A note on \(p\)-harmonic 1-forms on complete manifolds (Q2764796)

A result of \textit{R. E. Greene} and \textit{H. Wu} [Mich. Math. J. 28, 63-81 (1981; Zbl 0477.53058)] states that harmonic 1-forms in \(L^q, (1<q< +\infty)\) on a complete manifold with non-negative Ricci curvature must vanish. The purpose of this note is to extend this theorem to \(p\)-harmonic 1-forms (\(1<p\), \(0<q<+\infty\)), i.e. closed 1-forms \(\omega\) such that \(d^{*} (|\omega|^{p-2}\omega) = 0\). NEWLINENEWLINENEWLINEThe ingredients of the proof are a Sobolev inequality in the case of non-negative Ricci curvature, due to Saloff-Coste, a Bochner formula and a Moser iteration argument. The curvature condition is also used to deduce that the volume is infinite. NEWLINENEWLINENEWLINESince \(p\)-harmonic maps are, by definition, such that their differential is a \(p\)-harmonic 1-form, the author slightly adapts the previous proof to show that, for \(p>1\), a \(p\)-harmonic map from a complete non-compact Riemannian manifold with non-positive Ricci curvature into a complete Riemannian manifold with non-positive sectional curvature is a constant map if its differential is in \(L^q\) for some \(0<q< +\infty\). This generalises a result of Schoen-Yau on harmonic maps and a result of \textit{H. Takeuchi} [Jap. J. Math., New Ser. 17, No. 2, 317-332 (1991; Zbl 0754.58009)] on \(p\)-harmonic maps for \(p>2\) and \(q= 2p -2\).