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 eigenvalue \(\lambda _1(\Delta _f)\) of the weighted Laplacian \(\Delta _f\), i.e., \(\mathrm {Ric}_{m,n}\ge -a\lambda _1(\Delta _f)-b\) for \(0<a\le \dfrac{m}{m-1}, b\ge 0\). 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.
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Acknowledgements
The author would like to thank Prof. Jiayong Wu for useful suggestions and the anonymous referees for many valuable suggestions. J.R. Zhou is partially supported by a PRC grant NSFC 11771377 and the Natural Science Foundation of Jiangsu Province (BK20191435).
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Zhou, J. \(L^2_f\)-harmonic 1-forms on smooth metric measure spaces with positive \(\lambda _1(\Delta _f)\). Arch. Math. 116, 693–706 (2021). https://doi.org/10.1007/s00013-021-01588-y
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DOI: https://doi.org/10.1007/s00013-021-01588-y