Abstract

Given manifolds with a smooth measure (M, g, e −f dV), we consider gradient estimates for positive harmonic functions of the drifting Laplacian. If the ∞-Bakry-Emery Ricci tensor is bounded from below and \({|\nabla f|}\) is bounded, we obtain a Liouville-type theorem. This extends a classical result of Cheng and Yau.

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