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exhaustion function
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(Definition)
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For example  is pseudoconvex if and only if  has a continuous plurisubharmonic exhaustion function.
We can also define a bounded version.
Definition 2 Let
 be a domain and let
![$ f \colon G \to (-\infty,c]$](http://images.planetmath.org:8080/cache/objects/6092/l2h/img9.png "$ f \colon G \to (-\infty,c]$") for some
 , is called a bounded exhaustion function whenever
is relatively compact in  for all  .
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.
- 1
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"exhaustion function" is owned by jirka.
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(view preamble | get metadata)
See Also: pseudoconvex
| Also defines: |
bounded exhaustion function, hyperconvex |
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Cross-references: unbounded, Hartogs' triangle, bounded, plurisubharmonic, continuous, pseudoconvex, relatively compact, domain
There is 1 reference to this entry.
This is version 2 of exhaustion function, born on 2004-08-09, modified 2005-03-07.
Object id is 6092, canonical name is ExhaustionFunction.
Accessed 3352 times total.
Classification:
| AMS MSC: | 32T35 (Several complex variables and analytic spaces :: Pseudoconvex domains :: Exhaustion functions) | | | 32U10 (Several complex variables and analytic spaces :: Pluripotential theory :: Plurisubharmonic exhaustion functions) |
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Pending Errata and Addenda
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