Plurisubharmonic function

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In mathematics, plurisubharmonic functions form an important class of functions used in complex analysis. They are the higher dimensional generalization of subharmonic functions. They can be defined in full generality on complex spaces.

1 Formal definition
- 1.1 Differentiable plurisubharmonic functions
2 Properties
3 Applications
4 References

[edit] Formal definition

A function

$f \colon G \to {\mathbb{R}}\cup\{-\infty\}$ ,

with domain $G \subset {\mathbb{C}}^n$ is called plurisubharmonic if it is upper semi-continuous, and for every complex line

$\{ a + b z \mid z \in {\mathbb{C}} \}\subset {\mathbb{C}}^n$ with $a, b \in {\mathbb{C}}^n$

the function $z \mapsto f(a + bz)$ is a subharmonic function on the set

$\{ z \in {\mathbb{C}} \mid a + b z \in G \}.$

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

$f \colon X \to {\mathbb{R}} \cup \{ - \infty \}$

is said plurisubharmonic if and only if for any holomorphic map $\varphi\colon\Delta\to X$ the function

$f\circ\varphi \colon \Delta \to {\mathbb{R}} \cup \{ - \infty \}$

is subharmonic, where $\Delta\subset{\mathbb{C}}$ denotes the unit disk.

[edit] Differentiable plurisubharmonic functions

If $f$ is of (differentiability) class $C 2$ , then $f$ is plurisubharmonic, if and only if the hermitian matrix $L f = (λ i j)$ , called Levi matrix, with entries

$\lambda_{ij}=\frac{\partial^2f}{\partial z_i\partial\bar z_j}$

is positive definite.

[edit] Properties

The set of plurisubharmonic functions form a convex cone in the vector space of semicontinuous functions, i.e.

if $f$ is a plurisubharmonic function and $c > 0$ a positive real number, then the function $c\cdot f$ is plurisubharmonic,
if $f 1$ and $f 2$ are plurisubharmonic functions, then the sum $f 1 + f 2$ 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 $f$ is plurisubharmonic and $\phi:\mathbb{R}\to\mathbb{R}$ a monotonically increasing, convex function then $\phi\circ f$ is plurisubharmonic.
If $f 1$ and $f 2$ are plurisubharmonic functions, then the function $f(x):=\sup(f_1(x),f_2(x))$ is plurisubharmonic.
If $f 1, f 2,...$ is a monotonically decreasing sequence of plurisubharmonic functions

then so is $f(x):=\lim_{n\to\infty}f_n(x)$ .

The inequality in the usual semi-continuity condition holds as equality, i.e. if $f$ is plurisubharmonic then

$\limsup_{x\to x_0}f(x) =f(x_0)$

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

Plurisubharmonic functions satisfy the maximum principle, i.e. if $f$ is plurisubharmonic on the connected domain $D$ and

$\sup_{x\in D}f(x) =f(x_0)$

for some point $x_0\in D$ then $f$ is constant.

[edit] Applications

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

[edit] 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.

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Categories: Subharmonic functions | Several complex variables