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Elliptic gradient estimates for a weighted heat equation and applications

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Abstract

We obtain two elliptic gradient estimates for positive solutions to the \(f\)-heat equation on a complete smooth metric measure space with only Bakry–Émery Ricci tensor bounded below. One is a local sharp Souplet–Zhang’s type and the other is a global Hamilton’s type. As applications, we prove parabolic Liouville theorems for ancient solutions satisfying some growth restriction near infinity. In particular the Liouville results are suitable for the gradient shrinking or steady Ricci solitons. The estimates of derivation of the \(f\)-heat kernel are also obtained.

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Acknowledgments

The author thanks Professors Xiaodong Cao, Peng Wu and Qi S. Zhang for their valuable comments and suggestions on an earlier version of this paper. This work is partially supported by NSFC (11101267, 11271132) and the China Scholarship Council (201208310431).

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Maritime University, 1550 Haigang Avenue, Shanghai, 201306, People’s Republic of China

    Jia-Yong Wu

  2. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Jia-Yong Wu

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  1. Jia-Yong Wu
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Wu, JY. Elliptic gradient estimates for a weighted heat equation and applications. Math. Z. 280, 451–468 (2015). https://doi.org/10.1007/s00209-015-1432-9

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  • DOI: https://doi.org/10.1007/s00209-015-1432-9

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