The Bergman metric on complete Kähler manifolds
The author studies the Bergman completeness of complete Kähler manifolds. The main result of the paper is the following theorem. Let \((M,\omega)\) be a complete Kähler manifold with non-positive sectional curvature and the injectivity radius bounded below by a constant \(\tau>0\). Assume that for each \(y\in M\) there exists a plurisubharmonic function \(\lambda_y:M\rightarrow[-1,0)\) such that \(\partial\overline\partial\lambda_y\geq C'\omega\) on the geodesic ball \(B(y,\tau)\), and there exists a plurisubharmonic exhaustion function \(\Phi:M\rightarrow[-1,0)\) such that \(| \Phi(x)-\Phi(y)| \leq c\rho^\alpha(x,y)\), \(x,y\in M\), where \(\rho\) is the distance on \(M\). Then for every \(x_0\in M\) there exists a constant \(C>0\) such that \(\text{dist}_\beta(x_0,x)\geq C\log(-\log(-\Phi(x)))\), \(x\in M\), where dist\({}_\beta\) is the Bergman distance on \(M\). In particular, if \(M\) is a simply connected complete Kähler manifold with sectional curvature bounded above by a negative constant, then for every \(x_0\in M\) there exists a constant \(C>0\) such that dist\({}_\beta(x_0,x)\geq C\log\rho(x_0,x)\), \(x\in M\).
scientific article; zbMATH DE number 2018274