sub mean value property of plurisubharmonic function

Question

It is well known that a plurisubharmonic function $\varphi$ defined in a domain $\Omega\subset \mathbb C^n$ satisfies the sub mean value property. Now if $\varphi$ is defined on a complex manifold $X$, does it still have such a sub mean value property?

Answer 1

For a general complex manifold $X$, there isn't really any canonical way to define something like "small circles or balls centered at a point $x\in X$.

There is still some sub-mean value properties here though. Recall that a function $u : X \to \mathbb{R} \cup \{ -\infty \}$ is called plurisubharmonic if (and only if) for each holorphic disc $\phi : \mathbb{D} \to X$, the function $$ u \circ \phi : \mathbb{D} \to \mathbb{R} \cup \{ -\infty \} $$ is subharmonic. In particular, for each such disc $\phi$, we have that (with $x = \phi(0)$) $$ u(x) \le \max_{\xi \in \phi(\partial\mathbb{D}} u(\xi), $$ which gives you a "maximum principle". If you want a mean-value property instead, replace the right hand side with the suitable integral over $\phi(\partial\mathbb{D})$.

Another viewpoint: If $X$ is Kähler, then every plurisubharmonic function is subharmonic with respect to the Kähler metric, which gives you a more canonical sub-mean value inequality.