Plurisubharmonic Function

A plurisubharmonic function is a real-valued function of complex variables on a domain of complex space , , satisfying the following conditions:

1. is upper semicontinuous everywhere in ;
2. is a subharmonic function of the variable in each connected component of the open set for any fixed points , .

Related definitions

A function is called a plurisuperharmonic function if is a plurisubharmonic function.

Properties

Plurisubharmonic functions are subharmonic, but for the converse is not true.

In addition to the general properties of subharmonic functions, plurisubharmonic functions have the following properties:

• is a plurisubharmonic function in a domain if and only if is a plurisubharmonic function in a neighborhood of each point ;
• a linear combination of plurisubharmonic functions with positive coefficients is a plurisubharmonic function;
• the limits of uniformly convergent and monotonically decreasing sequences of plurisubharmonic functions are plurisubharmonic;
• for any point the mean value

over the sphere of radius is an increasing function of , convex with respect to on the interval , if the ball lies in ;

• under holomorphic maps, a plurisubharmonic function is carried to a plurisubharmonic function;
• if is a continuous plurisubharmonic function in a domain , is a closed connected analytic subset of , and the restriction attains a maximum on , then on .

Examples

, for , where is a holomorphic function in .

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

for some Kähler form , then is plurisubharmonic, and is called a Kähler potential.

Relation to the Dirac delta: on one-dimensional complex Euclidean space , is plurisubharmonic. Ifis a C ^∞ function with compact support , then the Cauchy integral formula says

which can be rewritten as

.

This is nothing other than the 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

Applications

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

Although the notion of a plurisubharmonic function can be hard to grasp, it has found applications in various areas of mathematics and physics. Here are a few examples:

Potential theory: Plurisubharmonic functions are used to study potential fields arising in various physical problems, such as electrostatics, gravitation and heat conduction.

Complex analysis: Plurisubharmonic functions are important in the study of analytic functions, especially in the theory of harmonic functions. They help to study the characteristics of the boundaries and integrals of analytic functions.

Probability theory: Plurisubharmonic functions have found application in the study of stochastic processes and probability theory. They are related to certain classes of martingales and can be used to model various random phenomena.

Geometry: Plurisubharmonic functions are used to study the properties of various classes of geometric objects, such as conformal maps, harmonic measures and harmonic functions at different levels of the complex plane.

These are only a few examples of applications of plurisubharmonic functions. In general, they are important tools in various areas of mathematics, physics, probabilistic modeling and geometry, making it possible to study and analyze various mathematical structures and physical phenomena.

Oka's theorem

The main geometric application of the theory of plurisubharmonic functions is a famous theorem proved by Kiyoshi Oka in 1942 ^.

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 .

Oka's theorem: Let M be a complex manifold admitting a smooth, exhaustive, strongly plurisubharmonic function. Then M is Stein . Conversely, every Stein manifold admits such a function.

See also

• Pluriharmonic function
• Hessian of a function