aarjaneiro/planetmath huggingface dataset:
analytic polyhedron

analytic polyhedron

Definition

.

Suppose

G

⊂

ℂ

n

is a domain and let

W

⊂

G

be an

open set

. Let

f

1

,

…

,

f

k

:

W

→

ℂ

be

holomorphic
functions

. Then if the set

Ω

:=

{

z

∈

W

∣

|

f

j

⁢

(

z

)

|

<

1

,

j

=

1

,

…

,

k

}

is relatively

compact

in

W

, we say that

Ω

is an

analytic polyhedron

in

G

. Sometimes it is denoted

Ω

⁢

(

f

1

,

…

,

f

k

)

. Further

(

W

,

f

1

,

…

,

f

k

)

is called the

of the analytic polyhedron

.

An analytic polyhedron is automatically a

domain of holomorphy

by using
the

functions

that define it as

g

⁢

(

z

)

:=

1

e

i

⁢

θ

-

f

j

⁢

(

z

)

to
show that

g

cannot be extended beyond a point where

f

j

⁢

(

z

)

=

e

i

⁢

θ

.
Every boundary point of

Ω

is of that form for some

f

j

.

Furthermore every domain of holomorphy can be exhausted by analytic polyhedra (that is, every compact subset is contained in an analytic polyhedron)
and in fact only domains of holomorphy can be exhausted by analytic
polyhedra, see the

Behnke-Stein theorem

.

Note that sometimes

W

is required to be

homeomorphic

to the

unit ball

.

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

analytic polyhedron

Canonical name

AnalyticPolyhedron

Date of creation

2013-03-22 14:32:39

Last modified on

2013-03-22 14:32:39

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32T05

Classification

msc 32A07

Synonym

analytic polyhedra

Defines

frame of an analytic polyhedron

Generated on Fri Feb 9 20:33:24 2018 by

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Behnke-Stein theorem

Behnke-Stein theorem

Theorem

(Behnke-Stein).

Suppose that

G

k

⊂

C

n

are

domains of holomorphy

such that

G

k

⊂

G

j

whenever

k

<

j

. Then

⋃

k

=

1

∞

G

k

is a domain of holomorphy.

This is related to the fact that an increasing union of

pseudoconvex

domains is pseudoconvex and so it can be proven using that fact and the solution of the

Levi problem

. Though historically this theorem was in fact used to solve
the Levi problem.

References

1

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

Title

Behnke-Stein theorem

Canonical name

BehnkeSteinTheorem

Date of creation

2013-03-22 14:31:13

Last modified on

2013-03-22 14:31:13

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32T05

Synonym

increasing union of domains of holomorphy is a domain of holomorphy

Generated on Fri Feb 9 20:33:25 2018 by

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domain of holomorphy

domain of holomorphy

Definition

.

An open set

Ω

⊂

ℂ

n

is called a

domain of holomorphy

if there do not exist non-empty open sets

U

⊂

Ω

and

V

⊂

ℂ

n

where

V

is

connected

,

V

⊄

Ω

and

U

⊂

Ω

∩

V

such that for every

holomorphic function

f

on

Ω

there exists
a holomorphic function

g

on

V

such that

f

=

g

on

U

.

When

n

=

1

, then every open set is a domain of holomorphy. For an example, assume that the boundary of

Ω

⊂

ℂ

is a

Jordan curve

for

simplicity

. We can define a holomorphic function which has zeros which accumulate on the boundary of the domain and thus the

function

cannot be continued past any point in the boundary.
If you could extend the function, it would be identically zero.

Alternatively given any open set

Ω

⊂

ℂ

and any point

p

∈

∂

⁡

Ω

,

the function

z

↦

1

z

-

p

is holomorphic in

Ω

, but cannot be continued past

p

.

For

n

≥

2

many domains are not domains of holomorphy. For example if you take

ℂ

2

∖

{

0

}

,

this is no longer a domain of holomorphy by Hartogs’s theorem (

http://planetmath.org/HartogsTheorem

). It turns out that a domain is a domain of holomorphy if and only if the boundary is

pseudoconvex

. In particular, every convex (in the classical sense)
domain is a domain of holomorphy. examples of domains of holomorphy are

ℂ

n

,

an open ball, or a

polydisc

.

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

domain of holomorphy

Canonical name

DomainOfHolomorphy

Date of creation

2013-03-22 14:29:29

Last modified on

2013-03-22 14:29:29

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32T05

Classification

msc 32A10

Related topic

LeviPseudoconvex

Related topic

SolutionOfTheLeviProblem

Related topic

SteinManifold

Generated on Fri Feb 9 20:33:27 2018 by

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Hartogs triangle

Hartogs triangle

A non-trivial example of

domain of holomorphy

that has some
interesting
non-obvious properties is the

Hartogs triangle

which is the set

{

(

z

,

w

)

∈

ℂ

2

∣

|

z

|

<

|

w

|

<

1

}

.

Since it is a

Reinhardt domain

it can be represented
by plotting it on the plane

|

z

|

×

|

w

|

as follows.

Figure 1: Hartogs triangle

It is obvious then where the name comes from. To see that this is a domain of holomorphy, then given a boundary point we wish to exhibit a

holomorphic function

on the whole Hartogs triangle which does not extend beyond that point. First note that on the top boundary

z

is anything and

w

=

e

i

⁢

θ

for some

θ

, so

f

⁢

(

z

,

w

)

=

1

w

-

e

i

⁢

θ

will not extend beyond

(

z

,

e

i

⁢

θ

)

.
Now for the diagonal boundary this is where

|

z

|

=

|

w

|

,
that is

z

=

e

i

⁢

θ

⁢

w

for some

θ

, so

f

⁢

(

z

,

w

)

=

1

z

-

e

i

⁢

θ

⁢

w

will do not extend beyond

(

e

i

⁢

θ

⁢

w

,

w

)

.

One of the many properties of this domain is that if

U

is the Hartogs
triangle, then it is a domain of holomorphy, but if we take a sufficently
small neighbourhood

V

of

U

¯

(the closure of

U

),
then any

function

holomorphic on

V

is holomorphic on the

polydisc

D

2

⁢

(

0

,

1

)

(just fill in everything below the triangle in Figure 1). So if

V

does not include all of

D

2

⁢

(

0

,

1

)

then it is not a domain of
holomorphy. This is because a Reinhardt domain that contains zero (the point

(

0

,

0

)

) is a domain
of holomorphy if and only if it is a

logarithmically convex set

and any neighbourhood of

U

¯

does contain zero while

U

itself does not.

References

1

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

Title

Hartogs triangle

Canonical name

HartogsTriangle

Date of creation

2013-03-22 14:31:08

Last modified on

2013-03-22 14:31:08

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Example

Classification

msc 32T05

Generated on Fri Feb 9 20:33:29 2018 by

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Kontinuitätssatz

Kontinuitätssatz

Theorem

.

G

⊂

ℂ

n

is

pseudoconvex

if and only if
for any family of

closed analytic discs

{

d

α

}

α

∈

I

in

G

with

∪

α

∈

I

∂

⁡

d

α

being
a relatively compact set in

G

then

∪

α

∈

I

d

α

is also
a relatively compact set in

G

.

This is the analogue of one of the

definitions

of a convex set. Just replace
pseudoconvex with convex and closed analytic discs with

closed line segments

.

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

Kontinuitätssatz

Canonical

name

Kontinuitatssatz

Date of creation

2013-03-22 15:49:15

Last modified on

2013-03-22 15:49:15

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32T05

Synonym

Hartogs Kontinuitätssatz

Synonym

Kontinuitatssatz

Generated on Fri Feb 9 20:33:31 2018 by

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solution of the Levi problem

solution of the Levi problem

The

Levi problem

is the problem of characterizing

domains of
holomorphy

by a local condition on the boundary that does not involve

holomorphic functions

themselves. This condition turned out to
be pseudoconvexity.

Theorem

.

An open set

G

⊂

C

n

is a domain of holomorphy if
and only if

G

is

pseudoconvex

.

The forward direction (domain of holomorphy implies pseudoconvexity) is
not hard to prove and was known for a long time. The opposite direction
is really what’s called the solution to the Levi problem.

References

1

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

Title

solution of the Levi problem

Canonical name

SolutionOfTheLeviProblem

Date of creation

2013-03-22 14:31:11

Last modified on

2013-03-22 14:31:11

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32T05

Classification

msc 32E40

Related topic

Pseudoconvex

Related topic

DomainOfHolomorphy

Defines

Levi problem

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

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biholomorphisms of strongly pseudoconvex domains extend to the boundary

biholomorphisms of strongly pseudoconvex domains extend to the boundary

It is a basic question in complex analysis to ask when does a

biholomorphic mapping

of two domains extend to the boundary. The following is a celebrated theorem of Fefferman for

strongly pseudoconvex

domains.

Theorem

(Fefferman).

Let

U

,

V

⊂

C

n

(

n

≥

2

) be two strongly pseudoconvex domains with smooth (

C

∞

) boundaries (the boundaries are

smooth submanifolds

). Let

f

:

U

→

V

be a biholomorphism.
Then

f

extends to a smooth

diffeomorphism

of

U

¯

to

V

¯

.

References

1

Fefferman, Charles.
.
Invent. Math.

26

(1974), 1–65.

2

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

Title

biholomorphisms of strongly pseudoconvex domains extend to the boundary

Canonical name

BiholomorphismsOfStronglyPseudoconvexDomainsExtendToTheBoundary

Date of creation

2013-03-22 16:44:30

Last modified on

2013-03-22 16:44:30

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32T15

Related topic

LeviPseudoconvex

Generated on Fri Feb 9 20:33:34 2018 by

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pseudoconvex

pseudoconvex

Definition

.

Let

G

⊂

ℂ

n

be a domain (open

connected

subset).
We say

G

is

pseudoconvex

(or

Hartogs pseudoconvex

) if there exists a

continuous

plurisubharmonic function

φ

on

G

such that
the sets

{

z

∈

G

∣

φ

⁢

(

z

)

<

x

}

are relatively compact
subsets of

G

for all

x

∈

ℝ

. That is we say that

G

has a continuous plurisubharmonic

exhaustion function

.

When

G

has a

C

2

(twice

continuously differentiable

) boundary then this
notion is the same as Levi pseudoconvexity (

http://planetmath.org/LeviPseudoconvex

), which
is easier to work with if you have such nice boundaries. If you don’t have
nice boundaries then the following approximation result can come in useful.

Proposition

.

If

G

⊂

C

n

is pseudoconvex then there exist bounded,

strongly Levi pseudoconvex

domains

G

k

⊂

G

with

C

∞

(smooth)
boundary which are relatively compact
in

G

, such that

G

=

⋃

k

=

1

∞

G

k

.

This is because once we have a

φ

as in the definition we can actually find a

C

∞

exhaustion function.

The reason for the definition of pseudoconvexity is that it classifies

domains of holomorphy

. One thing to note then is that every open domain in one complex
dimension (in the complex plane

ℂ

) is then pseudoconvex.

References

1

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

Title

pseudoconvex

Canonical name

Pseudoconvex

Date of creation

2013-03-22 14:30:58

Last modified on

2013-03-22 14:30:58

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32T15

Classification

msc 32T05

Synonym

Hartogs pseudoconvex

Related topic

LeviPseudoconvex

Related topic

SolutionOfTheLeviProblem

Related topic

ExhaustionFunction

Generated on Fri Feb 9 20:33:39 2018 by

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method of exhaustion

method of exhaustion

The

method of exhaustion

is calculating an area by approximating it by the areas of a sequence of polygons.

For example, filling up the interior of a circle by inscribing polygons with more and more sides.

Title

method of exhaustion

Canonical

name

MethodOfExhaustion

Date of creation

2013-03-22 13:54:46

Last modified on

2013-03-22 13:54:46

Owner

vmoraru (1243)

Last modified by

vmoraru (1243)

Numerical id

5

Author

vmoraru (1243)

Entry type

Definition

Classification

msc 32T35

Related topic

ExhaustionByCompactSets

Generated on Fri Feb 9 20:33:41 2018 by

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Hartogs functions

Hartogs functions

Definition

.

Let

G

⊂

ℂ

n

be an open set and let

ℱ

G

be the smallest class of

functions

on

G

to

ℝ

∪

{

-

∞

}

that contains all of the functions

z

↦

log

⁡

|

f

⁢

(

z

)

|

where

f

is

holomorphic

on

G

and such
that

ℱ

G

is closed with respect to the following conditions:

•

If

φ

1

,

φ

2

∈

ℱ

G

, then

φ

1

+

φ

2

∈

ℱ

G

.

•

If

φ

∈

ℱ

G

then

a

⁢

φ

1

∈

ℱ

G

for all

a

≥

0

.

•

If

{

φ

k

}

∈

ℱ

G

and

φ

1

≥

φ

2

≥

…

,
then

lim

k

→

∞

⁡

φ

k

∈

ℱ

G

.

•

If

{

φ

k

}

∈

ℱ

G

and
the sequence is uniformly bounded above on compact sets, then

sup

k

⁡

φ

k

∈

ℱ

G

.

•

If

φ

∈

ℱ

G

and

φ

^

⁢

(

w

)

:=

lim sup

w

→

z

⁡

φ

⁢

(

w

)

,
then

φ

^

∈

ℱ

G

•

If

φ

|

U

∈

ℱ

U

for all

U

⊂

G

where

U

is relatively

compact

(the closure
of

U

is compact), then

φ

∈

ℱ

G

.

These functions are called the

Hartogs functions

.

It is known that if

n

=

1

then the upper semi-continuous Hartogs functions
are precisely the

subharmonic functions

on

G

.

Theorem

(H. Bremerman).

All

plurisubharmonic functions

are Hartogs functions if

G

is a

domain of holomorphy

.

References

1

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

Title

Hartogs functions

Canonical name

HartogsFunctions

Date of creation

2013-03-22 14:29:27

Last modified on

2013-03-22 14:29:27

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32U05

Generated on Fri Feb 9 20:33:43 2018 by

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pluripolar set

pluripolar set

Definition

.

Let

G

⊂

ℂ

n

and let

f

:

G

→

ℝ

∪

{

-

∞

}

be a

plurisubharmonic
function

which is not identically

-

∞

.
The set

𝒫

:=

{

z

∈

G

∣

f

⁢

(

z

)

=

-

∞

}

is
called a

pluripolar set

.

If

f

is a

holomorphic function

then

log

⁡

|

f

|

is a plurisubharmonic function. The zero set of

f

is then
a pluripolar set.

References

1

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

Title

pluripolar set

Canonical name

PluripolarSet

Date of creation

2013-03-22 14:29:15

Last modified on

2013-03-22 14:29:15

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32U05

Classification

msc 31C10

Generated on Fri Feb 9 20:33:44 2018 by

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exhaustion function

exhaustion function

Definition

.

Let

G

⊂

ℂ

n

be a domain and let

f

:

G

→

ℝ

is called an

exhaustion function

whenever

{

z

∈

G

∣

f

⁢

(

z

)

<

r

}

is relatively compact in

G

for all

r

∈

ℝ

.

For example

G

is

pseudoconvex

if and only if

G

has a

continuous

plurisubharmonic

exhaustion function.

We can also define a bounded version.

Definition

.

Let

G

⊂

ℂ

n

be a domain and let

f

:

G

→

(

-

∞

,

c

]

for some

c

∈

ℝ

,
is called a

bounded exhaustion function

whenever

{

z

∈

G

∣

f

⁢

(

z

)

<

r

}

is relatively compact in

G

for all

r

<

c

.

A domain which has a bounded plurisubharmonic exhaustion function is usually
referred to as a

hyperconvex

domain. Note that not all pseudoconvex
domains have a bounded plurisubharmonic exhaustion function. For example
the Hartogs’s triangle does not, though it does have an unbounded one.

References

1

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

Title

exhaustion function

Canonical name

ExhaustionFunction

Date of creation

2013-03-22 14:32:41

Last modified on

2013-03-22 14:32:41

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32U10

Classification

msc 32T35

Related topic

Pseudoconvex

Defines

bounded exhaustion function

Defines

hyperconvex

Generated on Fri Feb 9 20:33:46 2018 by

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generic manifold

generic manifold

Definition

.

Let

M

⊂

ℂ

N

be a

real submanifold

of real dimension

n

. We say that

M

is a

generic manifold

if for every

x

∈

M

we have

T

x

⁢

(

M

)

+

J

⁢

T

x

⁢

(

M

)

=

T

x

⁢

(

ℂ

N

)

,

where

J

denotes the operator of multiplication by the

imaginary unit

in

T

x

⁢

(

ℂ

N

)

. That is every vector in

T

x

⁢

(

ℂ

N

)

can be written as

X

+

J

⁢

Y

where

X

,

Y

∈

T

x

⁢

(

M

)

.

For more details about the tangent spaces and the

J

operator see the
entry on

CR manifolds

(

http://planetmath.org/CRSubmanifold

). In fact every generic manifold is
also CR manifold (the converse is not true however). A basic important result
about generic submanifolds is.

Theorem

.

Let

M

⊂

C

N

be a generic submanifold and let

f

:

U

⊂

C

N

→

C

be a

holomorphic function

where

U

is a

connected

open set such that

M

∩

U

≠

∅

, and further
suppose that

f

⁢

(

M

∩

U

)

=

{

0

}

, that is

f

is zero when restricted
to

M

. Then in fact

f

≡

0

on

U

.

For example in

ℂ

1

the real line is a generic submanifold, and any holomorphic function which is zero on the real line is zero everywhere (if the
domain of the

function

is connected and intersects the real line of course). There are of course much stronger uniqueness results for the

complex plane

so the above is mostly useful for higher dimensions.

References

1

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

Title

generic manifold

Canonical name

GenericManifold

Date of creation

2013-03-22 14:56:03

Last modified on

2013-03-22 14:56:03

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32V05

Synonym

generic submanifold

Related topic

CRSubmanifold

Related topic

TotallyRealSubmanifold

Related topic

TangentialCauchyRiemannComplexOfCinftySmoothForms

Related topic

ACRcomplex

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Levi flat

Levi flat

Suppose

M

⊂

ℂ

n

is at least a

C

2

hypersurface

.

Definition

.

M

is

Levi-flat

if it is

pseudoconvex

from both sides, or equivalently if and only if the

Levi form

of

M

vanishes

identically.

Suppose

M

is locally defined by

ρ

=

0

.
The vanishing of the Levi form is equivalent to the

complex

Hessian

of

ρ

vanishing on all

holomorphic

vectors

tangent

to the hypersurface.
Hence

M

is Levi-flat if and only if the complex bordered Hessian of

ρ

is of rank two on the hypersurface. In other words, it is not hard to see that

M

is Levi-flat if and only if

rank

⁡

[

ρ

ρ

z

ρ

z

¯

ρ

z

⁢

z

¯

]

=

2

for all points on

{

ρ

=

0

}

.

Here

ρ

z

is the row vector

[

∂

⁡

ρ

∂

⁡

z

1

,

…

,

∂

⁡

ρ

∂

⁡

z

n

]

,

ρ

z

¯

is the column vector

[

∂

⁡

ρ

∂

⁡

z

1

,

…

,

∂

⁡

ρ

∂

⁡

z

n

]

T

,

and

ρ

z

⁢

z

¯

is the complex Hessian

[

∂

2

⁡

ρ

∂

⁡

z

i

⁢

∂

⁡

z

¯

j

]

i

⁢

j

.

Let

T

c

⁢

M

be the

complex tangent space

of

M

,

that is at each point

p

∈

M

,

define

T

p

c

⁢

M

=

J

⁢

(

T

p

⁢

M

)

∩

T

p

⁢

M

,

where

J

is the complex structure.
Since

M

is a hypersurface the

dimension

of

T

p

c

⁢

M

is always

2

⁢

n

-

2

,

and so

T

c

⁢

M

is a subbundle of

T

⁢

M

.

M

is Levi-flat
if and only if

T

c

⁢

M

is involutive. Since the leaves are graphs of functions that satisfy
the

Cauchy-Riemann equations

, the leaves are

complex analytic

. Hence,

M

is Levi-flat, if and only if it is foliated by complex hypersurfaces.

The cannonical example of a Levi-flat hypersurface is the hypersurface defined in

ℂ

n

by
the equation

Im

⁡

z

1

=

0

. In fact, locally, all real analytic Levi-flat hypersurfaces
are

biholomorphic

to this example.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.
Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

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

Title

Levi flat

Canonical

name

LeviFlat

Date of creation

2013-03-22 17:39:41

Last modified on

2013-03-22 17:39:41

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Definition

Classification

msc 32V05

Synonym

Levi-flat

Generated on Fri Feb 9 20:33:55 2018 by

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totally real submanifold

totally real submanifold

Definition

.

Suppose that

M

⊂

ℂ

N

is a

CR submanifold

. If the CR dimension of

M

is 0, we say that

M

is

totally real

. If in

addition

M

is generic (

http://planetmath.org/GenericManifold

), then

M

is said to be

maximally totally real

(or sometimes just

maximally real

).

Note that if

M

is maximally totally real, then the real dimension is automatically

N

, this is because

T

x

c

⁢

(

M

)

=

T

x

⁢

(

M

)

∩

J

⁢

T

x

⁢

(

M

)

(the complex tangent space) is of

dimension

0, and thus

T

x

⁢

(

M

)

must be of real dimension

N

if

M

is to be a

generic manifold

.

References

1

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

Title

totally real submanifold

Canonical

name

TotallyRealSubmanifold

Date of creation

2013-03-22 14:56:05

Last modified on

2013-03-22 14:56:05

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32V05

Synonym

totally real manifold

Related topic

CRSubmanifold

Related topic

GenericManifold

Defines

maximally totally real submanifold

Defines

maximally totally real manifold

Defines

maximally real manifold

Generated on Fri Feb 9 20:33:56 2018 by

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CR function

CR function

Definition

.

Let

M

⊂

ℂ

N

be a

CR submanifold

and let

f

be a

C

k

⁢

(

M

)

(

k

times

continuously differentiable

)

function

to

ℂ

where

k

≥

1

. Then

f

is a

CR function

if for every CR vector
field

L

on

M

we have

L

⁢

f

≡

0

. A

distribution

(

http://planetmath.org/Distribution4

)

f

on

M

is called a

CR distribution

if similarly every CR vector field annihilates

f

.

For example

restrictions

of

holomorphic functions

in

ℂ

N

to

M

are CR functions. The

converse

is not always true and is not easy to
see. For example the following basic

theorem

is very useful when you have

real analytic submanifolds

.

Theorem

.

Let

M

⊂

C

N

be a

generic submanifold

which is

real
analytic

(the defining function is real analytic). And let

f

:

M

→

C

be a real analytic function. Then

f

is a CR function if
and only if

f

is a restriction to

M

of a holomorphic function
defined in an open neighbourhood of

M

in

C

N

.

References

1

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

Title

CR function

Canonical name

CRFunction

Date of creation

2013-03-22 14:57:10

Last modified on

2013-03-22 14:57:10

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32V10

Defines

CR distribution

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