Graham Theorem on Bounded Symmetric Domains

TL;DR

This paper extends Graham's theorem, which characterizes invariant harmonic functions as pluriharmonic on the unit ball, to classical bounded symmetric domains, revealing a broader rigidity phenomenon in complex analysis.

Contribution

The paper proves Graham's theorem on invariant harmonic functions holds on all classical bounded symmetric domains, including various types and special cases.

Findings

Graham's theorem is valid on classical bounded symmetric domains.

Invariant harmonic functions are pluriharmonic on these domains.

The result broadens understanding of harmonic function behavior in complex 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.