Weighted cp-harmonic functions and rirgidity of smooth metric measure spaces.

Let (Mn, g, e- fdv) be a smooth metric measure space of dimensional n. Suppose that v is a positive weighted p-eigenfunction associated to the eigenvalue λ 1, p on M, namelyefdiv (e- f ⊥ v p- 2⊥ v) =- λ 1, pvp- 1, in the distribution sense. We first give a local gradient estimate for v provided the m-dimensional Bakry-Émery curvature Ricfm bounded from below. Consequently, we show that when Ricfm≥; 0 then v is constant if v is of sublinear growth. At the same time, we prove a Harnack inequality for weighted p-harmonic functions. Moreover, we show global sharp gradient estimates for weighted p-eigenfunctions. Then we use these estimates to study geometric structures at infinity when the first eigenvalue λ 1, p is maximal. Our achievements generalize several results proved earlier by Li-Wang, Munteanu-Wang ( [11,12,17,18]). [ABSTRACT FROM AUTHOR]