Resumen de Two results on the weighted Poincar\'{e} inequality on complete K\"{a}hler manifolds

Ovidiu Munteanu

• In this paper we consider complete noncompact K\"{a}hler manifolds $M^m$ that satisfy the weighted Poincar\'{e} 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 K\"{a}hler version of Theorem 6.3 and Theorem 7.2 in \cite{L-W4} and improve some results in \cite{L-W}.