Graham Theorem on Bounded Symmetric Domains

Abstract

Graham Theorem on the unit ball $B_{n}$ in $\mathbb{C}^{n}$ states that every invariant harmonic function $u\in C^{n}(\overline{B}_{n})$ must be pluriharmonic in $B_{n}$. This rigidity phenomenon of Graham have been studied by many authors. In this paper, we prove that Graham theorem holds on classical bounded symmetric domains. Which include Type I domains, Type II domains, Type III domains III(n) with even $n$ and some special Type IV domains.