Slits, spirals and Loewner hulls

1 Introduction and Results

Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation

d

dtgt(z) =

2

gt(z) − λ(t), g0(z) = z ∈ H, (1.1)

where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and

γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√

1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ

√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).

It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4

√1 − t is very similar to the infinitely spiraling prime

end, and that this is reflected in the driving terms:

Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies

limt→T

|λ(T ) − λ(t)|√T − t

= 4.

Figure 1: An infinite spiral converging towards a star.

See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving

Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.

The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√

1 − t for κ > 4 :

2

Joan Lind

University of Tennessee

The story begins

Bieberbach conjecture (1916):For f (z) = z + a2z

2 + a3z3 + · · · conformal on D, then |an| ≤ n.

Charles Loewner introduced the Loewner equation in 1923 to provethe n = 3 case.

Louis des Branges again used the Loewner equation when heproved the conjecture in 1985.

The story begins

Bieberbach conjecture (1916):For f (z) = z + a2z

2 + a3z3 + · · · conformal on D, then |an| ≤ n.

Charles Loewner introduced the Loewner equation in 1923 to provethe n = 3 case.

Louis des Branges again used the Loewner equation when heproved the conjecture in 1985.

The story begins

Bieberbach conjecture (1916):For f (z) = z + a2z

2 + a3z3 + · · · conformal on D, then |an| ≤ n.

Charles Loewner introduced the Loewner equation in 1923 to provethe n = 3 case.

Louis des Branges again used the Loewner equation when heproved the conjecture in 1985.

The story continues

0

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 (suchas SAW, LERW, etc.) have conformally invariant scaling limits?

0

The story continues

In 2000, Oded Schramm introduced SLEκ, a family of randomprocesses that contain the possible conformally invariant scalinglimits.

Through the Loewner equation, SLEκ correspond to√κBt .

The story continues

In 2000, Oded Schramm introduced SLEκ, a family of randomprocesses that contain the possible conformally invariant scalinglimits.

Through the Loewner equation, SLEκ correspond to√κBt .

The Loewner equation

L Equ

!

1

growing families of 2-d sets real-valued functions

From curves to functions

�(t)

�(t)

�(t)

gt

0 �(t)

1

From curves to functions

�(t)

�(t)

�(t)

gt

0 �(t)

1

From curves to functions

�(t)

�(t)

�(t)

gt

�(t)

1

From curves to functions

�(t)

�(t)

�(t)

gt

�(t)

1

gt(z) = z +ctz

+ O

(1

z2

)for z near infinity

From curves to functions

�(t)

�(t)

�(t)

gt

�(t)

1

gt(z) = z +2t

z+ O

(1

z2

)for z near infinity

From curves to functions

�(t)

�(t)

�(t)

gt

�(t)

1

∂

∂tgt(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

∂

∂tgt(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.

From functions to growing families of sets

For a continuous, real-valued function λ(t) and z ∈ H, consider

∂

∂tgt(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.

From functions to growing families of sets

For a continuous, real-valued function λ(t) and z ∈ H, consider

∂

∂tgt(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

∂

∂tgt(z) = 2

Re gt(z)− λ(t)

|gt(z)− λ(t)|2 − 2iIm 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

!

1

growing families of 2-d sets real-valued functions

The geometry of SLE curves

L Equ

SLEκ ←→ √κBt

0 ≤ κ ≤ 4 4 < κ < 8 8 ≤ κ

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 generatea 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 forλ ∈ Lip(1/2) with ||λ||1/2 < C0, then the Loewner hull is aquasislit γ.

Simple curve Loewner hulls

Question: When does the deterministic Loewner equation generatea 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 forλ ∈ Lip(1/2) with ||λ||1/2 < C0, then the Loewner hull is aquasislit γ.

Simple curve Loewner hulls

Question: When does the deterministic Loewner equation generatea 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 forλ ∈ Lip(1/2) with ||λ||1/2 < C0, then the Loewner hull is aquasislit γ.

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 (acurve that spirals around a disc) that is generated by a Lip(1/2)driving function.

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 (acurve 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-Rohdetheorem?

Answer: (L) C0 = 4.

Key examples: λ(t) = −c√

1− t

c = 3 c = 5

Simple curve Loewner hulls

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

Answer: (L) C0 = 4.

Key examples: λ(t) = −c√

1− t

c = 3 c = 5

Simple curve Loewner hulls

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

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?

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 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 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?

Answer: N0.

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

1 Introduction and Results

Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation

d

dtgt(z) =

2

gt(z) − λ(t), g0(z) = z ∈ H, (1.1)

where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and

γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√

1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ

√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).

It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4

√1 − t is very similar to the infinitely spiraling prime

end, and that this is reflected in the driving terms:

Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies

limt→T

|λ(T ) − λ(t)|√T − t

= 4.

Figure 1: An infinite spiral converging towards a star.

See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving

Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.

The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√

1 − t for κ > 4 :

2

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 rλgenerates a “bubble” curve.

Spirals and collisions

Theorem (L, Marshall, Rohde)

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

limt→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

limt→T

|λ(T )− λ(t)|√T − t

= κ > 4,

then γ(T ) = limt→T

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

same angle as the trace for κ√

1− t.

Spirals and collisions

Theorem (L, Marshall, Rohde)

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

limt→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

limt→T

|λ(T )− λ(t)|√T − t

= κ > 4,

then γ(T ) = limt→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:

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

Example: λ(t) = c√t

Spirals and collisions

Scaling Property:

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

Example: λ(t) = c√t

Spirals and collisions

Concatenation Property:

I 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

Concatenation Property:

I 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.

What do we obtain? The same hull that we started with.

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

limt→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 thehull generated by ±κ

√1− t.

Spirals and collisions

Start with λ satisfying

limt→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 thehull generated by ±κ

√1− t.

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.

1 Introduction and Results

Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation

d

dtgt(z) =

2

gt(z) − λ(t), g0(z) = z ∈ H, (1.1)

where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and

γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√

1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ

√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).

It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4

√1 − t is very similar to the infinitely spiraling prime

end, and that this is reflected in the driving terms:

Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies

limt→T

|λ(T ) − λ(t)|√T − t

= 4.

Figure 1: An infinite spiral converging towards a star.

See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving

Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.

The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√

1 − t for κ > 4 :

2

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.

1 Introduction and Results

Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation

d

dtgt(z) =

2

gt(z) − λ(t), g0(z) = z ∈ H, (1.1)

where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and

γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√

1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ

√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).

It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4

√1 − t is very similar to the infinitely spiraling prime

end, and that this is reflected in the driving terms:

Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies

limt→T

|λ(T ) − λ(t)|√T − t

= 4.

Figure 1: An infinite spiral converging towards a star.

See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving

Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.

The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√

1 − t for κ > 4 :

2

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.

1 Introduction and Results

Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation

d

dtgt(z) =

2

gt(z) − λ(t), g0(z) = z ∈ H, (1.1)

where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and

γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√

1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ

√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).

It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4

√1 − t is very similar to the infinitely spiraling prime

end, and that this is reflected in the driving terms:

Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies

limt→T

|λ(T ) − λ(t)|√T − t

= 4.

Figure 1: An infinite spiral converging towards a star.

See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving

Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.

The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√

1 − t for κ > 4 :

2

Recap: the Loewner equation question

L Equ

!

1

growing families of 2-d sets real-valued functions

Question: How do properties of the functions relate to geometriccharacteristics of the sets?

Sample of other Loewner explorations

I Spacefilling curves (L, Rohde)

I Driving functions with higher regularity (Wong, L, Tran)

I Holder domains (Kinneberg)

I Complex-valued driving functions (Tran)

I Multiple Loewner hulls (Roth, Schleissinger, Starnes)

I Trees (Healey)

I Trace existence (Zhang, Zinsmeister)

Sample of other Loewner explorations

I Spacefilling curves (L, Rohde)

I Driving functions with higher regularity (Wong, L, Tran)

I Holder domains (Kinneberg)

I Complex-valued driving functions (Tran)

I Multiple Loewner hulls (Roth, Schleissinger, Starnes)

I Trees (Healey)

I Trace existence (Zhang, Zinsmeister)

Sample of other Loewner explorations

I Spacefilling curves (L, Rohde)

I Driving functions with higher regularity (Wong, L, Tran)

I Holder domains (Kinneberg)

I Complex-valued driving functions (Tran)

I Multiple Loewner hulls (Roth, Schleissinger, Starnes)

I Trees (Healey)

I Trace existence (Zhang, Zinsmeister)

Sample of other Loewner explorations

I Spacefilling curves (L, Rohde)

I Driving functions with higher regularity (Wong, L, Tran)

I Holder domains (Kinneberg)

I Complex-valued driving functions (Tran)

I Multiple Loewner hulls (Roth, Schleissinger, Starnes)

I Trees (Healey)

I Trace existence (Zhang, Zinsmeister)

Sample of other Loewner explorations

I Spacefilling curves (L, Rohde)

I Driving functions with higher regularity (Wong, L, Tran)

I Holder domains (Kinneberg)

I Complex-valued driving functions (Tran)

I Multiple Loewner hulls (Roth, Schleissinger, Starnes)

I Trees (Healey)

I Trace existence (Zhang, Zinsmeister)

Sample of other Loewner explorations

I Spacefilling curves (L, Rohde)

I Driving functions with higher regularity (Wong, L, Tran)

I Holder domains (Kinneberg)

I Complex-valued driving functions (Tran)

I Multiple Loewner hulls (Roth, Schleissinger, Starnes)

I Trees (Healey)

I Trace existence (Zhang, Zinsmeister)

Sample of other Loewner explorations

I Spacefilling curves (L, Rohde)

I Driving functions with higher regularity (Wong, L, Tran)

I Holder domains (Kinneberg)

I Complex-valued driving functions (Tran)

I Multiple Loewner hulls (Roth, Schleissinger, Starnes)

I Trees (Healey)

I Trace existence (Zhang, Zinsmeister)

Sample of other Loewner explorations

I Spacefilling curves (L, Rohde)

I Driving functions with higher regularity (Wong, L, Tran)

I Holder domains (Kinneberg)

I Complex-valued driving functions (Tran)

I Multiple Loewner hulls (Roth, Schleissinger, Starnes)

I Trees (Healey)

I Trace existence (Zhang, Zinsmeister)

Thank you

1 Introduction and Results

Let λ(t) be continuous and real valued and let gt : H \ Kt → H be the solution to the Loewnerequation

d

dtgt(z) =

2

gt(z) − λ(t), g0(z) = z ∈ H, (1.1)

where H is the upper half-plane. It was shown in [MR1] and [Li] that if λ is Holder continuouswith exponent 1/2 and if ||λ||1/2 < 4, then there is a simple curve γ with γ[0, t] = Kt and

γ \ γ(0) ⊂ H. The norm 4 is sharp as the examples λ(t) = κ√

1 − t show: Indeed, by [KNK],γ touches back on the real line if κ ≥ 4 (hence the driving term λ(t) = κ for 0 ≤ t ≤ t0 andλ(t) = κ

√t0 + 1 − t for t0 ≤ t ≤ t0 + 1 has a self-intersection in H for t0 sufficiently large).

It was also shown in [MR1] that there is a λ with ||λ||1/2 < ∞ such that K1 spirals infinitelyoften around some disc, and hence is not locally connected. The starting point of this paperis the observation that from the conformal mapping point of view, the zero angle cusp at thetangential self-intersection for λ(t) = 4

√1 − t is very similar to the infinitely spiraling prime

end, and that this is reflected in the driving terms:

Theorem 1.1. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , or if γ hasa tangential self-intersection, then its driving term λ satisfies

limt→T

|λ(T ) − λ(t)|√T − t

= 4.

Figure 1: An infinite spiral converging towards a star.

See Sections 2.1 and 5 for the definitions and precise statements. In Section 5 we show that forevery compact connected set A ⊂ H with connected complement, there is a sufficiently smoothinfinite spiral winding infinitely often around A with limit set ∂A; see Figure 1. The followingnatural question has been asked by Omer Angel: If the hull of λ is generated by a continuouscurve γ and if r < 1, is it true that the hull of rλ is generated by a continuous curve, too? Inother words, is the space of driving terms of continuous curves starlike? We answer this questionin the negative by proving

Theorem 1.2. If γ is a sufficiently smooth infinite spiral of half-plane capacity T , and if λ isits driving term, then the trace of rλ is continuous on the closed interval [0, T ] for all r = ±1.

The main work is in proving a form of stability of the (nontangential) self-intersection of λ(t) =κ√

1 − t for κ > 4 :

2