Plurisubharmonic function

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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[edit]

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[edit]

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[edit]

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 .


More Examples

  • If is an analytic function on an open set, then is plurisubharmonic on that open set.
  • Convex functions are plurisubharmonic
  • If is a Domain of Holomorphy then is plurisubharmonic
  • Harmonic functions are not necessarily plurisubharmonic

History[edit]

Plurisubharmonic functions were defined in 1942 by Kiyoshi Oka [1] and Pierre Lelong.[2]

Properties[edit]

  • 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).

for some point then is constant.

Applications[edit]

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

Oka theorem[edit]

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[edit]

  • 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[edit]

  • Hazewinkel, Michiel, ed. (2001) [1994], "Plurisubharmonic function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

Notes[edit]

  1. ^ a b K. 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.