Two results on the weighted Poincaré inequality on complete Kähler manifolds

In this paper we consider complete noncompact Kähler manifolds $M^m$ that satisfy the weighted Poincaré inequality with a weight function $\rho (x)$ that has limit zero at infinity of $M$. We prove that if the Ricci curvature lower bound $Ric_M(x)\geq -4\rho (x)$ holds on $M$ then the manifold has one nonparabolic end and if the bisectional curvature is bounded from below by $BK_M(x)\geq -\frac{\rho (x)}{m^2}$ then the manifold has one end, thus it is connected at infinity. The two results that we prove are the Kähler version of Theorem 6.3 and Theorem 7.2 in \cite{L-W4} and improve some results in \cite{L-W}.

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Published 1 January 2007