Archiv der Mathematik

, Volume 96, Issue 4, pp 379–385 | Cite as

Liouville-type theorem for the drifting Laplacian operator

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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.

Mathematics Subject Classification (2000)

Primary 58J05 Secondary 58J35 

Keywords

Gradient estimates Positive solution Bakry-Emery Ricci tensor 
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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China
  2. 2.Department of MathematicsHenan Normal UniversityXinxiangPeople’s Republic of China

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