Topology of steady and expanding gradient Ricci solitons via f-harmonic maps

In this paper we give some results on the topology of manifolds with $\mathrm{\infty }$-Bakry-'Emery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifolds. Notably, we prove existence and vanishing results which generalize to the weighted setting part of Schoen and Yau’s theory of harmonic maps.