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

In this paper, we show sharp estimates for the first eigenvalue λ 1 , p \lambda _{1, p} of the weighted p p -Laplacian on smooth metric measure spaces ( M , g , e − f d v ) (M, g, e^{-f}dv) . When the Bakry-Émery curvature R i c f Ric_f is bounded from below and the weighted function f f is of sublinear growth, we prove some rigidity properties provided that the first eigenvalue λ 1 , p \lambda _{1, p} obtains its optimal value.