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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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