aarjaneiro/planetmath huggingface dataset:
Oka coherence theorem

Oka coherence theorem

Let

M

be a

complex manifold

.

Theorem

.

Suppose

F

is an

analytic sheaf

over

M

, that is a subsheaf of

O

k

.
If

F

is a

locally finitely generated sheaf

, then

F

is a coherent analytic sheaf.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Oka coherence theorem

Canonical name

OkaCoherenceTheorem

Date of creation

2013-03-22 17:39:08

Last modified on

2013-03-22 17:39:08

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32C35

Synonym

Oka theorem

Synonym

Oka’s theorem

Synonym

Oka’s coherence theorem

Generated on Fri Feb 9 20:32:49 2018 by

LaTeXML
holomorphically convex

holomorphically convex

Let

G

⊂

ℂ

n

be a domain, or alternatively for a more general

definition

let

G

be an

n

dimensional

complex analytic manifold

.
Further let

𝒪

⁢

(

G

)

stand for the set of

holomorphic functions

on

G

.

Definition

.

Let

K

⊂

G

be a compact set.
We define the

holomorphically

of

K

as

K

^

G

:=

{

z

∈

G

∣

|

f

⁢

(

z

)

|

≤

sup

w

∈

K

⁡

|

f

⁢

(

w

)

|

⁢

for all

⁢

f

∈

𝒪

⁢

(

G

)

}

.

The domain

G

is called

holomorphically convex

if for every

K

⊂

G

compact

in

G

,

K

^

G

is also compact in

G

. Sometimes this is just abbreviated as

holomorph-convex

.

Note that when

n

=

1

, any domain

G

is holomorphically convex since when

n

=

1

K

^

G

=

K

for all compact

K

⊂

G

. Also note that this is the same as being a

domain of holomorphy

.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

holomorphically convex

Canonical

name

HolomorphicallyConvex

Date of creation

2013-03-22 15:04:33

Last modified on

2013-03-22 15:04:33

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Definition

Classification

msc 32E05

Synonym

holomorph-convex

Related topic

PolynomiallyConvexHull

Related topic

SteinManifold

Defines

holomorphically convex hull

Generated on Fri Feb 9 20:32:50 2018 by

LaTeXML
Stein manifold

Stein manifold

Definition

.

A

complex manifold

M

of complex dimension

n

is a

Stein manifold

if it satisfies the following

properties

1.

M

is

holomorphically convex

,

2.

if

z

,

w

∈

M

and

z

≠

w

then

f

⁢

(

z

)

≠

f

⁢

(

w

)

for some

function

f

holomorphic

on

M

(i.e.

M

is

holomorphically separable

),

3.

for every

z

∈

M

there are

holomorphic functions

f

1

,

…

,

f

n

which form a

coordinate system

at

z

(i.e.

M

is

holomorphically spreadable

).

Stein manifold is a generalization of the concept of the

domain of holomorphy

to manifolds. Furthermore, Stein manifolds are the generalizations of

Riemann surfaces

in higher

dimensions

. Every noncompact Riemann surface is a Stein manifold
by a

theorem

of Behnke and Stein.
Note that every domain of holomorphy in

ℂ

n

is a Stein manifold.
It is not hard to see that every closed complex submanifold of a Stein manifold is Stein.

Theorem

(Remmert, Narasimhan, Bishop).

If

M

is a Stein manifold of dimension

n

. There exists a proper (

http://planetmath.org/ProperMap

) holomorphic embedding of

M

into

C

2

⁢

n

+

1

.

Note that no

compact

complex manifold can be Stein since compact complex manifolds have no holomorphic functions. On the other hand, every compact complex manifold is holomorphically convex.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Stein manifold

Canonical

name

SteinManifold

Date of creation

2013-03-22 15:04:37

Last modified on

2013-03-22 15:04:37

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32E10

Related topic

HolomorphicallyConvex

Related topic

DomainOfHolomorphy

Defines

holomorphically separable

Defines

holomorphically spreadable

Generated on Fri Feb 9 20:32:52 2018 by

LaTeXML
Bergman metric

Bergman metric

Definition

.

Let

G

⊂

ℂ

n

be a domain and let

K

⁢

(

z

,

w

)

be the

Bergman kernel

on

G

. We define a Hermitian metric on the tangent bundle

T

z

⁢

ℂ

n

by

g

i

⁢

j

⁢

(

z

)

:=

∂

2

∂

⁡

z

i

⁢

∂

⁡

z

¯

j

⁢

log

⁡

K

⁢

(

z

,

z

)

,

for

z

∈

G

. Then the length of a

tangent vector

ξ

∈

T

z

⁢

ℂ

n

is then
given by

|

ξ

|

B

,

z

:=

∑

i

,

j

=

1

n

g

i

⁢

j

⁢

(

z

)

⁢

ξ

i

⁢

ξ

¯

j

.

This metric is called the

Bergman metric

on

G

.

The length of a (

piecewise

)

C

1

curve

γ

:

[

0

,

1

]

→

ℂ

n

is
then computed as

ℓ

⁢

(

γ

)

=

∫

0

1

|

∂

⁡

γ

∂

⁡

t

⁢

(

t

)

|

B

,

γ

⁢

(

t

)

⁢

𝑑

t

.

The

distance

d

G

⁢

(

p

,

q

)

of two points

p

,

q

∈

G

is then defined as

d

G

⁢

(

p

,

q

)

:=

inf

⁡

{

ℓ

⁢

(

γ

)

∣

all piecewise

C

1

curves

γ

such that

γ

⁢

(

0

)

=

p

and

γ

⁢

(

1

)

=

q

}

.

The distance

d

G

is called the

Bergman distance

.

The Bergman metric is in fact a

positive definite matrix

at each point if

G

is a bounded domain. More importantly, the distance

d

G

is

invariant

under

biholomorphic mappings

of

G

to another domain

G

′

. That is if

f

is a biholomorphism of

G

and

G

′

, then

d

G

⁢

(

p

,

q

)

=

d

G

′

⁢

(

f

⁢

(

p

)

,

f

⁢

(

q

)

)

.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Bergman metric

Canonical name

BergmanMetric

Date of creation

2013-03-22 15:04:49

Last modified on

2013-03-22 15:04:49

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 32F45

Related topic

BergmanKernel

Defines

Bergman distance

Generated on Fri Feb 9 20:32:54 2018 by

LaTeXML
biholomorphically equivalent

biholomorphically equivalent

Definition

.

Let

U

,

V

⊂

ℂ

n

. If there exists a one-to-one and onto

holomorphic

mapping

ϕ

:

U

→

V

such that the

inverse

ϕ

-

1

exists and is also holomorphic, then we say that

U

and

V

are

biholomorphically equivalent

or that they are

biholomorphic

. The mapping

ϕ

is called a

biholomorphic mapping

.

It is not an

obvious

fact, but if the source and target

dimension

are the same then every one-to-one holomorphic mapping is biholomorphic as a one-to-one holomorphic map has a nonvanishing jacobian.

When

n

=

1

biholomorphic equivalence is often called

conformal equivalence

(

http://planetmath.org/ConformallyEquivalent

), since in one complex
dimension, the one-to-one holomorphic mappings are

conformal mappings

.

Further if

n

=

1

then there are plenty of

conformal

(biholomorhic) equivalences,
since for example every simply connected domain (

http://planetmath.org/Domain2

) other than the whole

complex plane

is conformally
equivalent to the

unit disc

. On the other hand,
when

n

>

1

then the open unit ball and open unit

polydisc

are not biholomorphically equivalent. In fact there does not exist
a proper (

http://planetmath.org/ProperMap

) holomorphic mapping from one to the other.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

biholomorphically equivalent

Canonical

name

BiholomorphicallyEquivalent

Date of creation

2013-03-22 14:29:47

Last modified on

2013-03-22 14:29:47

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32H02

Synonym

biholomorphic

Synonym

biholomorphic equivalence

Defines

biholomorphic mapping

Generated on Fri Feb 9 20:32:56 2018 by

LaTeXML
Bloch’s constant

Bloch’s constant

Bloch’s theorem can be stated in the following way:

Bloch’s Theorem

.

Let

F

be the set of all

functions

f

holomorphic on a region containing the
closure of the disk

D

=

{

z

∈

C

:

|

z

|

<

1

}

and satisfying

f

⁢

(

0

)

=

0

and

f

′

⁢

(

0

)

=

1

. For each

f

∈

F

let

β

⁢

(

f

)

be the supremum of all numbers

r

such that there is a disk

S

⊂

D

on which

f

is injective and

f

⁢

(

S

)

contains a disk
of radius

r

. Let

B

be the infimum of all

β

⁢

(

f

)

, for

f

∈

F

. Then

B

≥

1

/

72

.

The constant

B

is usually referred to as Bloch’s constant.
Nowadays, better bounds are known and, in fact, it has been
conjectured that

B

has the following tantalizing form

B

=

Γ

⁢

(

1

/

3

)

⋅

Γ

⁢

(

11

/

12

)

(

1

+

3

)

⋅

Γ

⁢

(

1

/

4

)

where

Γ

⁢

(

x

)

is the

gamma function

.

References

1

John B. Conway,

Functions of One

Complex
Variable

I

, Second Edition, 1978, Springer-Verlag, New York.

Title

Bloch’s constant

Canonical name

BlochsConstant

Date of creation

2013-03-22 15:58:04

Last modified on

2013-03-22 15:58:04

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 32H02

Related topic

LandausConstant

Generated on Fri Feb 9 20:32:58 2018 by

LaTeXML
Bloch’s theorem

Bloch’s theorem

Let

f

be an

holomorphic function

on a region containing the closure of the disk

D

=

{

z

∈

ℂ

:

|

z

|

<

1

}

, such that

f

⁢

(

0

)

=

0

and

f

′

⁢

(

0

)

=

1

. Then there is a disk

S

⊂

D

such that

f

is injective on

S

and

f

⁢

(

S

)

contains a disk of radius

1

72

.

Title

Bloch’s theorem

Canonical name

BlochsTheorem

Date of creation

2013-03-22 13:15:17

Last modified on

2013-03-22 13:15:17

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Theorem

Classification

msc 32H02

Generated on Fri Feb 9 20:32:59 2018 by

LaTeXML
failure of Hartogs’ theorem in one dimension

failure of Hartogs’ theorem in one dimension

It is instructive to see an example where Hartogs’

theorem

fails in one

dimension

.
Take

U

=

ℂ

and let

K

=

{

0

}

.

The

function

1

z

is holomorphic in

U

∖

K

,

but cannot be extended to

U

.

To understand the example and failure of the theorem it is important to understand the proof (

http://planetmath.org/ProofOfHartogsTheorem

). In the proof, the way we construct an

extension

is that we start with a function holomorphic in

U

∖

K

,

modify it in a

neighbourhood

of

K

to be zero, hence extending as a

smooth function

through

K

.

Then we solve the

∂

¯

operator

(

http://planetmath.org/BarPartialOperator

) inhomogeneous equation

∂

¯

⁢

ψ

=

g

to “correct” our extension to be holomorphic.
The key point is that

g

has compact support allowing us to solve the equation and find a

ψ

with compact support. This fails in dimension 1. While we always get a solution

ψ

,

the solution can never have compact support. Hence, if we tried the proof with

1

z

,

the new function we obtain in the proof does not agree with

1

z

on any

open set

and hence is not an extension.

Title

failure of Hartogs’ theorem in one dimension

Canonical

name

FailureOfHartogsTheoremInOneDimension

Date of creation

2013-03-22 17:46:57

Last modified on

2013-03-22 17:46:57

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Example

Classification

msc 32H02

Generated on Fri Feb 9 20:33:01 2018 by

LaTeXML
Hartogs extension theorem

Hartogs extension theorem

Theorem

.

Suppose

V

is an

analytic variety

in an

open set

U

⊂

C

n

(

n

≥

2

)
of

dimension

at most

n

-

2

and suppose that

f

:

U

\

V

→

C

is

holomorphic

. Then there exists a unique holomorphic
extention of

f

to all of

U

.

Note that when

V

is 0 dimensional (a point) then this is just a special
case of the Hartogs’ phenomenon. Also note the similarity to Riemann’s

removable singularity

theorem in several variables, where however we also assume that

f

is

locally bounded

.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

2

Hassler Whitney.
.
Addison-Wesley, Philippines, 1972.

Title

Hartogs extension theorem

Canonical name

HartogsExtensionTheorem

Date of creation

2013-03-22 15:34:54

Last modified on

2013-03-22 15:34:54

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32H02

Generated on Fri Feb 9 20:33:03 2018 by

LaTeXML
Hartogs’ theorem

Hartogs’ theorem

Let

U

⊂

ℂ

n

(

n

>

1

) be an open set, and let

K

be a compact subset of

U

such that

U

-

K

is

connected

. Then any

holomorphic function

on

U

-

K

extends uniquely to a holomorphic function on

U

.

Title

Hartogs’ theorem

Canonical name

HartogsTheorem

Date of creation

2013-03-22 13:21:57

Last modified on

2013-03-22 13:21:57

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 32H02

Synonym

Hartogs’ Phenomenon

Generated on Fri Feb 9 20:33:04 2018 by

LaTeXML
Landau’s constant

Landau’s constant

We suggest that the reader reads first the entry on Bloch’s

constant

. Let

ℱ

be the set of all

functions

f

holomorphic on a region containing the

closure

of the disk

D

=

{

z

∈

ℂ

:

|

z

|

<

1

}

and satisfying

f

⁢

(

0

)

=

0

and

f

′

⁢

(

0

)

=

1

. For each

f

∈

ℱ

let

λ

⁢

(

f

)

be the

supremum

of all numbers

r

such that there is a disk

S

⊂

D

such that

f

⁢

(

S

)

contains a disk of radius

r

(notice that here we don’t require

f

to be

injective

on

S

).

Definition

.

Landau’s constant

L

is defined by

L

=

inf

⁡

{

λ

⁢

(

f

)

:

f

∈

ℱ

}

.

Let

B

be Bloch’s constant. Then, clearly,

L

≥

B

. The exact value of

L

(as that of

B

) is not known but it has been shown that

0.5

≤

L

≤

0.56

. In particular, it is known that

L

is strictly greater than

B

.

References

1

John B. Conway,

Functions of One

Complex
Variable

I

, Second Edition, 1978, Springer-Verlag, New York.

Title

Landau’s constant

Canonical name

LandausConstant

Date of creation

2013-03-22 15:58:07

Last modified on

2013-03-22 15:58:07

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 32H02

Related topic

BlochsConstant

Generated on Fri Feb 9 20:33:06 2018 by

LaTeXML
little Picard theorem

little Picard theorem

The range of a nonconstant

entire function

is either the whole

complex plane

ℂ

, or the complex plane with a single point removed.
In other words, if an entire function omits two or more values, then it is a constant function.

Title

little Picard theorem

Canonical name

LittlePicardTheorem

Date of creation

2013-03-22 13:15:20

Last modified on

2013-03-22 13:15:20

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Theorem

Classification

msc 32H25

Generated on Fri Feb 9 20:33:11 2018 by

LaTeXML
Picard’s theorem

Picard’s theorem

Let

f

be an

holomorphic function

with an

essential singularity

at

w

∈

ℂ

. Then there is a number

z

0

∈

ℂ

such that the image of any neighborhood of

w

by

f

contains

ℂ

-

{

z

0

}

. In other words,

f

assumes every

complex

value, with the possible exception of

z

0

, in any neighborhood of

w

.

Remark.

Little Picard theorem

follows as a corollary:
Given a nonconstant

entire function

f

, if it is a

polynomial

, it assumes every value in

ℂ

as a consequence of the

fundamental theorem of algebra

. If

f

is not a polynomial, then

g

⁢

(

z

)

=

f

⁢

(

1

/

z

)

has an essential singularity at

0

; Picard’s theorem implies that

g

(and thus

f

) assumes every complex value, with one possible exception.

Title

Picard’s theorem

Canonical name

PicardsTheorem

Date of creation

2013-03-22 13:15:23

Last modified on

2013-03-22 13:15:23

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Theorem

Classification

msc 32H25

Synonym

great Picard theorem

Related topic

EssentialSingularity

Related topic

CasoratiWeierstrassTheorem

Related topic

ProofOfCasoratiWeierstrassTheorem

Generated on Fri Feb 9 20:33:13 2018 by

LaTeXML
Cartan theorem A

Cartan theorem A

Let

𝒪

z

denote the ring of germs of

holomorphic functions

at

z

Theorem

(Cartan).

Suppose

F

is a

coherent analytic sheaf

on a

Stein
manifold

M

.
For every

z

∈

M

, the the stalk

F

z

is generated as an

O

z

module by the germs at

z

of the sections (

http://planetmath.org/Sheaf

)

Γ

⁢

(

M

,

F

)

.

Philosophically, this theorem says that there is good supply of of a coherent analytic sheaf on a Stein manifold.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Cartan theorem

A

Canonical name

CartanTheoremA

Date of creation

2013-03-22 17:39:10

Last modified on

2013-03-22 17:39:10

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32Q28

Classification

msc 32C35

Synonym

Cartan’s theorem A

Related topic

CartanTheoremB

Generated on Fri Feb 9 20:33:15 2018 by

LaTeXML
Cartan theorem B

Cartan theorem B

Theorem

(Cartan).

Suppose

F

is a

coherent analytic sheaf

on a

Stein
manifold

M

.
Then for

k

>

0

, then

H

k

⁢

(

M

,

ℱ

)

=

0

.

Here,

H

k

⁢

(

M

,

ℱ

)

is the

k

th

cohomology group

valued in the sheaf

ℱ

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Cartan theorem B

Canonical name

CartanTheoremB

Date of creation

2013-03-22 17:39:13

Last modified on

2013-03-22 17:39:13

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32Q28

Classification

msc 32C35

Synonym

Cartan’s theorem B

Related topic

CartanTheoremA

Generated on Fri Feb 9 20:33:16 2018 by

LaTeXML
Newlander-Niremberg theorem

Newlander-Niremberg theorem

Let

M

be a

smooth manifold

of real dimension

m

=

2

⁢

n

. Then every integrable almost

complex structure

J

on

M

is defined by a unique

analytic structure

.

Title

Newlander-Niremberg theorem

Canonical name

NewlanderNirembergTheorem

Date of creation

2013-03-22 15:37:53

Last modified on

2013-03-22 15:37:53

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

5

Author

Simone (5904)

Entry type

Theorem

Classification

msc 32Q60

Generated on Fri Feb 9 20:33:18 2018 by

LaTeXML
complex analytic manifold

complex analytic manifold

Definition

.

A

manifold

M

is called a

complex analytic manifold

(or sometimes just
a

complex manifold

) if the

transition functions

are

holomorphic

.

Definition

.

A subset

N

⊂

M

is called a

complex analytic submanifold

of

M

if

N

is closed in

M

and if for every point

z

∈

N

there is a coordinate neighbourhood

U

in

M

with coordinates

z

1

,

…

,

z

n

such that

U

∩

N

=

{

p

∈

U

∣

z

d

+

1

⁢

(

p

)

=

…

=

z

n

⁢

(

p

)

}

for some integer

d

≤

n

.

Obviously

N

is now also a complex analytic manifold itself.

For a complex analytic manifold,

dimension

always means the complex dimension,
not the real dimension. That is

M

is of dimension

n

when there are neighbourhoods of every point homeomorphic to

ℂ

n

. Such a manifold is of real dimension

2

⁢

n

if we identify

ℂ

n

with

ℝ

2

⁢

n

.
Of course the

tangent bundle

is now also a complex vector space.

A

function

f

is said to be holomorphic on

M

if it is a

holomorphic function

when considered as a function of the local coordinates.

Examples of complex analytic manifolds are for example the

Stein manifolds

or the

Riemann surfaces

. Of course also any open set in

ℂ

n

is also a complex analytic manifold. Another example may be the set of

regular points

of an analytic set.

Complex analytic manifolds can also be considered as a special case of

CR manifolds

where the CR dimension is maximal.

Complex manifolds are sometimes described as manifolds carrying an or . This refers to the atlas and transition functions defined on the manifold.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

complex analytic manifold

Canonical

name

ComplexAnalyticManifold

Date of creation

2013-03-22 15:04:40

Last modified on

2013-03-22 15:04:40

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 32Q99

Synonym

complex manifold

Defines

complex analytic submanifold

Defines

complex submanifold

Defines

analytic structure

Defines

holomorphic structure

Generated on Fri Feb 9 20:33:20 2018 by

LaTeXML
analytic disc

analytic disc

Definition

.

Let

D

:=

{

z

∈

ℂ

∣

|

z

|

<

1

}

be the
open

unit disc

. A non-constant

holomorphic

mapping

φ

:

D

→

ℂ

n

is called an

analytic disc

in

ℂ

n

. The
really refers to both the embedding and the image.
If the mapping

φ

extends continuously to the closed unit disc

D

¯

, then

φ

⁢

(

D

¯

)

is called a

closed analytic disc

and

φ

⁢

(

∂

⁡

D

)

is called the

boundary of a closed analytic
disc

.

Analytic discs play in some sense a role of

line segments

in

ℂ

n

.
For example they give another way to see that a domain

G

⊂

ℂ

n

is

pseudoconvex

. See the

Hartogs Kontinuitatssatz

theorem.

Another use of analytic discs are as a technique for extending

CR functions

on

generic manifolds

[

1

]

. The idea here is that you can always extend a

function

from the boundary of a disc to the inside of the disc by solving the Dirichlet problem.

Definition

.

A closed analytic disc

φ

is said to be

attached

to a set

M

⊂

ℂ

n

if

φ

⁢

(

∂

⁡

D

)

⊂

M

, that is if

φ

maps the boundary of the unit disc to

M

.

Analytic discs are also used for defining the Kobayashi metric and thus plays a role in the study of invariant metrics.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

analytic disc

Canonical name

AnalyticDisc

Date of creation

2013-03-22 14:30:49

Last modified on

2013-03-22 14:30:49

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Definition

Classification

msc 32T05

Defines

closed analytic disc

Defines

boundary of a closed analytic disc

Defines

attached analytic disc

Generated on Fri Feb 9 20:33:22 2018 by

LaTeXML