Abstract. We prove that the Bergman metric on domains satisfying condition S is complete. This implies that any bounded pseudoconvex domain with Lipschitz boundary is complete with respect to the Bergman metric. We also show that bounded hyperconvex domains in the plane and convex domains in C n are Bergman comlete.
Abstract. In this article, we use the pluricomplex Green function to give a sufficient condition for the existence and the completeness of the Bergman metric. As a consequence, we proved that a simply connected complete Kähler manifold possesses a complete Bergman metric provided that the Riemann sectional curvature ≤ −A/ρ 2 , which implies a conjecture of Greene and Wu. Moreover, we obtain a sharp estimate for the Bergman distance on such manifolds.
Let Ω ⊂ C n be a bounded domain with the hyperconvexity index α(Ω) > 0. Let ̺ be the relative extremal function of a fixed closed ball in Ω and set µ := |̺|(1 + | log |̺||) −1 , ν := |̺|(1 + | log |̺||) n . We obtain the following estimates for the Bergman kernel: (1) For every 0 < α < α(Ω) and 2 ≤ p < 2+ 2α(Ω) 2n−α(Ω) , there exists a constant C > 0 such that Ω | K Ω (·,w) √K Ω (w)
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