ = −c√1 − t. For t0 ∈ (0, 1), map down by gt0. Rescale by 1/√1 − t0. What do we obtain? The same hull that we started with.

Spirals and collisions

Start with λ satisfying lim

t→1

|λ(1) − λ(t)| √1 − t = κ ≥ 4 Shift so that λ(1) = 0. For t0 ∈ (0, 1), map down by gt0. Rescale by 1/√1 − t0.

Spirals and collisions

Start with λ satisfying lim

t→1

|λ(1) − λ(t)| √1 − t = κ ≥ 4 Shift so that λ(1) = 0. For t0 ∈ (0, 1), map down by gt0. Rescale by 1/√1 − t0. What do we obtain? A hull that is getting closer and closer to the hull generated by ±κ√1 − t.

Spirals and collisions

What happens when κ = 4?

Spirals and collisions

What happens when κ = 4? 4√1 − t generates a hull that intersects the real line tangentially.

Spirals and collisions

What happens when κ = 4? 4√1 − t generates a hull that intersects the real line tangentially. The spiral behavior can be viewed as tangential behavior.

Recap: the Loewner equation question

L Equ

← !

growing families of 2-d sets real-valued functions Question: How do properties of the functions relate to geometric characteristics of the sets?

Sample of other Loewner explorations

Sample of other Loewner explorations

◮ Spacefilling curves (L, Rohde)

Sample of other Loewner explorations

◮ Spacefilling curves (L, Rohde) ◮ Driving functions with higher regularity (Wong, L, Tran)

Sample of other Loewner explorations

◮ Spacefilling curves (L, Rohde) ◮ Driving functions with higher regularity (Wong, L, Tran) ◮ H¨

• lder domains (Kinneberg)

Sample of other Loewner explorations

◮ Spacefilling curves (L, Rohde) ◮ Driving functions with higher regularity (Wong, L, Tran) ◮ H¨

• lder domains (Kinneberg)

◮ Complex-valued driving functions (Tran)

Sample of other Loewner explorations

◮ Spacefilling curves (L, Rohde) ◮ Driving functions with higher regularity (Wong, L, Tran) ◮ H¨

• lder domains (Kinneberg)

◮ Complex-valued driving functions (Tran) ◮ Multiple Loewner hulls (Roth, Schleissinger, Starnes)

Sample of other Loewner explorations

◮ Spacefilling curves (L, Rohde) ◮ Driving functions with higher regularity (Wong, L, Tran) ◮ H¨

• lder domains (Kinneberg)

◮ Complex-valued driving functions (Tran) ◮ Multiple Loewner hulls (Roth, Schleissinger, Starnes) ◮ Trees (Healey)

Sample of other Loewner explorations

◮ Spacefilling curves (L, Rohde) ◮ Driving functions with higher regularity (Wong, L, Tran) ◮ H¨

• lder domains (Kinneberg)

◮ Complex-valued driving functions (Tran) ◮ Multiple Loewner hulls (Roth, Schleissinger, Starnes) ◮ Trees (Healey) ◮ Trace existence (Zhang, Zinsmeister)

Thank you