On the rigidity theorem for harmonic functions in Kähler metric of Bergman type

Abstract

The paper gives a method to generate the potential functions which can induce Kähler metrics u = u $u_{i\bar j}$dz _i ⊗d $\bar z_j$ of Bergman type on the unit ball B _n in ℂⁿ. The paper proves that if h ∈ ℂⁿ($\bar B_n$) is harmonic in these metrics u (Δ _u h = 0) in B _n , then h must be pluriharmonic in B _n . In fact, it is a characterization theorem, as a consequence, the paper provides a way to construct many counter examples for the potential functions of the metric u so that the above theorem fails. The results in this paper generalize the theorems of Graham (1983) and examples constructed by Graham and Lee (1988).