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

TL;DR

This paper explores the topology of certain Ricci solitons by developing the theory of f-harmonic maps, providing new existence and vanishing results that extend classical harmonic map theory to weighted manifolds.

Contribution

It advances the understanding of Ricci solitons' topology by extending harmonic map theory to the weighted setting, including new existence and vanishing theorems.

Findings

Generalized Schoen-Yau theory to weighted manifolds

Proved existence of f-harmonic maps under certain conditions

Established vanishing results for f-harmonic maps

Abstract

In this paper we give some results on the topology of manifolds with $\infty$-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.