Weighted \(p\)-basic harmonic forms and its applications (Q6596117)

The study of weighted Riemannian manifolds can be traced back to the work of \textit{A. Lichnerowicz} [C. R. Acad. Sci., Paris, Sér. A 271, 650--653 (1970; Zbl 0208.50003)]. The paper under review introduces and studies harmonic \(p\)-forms on weighted Riemannian foliations and generalises certain vanishing theorems by \textit{X. Zhang} [Can. Math. Bull. 44, No. 3, 376--384 (2001; Zbl 0989.58006)], \textit{N. T. Dung} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 150, 138--150 (2017; Zbl 1355.53030)] and \textit{M. Vieira} [Geom. Dedicata 184, 175--191 (2016; Zbl 1353.53047)]. The principal result is that the presence of points with positive (generalised) weighted curvature forces all \(L^p\) weighted basic harmonic forms to vanish, for any \(p \geq 2\). As an application, the authors prove a Liouville-type theorem.