Refined Kato type inequalities and new vanishing theorems on complete K\"ahler and quaternionic K\"ahler manifolds

TL;DR

This paper extends vanishing theorems for harmonic functions and forms on complete Kähler and quaternionic Kähler manifolds using refined Kato inequalities and Sobolev inequalities, broadening previous results to $L^p$ contexts.

Contribution

It introduces generalized vanishing theorems for $L^p$-integrable pluriharmonic functions and harmonic 1-forms on complete Kähler and quaternionic Kähler manifolds, employing refined Kato inequalities.

Findings

Vanishing of $L^p$-integrable pluriharmonic functions under Sobolev inequalities.

Vanishing of harmonic 1-forms on complete Kähler and quaternionic Kähler manifolds.

Extension of classical results to broader $L^p$-integrability settings.

Abstract

Given a complete Riemannian manifold satisfying a weighted Poincar\'{e} inequality and having a bounded below Ricci curvature, various vanishing theorems for harmonic functions and harmonic 1-forms have been published. We generalized these results to $L^p$-integrable pluriharmonic functions and harmonic 1-forms on complete K\"{a}hler and quaternionic K\"{a}hler manifolds respectively by utilizing the B\"ochner technique and several refined Kato type inequalities. Moreover, we also prove the vanishing property of pluriharmonic functions with finite $L^p$ energy on complete K\"{a}hler manifolds satisfying a Sobolev type inequality.