Rigidity properties of smooth metric measure spaces via the weighted $p$-Laplacian
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Abstract:
In this paper, we show sharp estimates for the first eigenvalue $\lambda _{1, p}$ of the weighted $p$-Laplacian on smooth metric measure spaces $(M, g, e^{-f}dv)$. When the Bakry-Émery curvature $Ric_f$ is bounded from below and the weighted function $f$ is of sublinear growth, we prove some rigidity properties provided that the first eigenvalue $\lambda _{1, p}$ obtains its optimal value.References
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Additional Information
- Nguyen Thac Dung
- Affiliation: Department of Mathematics–Mechanics–Informatics (MIM), Hanoi University of Sciences (HUS-VNU), No. 334, Nguyen Trai Road, Thanh Xuan, Hanoi, Vietnam
- MR Author ID: 772632
- Email: dungmath@yahoo.co.uk, dungmath@gmail.com
- Received by editor(s): March 14, 2016
- Received by editor(s) in revised form: April 23, 2016, and May 4, 2016
- Published electronically: September 8, 2016
- Communicated by: Guofang Wei
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1287-1299
- MSC (2010): Primary 53C23, 53C24, 58J05
- DOI: https://doi.org/10.1090/proc/13285
- MathSciNet review: 3589326