conformal radius

conformal radius

Definition

.

Let

G

⊂

ℂ

be a

simply connected

region that is not the whole
plane and let

a

∈

G

be any point.
The

Riemann mapping theorem

tells us that there exists
a unique one-to-one and onto holomorphic map

f

:

𝔻

→

G

(where

𝔻

is the

unit disc

) such that

f

⁢

(

0

)

=

a

and

f

′

⁢

(

0

)

>

0

. Then
define the

conformal radius

r

⁢

(

G

,

a

)

=

f

′

⁢

(

0

)

.

Example

.

For example, take

G

=

B

⁢

(

0

,

δ

)

(the open ball of radius

δ

around 0) for some

δ

>

0

, then

r

⁢

(

G

,

0

)

=

δ

because
we have a map

f

⁢

(

z

)

=

δ

⋅

z

as our unique map.
And thus this definition
coincides with our definition of radius for this special case.

Example

.

For another example we look at how the conformal radius is affected by
the choice of the point

a

. So suppose that we take

G

to be the unit disc
(

D

)
itself and we take some point

a

∈

D

.
The unique map that takes 0 to

a

is
the map

f

⁢

(

z

)

=

z

+

a

1

+

a

¯

⁢

z

(where

a

¯

is the

complex conjugate

of

a

) and by the quotient rule we get that

f

′

⁢

(

z

)

=

1

-

|

a

|

2

(

1

+

a

¯

⁢

z

)

2

. And so

r

⁢

(

D

,

a

)

=

f

′

⁢

(

0

)

=

1

-

|

a

|

2

, so the conformal radius
of the unit disc goes to 0 as we move the point

a

towards the boundary
of the disc, and
it is largest (equal to 1) when

a

=

0

.

From the first example we can now see another way of characterizing the conformal
radius. Take the inverse map (inverses of

holomorphic

one-to-one functions are also always holomorphic) and call it

φ

:

G

→

𝔻

(the map such that

φ

⁢

(

f

⁢

(

z

)

)

=

z

).
We take the derivative (see the entry on

univalent functions

(

http://planetmath.org/UnivalentFunction

)) we get

φ

′

⁢

(

f

⁢

(

0

)

)

=

1

f

′

⁢

(

0

)

, that is

φ

′

⁢

(

a

)

=

1

r

(where we call

r

=

r

⁢

(

G

,

a

)

for brevity now).
If we multiply the map by
the conformal radius we get a map

γ

:

G

→

B

⁢

(

0

,

r

)

such that

γ

⁢

(

z

)

=

r

⋅

φ

⁢

(

z

)

and

γ

′

⁢

(

a

)

=

1

. By uniqueness of the
map arising from the Riemann mapping theorem we can see that

γ

is
also unique. Thus we could define the conformal radius as follows.

Definition

.

Let

G

⊂

ℂ

be a region and let

a

∈

G

be any
point. By application of Riemann mapping
theorem there exists a unique map

γ

:

G

→

B

⁢

(

0

,

r

)

for
some

r

>

0

, such that

γ

⁢

(

a

)

=

0

and

γ

′

⁢

(

a

)

=

1

. The

conformal radius

is then defined as

r

⁢

(

G

,

a

)

=

r

.

This definition gives more of an intuitive understanding of why we’d call this the conformal radius of

G

. We look at the unique map with

γ

′

⁢

(

a

)

=

1

, that is, the map that doesn’t “stretch” the set. So the radius of

G

with respect to

a

is really the radius of the unique ball around zero to which

G

is

conformally equivalent

without any “stretching” needed.

References

1

S. Rohde, M. Zinsmeister. ,

Journal d’Analyse

(to appear).
Available at

http://www.math.washington.edu/ rohde/papers/rozi.ps

http://www.math.washington.edu/ rohde/papers/rozi.ps

Title

conformal radius

Canonical name

ConformalRadius

Date of creation

2013-03-22 14:18:33

Last modified on

2013-03-22 14:18:33

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

9

Author

jirka (4157)

Entry type

Definition

Classification

msc 30C55

Related topic

RiemannMappingTheorem

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