Paper detail

$L^2_f$ harmonic 1-forms on smooth metric measure spaces with positive $λ_1(Δ_f)$

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 $λ_1(Δ_f)$ of the weighted Laplacian $Δ_f$, i.e. $\mathrm{Ric}_{m,n}\geq -aλ_1(Δ_f)-b$ for $0<a\leq\dfrac{m}{m-1}, b\geq0$. In particular, we consider three main cases for different $a$ and $b$ with or without conditions on $λ_1(Δ_f)$. These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung and Vieira.