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

Abstract

In this paper, we study vanishing and splitting results on a complete smooth metric measure space $(M^n,g,{\mathrm{e}}^{-f}\mathrm{d}v)$ with various negative $m$-Bakry-Émery-Ricci curvature lower bounds in terms of the first spectrum ${\lambda }_1({\Delta }_f)$ of the weighted Laplacian ${\Delta }_f$, i.e. ${\mathrm{Ric}}_{m,n}\ge -a{\lambda }_1({\Delta }_f)-b$ for $0<a\le \frac{m}{m-1},b\ge 0$. In particular, we consider three main cases for different $a$ and $b$ with or without conditions on ${\lambda }_1({\Delta }_f)$. These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung and Vieira.

13 pages, Theorem 1.5 is updated

References in corpus (1)

• L^2 harmonics forms on non compact manifolds

Cited by in corpus (1)

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