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arXiv · 2001.07037

$L^2_f$ harmonic 1-forms on smooth metric measure spaces with positive $\lambda_1(\Delta_f)$

Jiuru Zhou
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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-\'Emery-Ricci curvature lower bounds in terms of the first spectrum $\lambda_1(\Delta_f)$ of the weighted Laplacian $\Delta_f$, i.e. $\mathrm{Ric}_{m,n}\geq -a\lambda_1(\Delta_f)-b$ 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 $\lambda_1(\Delta_f)$. These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung and Vieira.

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Jiuru Zhou. 2020-01-20. $L^2_f$ harmonic 1-forms on smooth metric measure spaces with positive $\lambda_1(\Delta_f)$. https://arxiv.org/abs/2001.07037

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