1. Loewner Spacefilling Curves
Bridget E. Jones
University of Tennessee, Knoxville

2. Loewner Equation
1-dimensional
function
2-dimensional curve

3. Example 1
L Equ
!

4. Example 2
L Equ
!

5. What is a spacefilling curve?
Definition
A spacefilling curve is a curve fills a region entirely.
Goal: Understand driving functions when transformed into
spacefilling curves.

6. Our method of generating examples

7. Our method of generating examples
L Equ
!
L Equ
!
λ(t) = −10
√
1 − t
λ(t) =



−10
√
0.25 − t − 5 : 0 <
10
√
0.5 − t − 10 : 0.25 <
−10
√
0.75 − t − 5 : 0.5 <
−10
√
1 − t : 0.75

8. Our method of generating examples

9. Our method of generating examples

10. Parameters for functions

11. Lip(1
2) functions
Definition
λ is in Lip(1
2) means that there is a c > 0 such that
|λ(t) − λ(s)| ≤ c |t − s|
for all s and t in the domain of λ.
The smallest value of c is called the Lip(1
2) norm of λ.

12. Example 3
λ(t) = 8
√
1 − t has Lip(1/2) norm 8.

13. Example 4
λ(t) =



6
√
1 − t : 0 < t < 1
6
√
2 − t − 6 : 1 < t < 2
6
√
3 − t − 12 : 2 < t < 3
has Lip(1/2) norm 6
√
3.

14. n Curves in a row
λ(t) =



6
√
1 − t : 0 < t < 1
6
√
2 − t − 6 : 1 < t < 2
6
√
3 − t − 12 : 2 < t < 3
6
√
4 − t − 18 : 3 < t < 4
6
√
5 − t − 24 : 4 < t < 5
has Lip(1/2) norm 6
√
5.

15. Example 5
λ(t) =
−8
√
1 − t : 0 < t < 1
8
√
2 − t − 8 : 1 < t < 2
has Lip(1/2) norm 8.

16. Properties of Lip(1
2) functions
Assume the Lip (1
2) norm of λ(t) = k
1. The Lip (1
2) norm of −λ(t) = k
2. The Lip (1
2) norm of λ(−t) = k
3. The Lip (1
2) norm of λ(t + c) = k
4. The Lip (1
2) norm of λ(t) + c = k
5. The Lip (1
2) norm of cλ(t) = |c|k
6. The Lip (1
2) norm of λ(ct) = k |c|

17. Proof of 1
Lip(1
2) norm is the smallest c such that
|λ(t) − λ(s)| ≤ c |t − s|.
Can be rewritten as:
c = sup {
λ(t) − λ(s)
|t − s|
}
By the definition Lip (1
2) norm, we have
|| − λ||1
2
= sup{
| − λ(t) − (−λ(s))|
|t − s|
}.
Then,
|| − λ||1
2
= sup{
|λ(t) − λ(s)|
|t − s|
}.
Thus,
|| − λ||1
2
= k.

18. Significance of the Lip (1
2) norm
The Lip (1
2) norm gives information about the geometry of a curve.
Theorem (Lind)
If the Lip (1
2) norm is less than 4, we have a simple curve.
Theorem (Lind)
If a curve is spacefilling, then the Lip (1
2) norm is greater than or
equal to 4.0001.
Note: 4.0001 is the wrong number. Our goal is to find out what
this number should be.

19. Lip(1
2) norm = 10 Lip(1
2) norm = 10
√
2
Lip(1
2) norm = 10
√
3 Lip(1
2) norm = 18.860925

20. Different Schemes Give Different Lip(1
2) norms

21. Data Results

22. Thank you for your attention!
And thanks to NSF for funding and to my advisor, Dr. Joan Lind.