Graham type theorem on classical 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 $$B_{n}$$ (Graham in Commun Partial Differ Equ 8(5):433–476, 1983). This rigidity phenomenon of Graham has been studied by many authors [see, for examples, Graham and Lee (Duke Math J 57:697–720, 1988), Li and Simon (Am J Math 124:1045–1057, 2002), Li and Wei (Sci China Math 53:779–790, 2010), etc]. In this paper, we prove that Graham theorem holds on classical bounded symmetric domains, which include Type I and Type II domains, Type III domain III(n) and Type IV domain IV(n) with even $$n\ge 4$$ .