Vanishing results from Lichnerowicz Laplacian on complete K\"{a}hler manifolds and applications

TL;DR

This paper establishes rigidity results for harmonic forms on complete Kähler manifolds and explores applications to non-compact Kähler and quaternion Kähler manifolds, extending previous work in the field.

Contribution

It extends existing results on harmonic forms to the setting of complete, non-compact Kähler manifolds, providing new rigidity theorems and applications.

Findings

Rigidity results for harmonic (p,q)-forms

Applications to manifolds with parallel Bochner tensor

Extensions of Petersen and Wink's results

Abstract

In this paper, we show several rigidity results for harmonic $(p,q)$-forms in complete K\"{a}hler manifolds. We also give several applications to study non-compact K\"{a}hler manifolds with parallel Bochner tensor or quaternion K\"{a}hler manifolds. Our results are natural extensions of Petersen and Wink's results in \cite{PW21, PW} in the setting of complete, non-compact K\"{a}hler manifolds.