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Rigidity properties of smooth metric measure spaces via the weighted $ p$-Laplacian


Author: Nguyen Thac Dung
Journal: Proc. Amer. Math. Soc. 145 (2017), 1287-1299
MSC (2010): Primary 53C23, 53C24, 58J05
DOI: https://doi.org/10.1090/proc/13285
Published electronically: September 8, 2016
MathSciNet review: 3589326
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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.


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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
Email: dungmath@yahoo.co.uk, dungmath@gmail.com

DOI: https://doi.org/10.1090/proc/13285
Keywords: Weighted $p$-harmonic functions, smooth metric measure spaces, the first eigenvalue, splitting theorems, gradient Ricci solitons
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
Article copyright: © Copyright 2016 American Mathematical Society