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A Liouville-Type Theorem for Smooth Metric Measure Spaces

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Abstract

For smooth metric measure spaces (M,g,e −f d vol ) we prove a Liouville-type theorem when the Bakry–Emery Ricci tensor is nonnegative. This generalizes a result of Yau, which is recovered in the case f is constant. This result follows from a gradient estimate for f-harmonic functions on smooth metric measure spaces with Bakry–Emery Ricci tensor bounded from below.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    Kevin Brighton

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  1. Kevin Brighton
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Correspondence to Kevin Brighton.

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Communicated by Kang-Tae Kim.

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Brighton, K. A Liouville-Type Theorem for Smooth Metric Measure Spaces. J Geom Anal 23, 562–570 (2013). https://doi.org/10.1007/s12220-011-9253-5

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  • DOI: https://doi.org/10.1007/s12220-011-9253-5

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