Graham type theorem on classical bounded symmetric domains

  • Ren-Yu ChenEmail author
  • Song-Ying Li


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 \(n\ge 4\).


Invariant harmonic Bounded symmetric domains Pluriharmonic Laplace–Beltrami operator 

Mathematics Subject Classification

Primary 32A50 Secondary 32T15 35J70 35R01 



The authors thank the referee for a very careful reading of the manuscript and providing a number of helpful comments and corrections.


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Authors and Affiliations

  1. 1.School of MathematicsTianjin UniversityTianjinPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of California IrvineIrvineUSA
  3. 3.College of Mathematics and InformaticsFujian Normal UniversityFuzhouPeople’s Republic of China

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