Two results on the weighted Poincaré inequality on complete Kähler manifolds (Q2474604)

A Riemannian manifold \(M^n\) is said to satisfy the weighted Poincaré inequality with a positive weight function \(\rho\) if \(\int_M \rho(x) \phi^2 dv(x)\leq \int_M | \nabla \phi| ^2(x) dv(x)\) for any compactly supported smooth function \(\phi\). This article studies the topology at infinity of complete Kähler manifolds with weighted Poincaré inequality relative to \(\rho\) where the curvature behavior is controlled by \(\rho\). This is a Kähler version of some results for complete Riemannian manifolds by \textit{P. Li} and \textit{J. Wang} [J. Differ. Geom. 69, No. 1, 43--74 (2005; Zbl 1087.53067)]. One theorem says the following: assume \(n\geq 2\) and the Ricci curvature is bounded from below by \(-4\rho(x)\). Then, if \(\lim_{x\rightarrow\infty}\rho(x)=0\), the manifold \(M\) has only one nonparabolic end at infinity.