A finiteness theorem for ends of weighted manifolds with a weighted Poincaré inequality
DOI:
https://doi.org/10.56764/hpu2.jos.2025.4.03.11-25Abstract
In this paper, we study complete weighted manifolds that satisfy a weighted Poincaré inequality, with the associated weight function assumed to be non-negative throughout the manifold. Our main focus is to study the geometric consequences of such an inequality on the global structure of the manifold, particularly at infinity. Specifically, we prove that such a manifold has only finitely many \(\phi\)-nonparabolic ends, provided that the Bakry-Émery Ricci curvature is bounded from below outside a compact subset with respect to the weight function. This result generalizes several existing theorems in the theory of Riemannian geometry and offers valuable insight into the interplay between curvature conditions and the topology of ends.
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