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The Bergman metric on a Stein manifold with a bounded plurisubharmonic function. (English) Zbl 0997.32011
Let $$M$$ be a Stein manifold such that for every $$y\in M$$ there exists an $$a>0$$ for which $$A(y,a):=\{x\in M: g_M(x,y)<-a\}\subset\subset M$$, where $$g_M(\cdot,y)$$ denotes the pluricomplex Green function with pole at $$y$$.
In the main result of the paper the authors prove that then the Bergman metric of $$M$$ is everywhere positive definite. If, moreover, for every sequence $$(y_k)_k\subset M$$, without accumulation point in $$M$$, there exist a subsequence $$(y_{k_j})_j$$ and an $$a>0$$ such that for any compact set $$K\subset M$$ we have $$A(y_{k_j},a)\cap K=\emptyset$$ for large $$j$$, then $$M$$ is Bergman complete. As a consequence the authors get a positive answer to the Greene-Wu conjecture: Let $$M$$ be a simply connected complete Kähler $$n$$-dimensional manifold with non-positive sectional curvature which is $$\leq -A/\rho^2$$ outside a compact subset of $$M$$ (with a constant $$A>0$$). Then the Bergman metric of $$M$$ is everywhere positive definite and $$M$$ is Bergman complete.

##### MSC:
 32E10 Stein spaces 32U35 Plurisubharmonic extremal functions, pluricomplex Green functions
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##### References:
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