Weighted p-harmonic functions and rigidity of smooth metric measure spaces

Let (Mn,g,e−fdv)(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 λ1,pλ1,p on M, namelyefdiv(e−f|∇v|p−2∇v)=−λ1,pvp−1,efdiv(e−f|∇v|p−2∇v)=−λ1,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 λ1,pλ1,p is maximal. Our achievements generalize several results proved earlier by Li–Wang, Munteanu–Wang ( [11], [12], [17] and [18]).