$L_f^p$ harmonic 1-forms on complete non-compact smooth metric measure spaces

Abstract

This paper studies complete non-compact smooth metric measure space $(M^n,g,{\mathrm{e}}^{-f}\mathrm{d}v)$ with positive first spectrum ${\lambda }_1({\Delta }_f)$ or satisfying a weighted Poincaré inequality with weight function $\rho$. We establish two splitting and vanishing theorems for $L_f^p$ harmonic $1$-forms under the assumption that $m$-Bakry-Émery Ricci curvature ${\mathrm{Ric}}_{m,n}\ge -a{\lambda }_1({\Delta }_f)$ or ${\mathrm{Ric}}_{m,n}\ge -a\rho -b$ for particular constants $a$ and $b>0$. These results are inspired by the work of Han-Lin and are $L_f^p$ generalizations of previous works by Dung-Sung and Vieira for $L^2$ harmonic $1$-forms.

9 pages

References in corpus (1)

• $L_f^2$ harmonic 1-forms on smooth metric measure spaces with positive ${\lambda }_1({\Delta }_f)$