Liouville theorems for harmonic 1-forms on gradient Ricci solitons

Abstract

We prove that there is no nontrivial $L^2$-integrable harmonic 1-form on noncompact complete gradient steady Ricci solitons or noncompact complete gradient shrinking Kähler-Ricci solitons. As an application, it can be used to distinguish certain flat vector bundles that arise from fundamental group representations into $SL(r,\mathbb{C})$.