Liouville-type theorems for weighted $p$-harmonic $1$-forms and weighted $p$-harmonic maps

Keomkyo Seo and Gabjin Yun

Vol. 305 (2020), No. 1, 291–310

DOI: 10.2140/pjm.2020.305.291

Abstract

In this paper, we obtain Bochner–Weitzenböck formulas for the weighted Hodge Laplacian operator acting on differential forms and more generally on vector bundle-valued weighted $p$-harmonic forms. Applying these formulas, we prove Liouville-type theorems for weighted $L^q$ $p$-harmonic $1$-forms and for weighted $p$-harmonic maps in a weighted complete noncompact manifold with nonnegative Bakry–Émery Ricci curvature, where $q=2p-2$ or $q=p$.

Keywords

harmonic form, harmonic map, Liouville-type theorem, weighted manifold

Mathematical Subject Classification 2010

Primary: 53C20

Secondary: 58A10, 58E20

Milestones

Received: 30 April 2018

Revised: 17 February 2019

Accepted: 31 October 2019

Published: 17 March 2020