Liouville theorems for harmonic 1-forms on gradient Ricci solitons

TL;DR

This paper proves the nonexistence of nontrivial $L^2$ harmonic 1-forms on certain noncompact gradient Ricci solitons, aiding in the classification of flat vector bundles from fundamental group representations.

Contribution

It establishes Liouville theorems for harmonic 1-forms on specific gradient Ricci solitons, providing new tools for geometric analysis and bundle classification.

Findings

No nontrivial $L^2$ harmonic 1-forms on steady Ricci solitons

No nontrivial $L^2$ harmonic 1-forms on shrinking Kähler-Ricci solitons

Application to distinguishing flat vector bundles from fundamental group representations

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 K\"{a}hler-Ricci solitons. As an application, it can be used to distinguish certain flat vector bundles that arise from fundamental group representations into $SL(r,\mathbb{C})$.