Pluriharmonic function

Definition

Let $U\subseteq\mathbb C^d$ be open. A real-valued function $u$ on $U$ is pluriharmonic if its restriction to every affine complex line is harmonic on each component of the intersection with $U$ .

Differential characterization

For $u\in C^2(U)$ , pluriharmonicity is equivalent to the vanishing of the entire Levi form:

\frac{\partial^2u}{\partial z_j\partial\bar z_k}=0 \qquad (1\le j,k\le d).

Thus both $u$ and $-u$ are plurisubharmonic. In particular, every pluriharmonic function is harmonic, but the converse fails in complex dimension greater than one.

Local holomorphic representation

Locally, a pluriharmonic function is the real part of a holomorphic function: near every point there is a holomorphic $F$ with $u=\operatorname{Re}F$ . On a non-simply-connected domain these local conjugates can have nontrivial periods, so a single global $F$ need not exist.

References

Lars Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd ed., North-Holland, 1990. Relevant: Chapter 2, plurisubharmonic and pluriharmonic functions.
Marek Klimek, Pluripotential Theory, Oxford University Press, 1991. Publisher record.