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In this article, we use the pluricomplex Green function to give a sucient condition for the existence and the completeness of the Bergman met- ric. As a consequence, we proved that a simply connected complete Kahler manifold possesses a complete Bergman metric provided that the Riemann sectional curvature A=2, which implies a conjecture of Greene and Wu. Moreover, we obtain a sharp estimate for the Bergman distance on such man- ifolds.

The Bergman metric on a Stein manifold with a bounded plurisubharmonic function