Levi condition

A condition, which can be effectively verified, for pseudo-convexity in the sense of Levi of domains in the complex space $ \mathbf C ^ {n} $. It was proposed by E.E. Levi

and consists of the following. Suppose that a domain $ D $ is specified in a neighbourhood $ U _ \zeta $ of a boundary point $ \zeta \in \partial D $ by the condition

$$ D \cap U _ \zeta = \{ {z = ( z _ {1} \dots z _ {n} ) \in U _ \zeta } : {\phi ( z) = \phi ( z , \overline{z}\; ) < 0 } \} , $$

where the real function $ \phi $ belongs to the class $ C ^ {2} ( U _ \zeta ) $ and $ \mathop{\rm grad} \phi ( \zeta ) \neq 0 $. If $ D $ is Levi pseudo-convex at $ \zeta $, then the (complex) Hessian

$$ \tag{1 } H ( \zeta ; \phi ) ( a , \overline{a}\; ) = \sum _ {j , k = 1 } ^ { n } \frac{\partial ^ {2} \phi }{\partial z _ {j} \partial {\overline{z}\; } _ {k} } ( \zeta ) a _ {j} {\overline{a}\; } _ {k} \geq 0 $$

is non-negative for all $ a = ( a _ {1} \dots a _ {n} ) \in \mathbf C ^ {n} $ that are complex orthogonal to $ \mathop{\rm grad} \phi ( \zeta ) $, that is, are such that

$$ \tag{2 } \sum _ { k= } 1 ^ { n } \frac{\partial \phi }{\partial z _ {k} } ( \zeta ) a _ {k} = 0 . $$

Conversely, if the condition

$$ \tag{3 } H ( \zeta ; \phi ) ( a , \overline{a}\; ) > 0 $$

is satisfied at the point $ \zeta \in \partial D $ for all $ a \neq 0 $ satisfying (2), then $ D $ is Levi pseudo-convex at $ \zeta $.

For $ n = 2 $ the inequalities (1) and (3) given above can be replaced by the simpler equivalent inequalities $ L ( \phi ) ( \zeta ) \geq 0 $ and $ L ( \phi ) ( \zeta ) > 0 $, respectively, where

$$ L ( \phi ) = - \left | \begin{array}{ccc} 0 & \frac{\partial \phi }{\partial z _ {1} } & \frac{\partial \phi }{\partial z _ {2} } \\ \frac{\partial \phi }{\partial {\overline{z}\; } _ {1} } & \frac{\partial ^ {2} \phi }{\partial z _ {1} \partial {\overline{z}\; } _ {1} } & \frac{\partial ^ {2} \phi }{\partial {\overline{z}\; } _ {1} {\partial z _ {2} } } \\ \frac{\partial \phi }{\partial {\overline{z}\; } _ {2} } & \frac{\partial ^ {2} \phi }{\partial z _ {1} \partial {\overline{z}\; } _ {2} } & \frac{\partial ^ {2} \phi }{\partial z _ {2} \partial \overline{ {z }}\; _ {2} } \\ \end{array} \right | $$

is the determinant of the Levi function $ \phi ( z) $.

The Levi condition (1)–(3) has also been generalized to domains on complex manifolds (see [4]).

References

[1a]	E.E. Levi, "Studii sui punti singolari essenziali delle funzioni analiticke de due o più variabili complesse" Ann. Mat. Pura Appl. , 17 (1910) pp. 61–87
[1b]	E.E. Levi, "Sulle ipersurficie dello spazio a 4 dimensione che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse" Ann. Mat. Pura Appl. , 18 (1911) pp. 69–79
[2]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[3]	B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)
[4]	R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)

Comments

By definition, $ D $ is Levi pseudo-convex at $ \zeta $ if (1) is satisfied for vectors that satisfy (2); $ D $ is called strictly (Levi) pseudo-convex at $ \zeta $ if (3) is satisfied for vectors that satisfy (2).

The domain $ D $ is called (Levi) pseudo-convex if it is Levi pseudo-convex at every boundary point.

For domains with $ C ^ {2} $ boundary, Levi pseudo-convexity is equivalent with any of the following:

a) $ \mathop{\rm log} d ( z) $ is plurisubharmonic on $ D $( i.e. $ D $ is Hartogs pseudo-convex), where $ d ( z) $ denotes the Euclidean distance of $ z $ to the boundary of $ D $.

b) $ K $ relatively compact in $ D $ implies $ \widehat{K} $ relatively compact in $ D $, where $ \widehat{K} = \{ {z \in D } : {p ( z) \leq \sup _ {z \in K } p( z) \textrm{ for every plurisubharmonic function } p \mathop{\rm on} D } \} $.

References

[a1]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)