Levi pseudoconvex

Levi pseudoconvex

Let

G

⊂

ℂ

n

be a domain (

http://planetmath.org/Domain2

) (open connected subset) with

C

2

boundary, that is the boundary is locally the graph of a twice

continuously
differentiable function

. Let

ρ

:

ℂ

n

→

ℝ

be a defining function of

G

,
that is

ρ

is a twice continuously differentiable function such that

grad

⁡

ρ

⁢

(

z

)

≠

0

for

z

∈

∂

⁡

G

and

G

=

{

z

∈

ℂ

n

∣

ρ

⁢

(

z

)

<

0

}

(such a function always exists).

Definition

.

Let

p

∈

∂

⁡

G

(boundary of

G

).
We call the space of vectors

w

=

(

w

1

,

…

,

w

n

)

∈

ℂ

n

such that

∑

k

=

1

n

∂

⁡

ρ

∂

⁡

z

k

⁢

(

p

)

⁢

w

k

=

0

,

the space of

holomorphic tangent vectors

at

p

and denote it

T

p

1

,

0

⁢

(

∂

⁡

G

)

.

T

p

1

,

0

⁢

(

∂

⁡

G

)

is an

n

-

1

dimensional

complex vector space

and is a

subspace

of the complexified real tangent space (

http://planetmath.org/TangentSpace

), that is

ℂ

⊗

ℝ

T

p

⁢

(

∂

⁡

G

)

.

Note that when

n

=

1

then the

complex
tangent space

contains just the

zero vector

.

Definition

.

The point

p

∈

∂

⁡

G

is called

Levi pseudoconvex

(or just

pseudoconvex

)
if

∑

j

,

k

=

1

n

∂

2

⁡

ρ

∂

⁡

z

j

⁢

∂

⁡

z

¯

k

⁢

(

p

)

⁢

w

j

⁢

w

¯

k

≥

0

,

for all

w

∈

T

p

1

,

0

⁢

(

∂

⁡

G

)

. The point is
called

strongly Levi pseudoconvex

(or just

strongly pseudoconvex

or also

strictly pseudoconvex

)
if the inequality above is strict. The

expression

on the left is
called the

Levi form

.

Note that if a point is not strongly Levi pseudoconvex then it is sometimes called a

weakly Levi pseudoconvex

point.

The Levi form really acts on an

n

-

1

dimensional space, so the expression above may be confusing as it only acts on

T

p

1

,

0

⁢

(

∂

⁡

G

)

and not on all
of

ℂ

n

.

Definition

.

The domain

G

is called Levi pseudoconvex
if every boundary point is
Levi pseudoconvex. Similarly

G

is called strongly Levi pseudoconvex
if every boundary point is
strongly Levi pseudoconvex.

Note that in particular all convex domains are pseudoconvex.

It turns out that

G

with

C

2

boundary is a

domain of holomorphy

if and only if

G

is Levi pseudoconvex.

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

Levi pseudoconvex

Canonical

name

LeviPseudoconvex

Date of creation

2013-03-22 14:30:37

Last modified on

2013-03-22 14:30:37

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

10

Author

jirka (4157)

Entry type

Definition

Classification

msc 32T15

Classification

msc 32T05

Related topic

DomainOfHolomorphy

Related topic

Pseudoconvex

Related topic

BiholomorphismsOfStronglyPseudoconvexDomainsExtendToTheBoundary

Defines

Levi form

Defines

strongly Levi pseudoconvex

Defines

strongly pseudoconvex

Defines

strictly pseudoconvex

Defines

weakly pseudoconvex

Defines

weakly Levi pseudoconvex

Defines

holomorphic tangent vector

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