“Plurisubharmonic function”的意思、由来-开放百科全书

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Plurisubharmonic function

释义

Formal definition
Differentiable plurisubharmonic functions
Examples
History
Properties
Applications
Oka theorem
References
External links
Notes

In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions (which are defined on a Riemannian manifold) plurisubharmonic functions can be defined in full generality on complex analytic spaces.

Formal definition

A function

with domain

is called plurisubharmonic if it is upper semi-continuous, and for every complex line

with

the function is a subharmonic function on the set

In full generality, the notion can be defined on an arbitrary complex manifold or even a Complex analytic space as follows. An upper semi-continuous function

is said to be plurisubharmonic if and only if for any holomorphic map

the function

is subharmonic, where denotes the unit disk.

Differentiable plurisubharmonic functions

If is of (differentiability) class , then is plurisubharmonic if and only if the hermitian matrix , called Levi matrix, with

entries

is positive semidefinite.

Equivalently, a -function f is plurisubharmonic if and only if is a positive (1,1)-form.

Examples

Relation to Kähler manifold: On n-dimensional complex Euclidean space , is plurisubharmonic. In fact, is equal to the standard Kähler form on up to constant multiples. More generally, if satisfies

for some Kähler form , then is plurisubharmonic, which is called Kähler potential.

Relation to Dirac Delta: On 1-dimensional complex Euclidean space , is plurisubharmonic. If is a C^∞-class function with compact support, then Cauchy integral formula says

which can be modified to

.

It is nothing but Dirac measure at the origin 0 .

History

Plurisubharmonic functions were defined in 1942 by

Kiyoshi Oka ^[1] and Pierre Lelong.^[2]

Properties

The set of plurisubharmonic functions form a convex cone in the vector space of semicontinuous functions, i.e.
if is a plurisubharmonic function and a positive real number, then the function is plurisubharmonic,
if and are plurisubharmonic functions, then the sum is a plurisubharmonic function.
Plurisubharmonicity is a local property, i.e. a function is plurisubharmonic if and only if it is plurisubharmonic in a neighborhood of each point.
If is plurisubharmonic and a monotonically increasing, convex function then is plurisubharmonic.
If and are plurisubharmonic functions, then the function is plurisubharmonic.
If is a monotonically decreasing sequence of plurisubharmonic functions

then is plurisubharmonic.

Every continuous plurisubharmonic function can be obtained as the limit of a monotonically decreasing sequence of smooth plurisubharmonic functions. Moreover, this sequence can be chosen uniformly convergent.^[3]
The inequality in the usual semi-continuity condition holds as equality, i.e. if is plurisubharmonic then

(see limit superior and limit inferior for the definition of lim sup).

Plurisubharmonic functions are subharmonic, for any Kähler metric.
Therefore, plurisubharmonic functions satisfy the maximum principle, i.e. if is plurisubharmonic on the connected open domain and

for some point then is constant.

Applications

In complex analysis, plurisubharmonic functions are used to describe pseudoconvex domains, domains of holomorphy and Stein manifolds.

Oka theorem

The main geometric application of the theory of plurisubharmonic functions is the famous theorem proven by Kiyoshi Oka in 1942.^[1]

A continuous function

is called exhaustive if the preimage

is compact for all . A plurisubharmonic

function f is called strongly plurisubharmonic

if the form

is positive, for some Kähler form

on M.

Theorem of Oka: Let M be a complex manifold,

admitting a smooth, exhaustive, strongly plurisubharmonic function.

Then M is Stein. Conversely, any

Stein manifold admits such a function.

References

Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Robert C. Gunning. Introduction to Holomorphic Functions in Several Variables, Wadsworth & Brooks/Cole.
Klimek, Pluripotential Theory, Clarendon Press 1992.

External links

{{springer|title=Plurisubharmonic function|id=p/p072930}}

Notes

1. ^¹K. Oka, Domaines pseudoconvexes, Tohoku Math. J. 49 (1942), 15–52.
2. ^P. Lelong, Definition des fonctions plurisousharmoniques, C. R. Acd. Sci. Paris 215 (1942), 398–400.
3. ^R. E. Greene and H. Wu, -approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Scient. Ec. Norm. Sup. 12 (1979), 47–84.

2 : Subharmonic functions|Several complex variables

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