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

The paper gives a method to generate the potential functions which can induce Kahler metrics u = u(i(j)) over bar dz(i) circle times d(z) over bar (j) of Bergman type on the unit ball B-n in C-n. The paper proves that if h is an element of C-n ((B) over bar (n)) is harmonic in these metrics u (Delta(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).