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Gradient estimates for f-exponentially harmonic functions on complete Riemannian manifolds

  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

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https://doi.org/10.3969/j.issn.0253-2778.2015.09.002
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  • Author Bio:

    XING Jie, male, born in 1992, master. Research field: differential geometry in the large. E-mail:xj3553@mail.ustc.edu.cn

  • Received Date: 09 March 2015
  • Accepted Date: 19 June 2015
  • Rev Recd Date: 19 June 2015
  • Publish Date: 30 September 2015
    Abstract

  • Abstract

    For smooth metric measure spaces (M,g,e-fdvol), the gradient estimates of positive solutions to the f-exponentially harmonic functions was considered by using the maximum principle. Then a Liouville type theorem was obtained when the Bakry-Emery Ricci tensor was nonnegtive and the sectional curvature was bounded by a negative constant. This generalizes a result in Ref.[Wu J, Ruan Q, Yang Y H. Gradient estimates for exponentially harmonic functions on complete Riemannian manifolds. Manuscripta Mathematica, 2014, 143(3-4): 483-489], which is covered in the case where f is a constant.

    Abstract

    For smooth metric measure spaces (M,g,e-fdvol), the gradient estimates of positive solutions to the f-exponentially harmonic functions was considered by using the maximum principle. Then a Liouville type theorem was obtained when the Bakry-Emery Ricci tensor was nonnegtive and the sectional curvature was bounded by a negative constant. This generalizes a result in Ref.[Wu J, Ruan Q, Yang Y H. Gradient estimates for exponentially harmonic functions on complete Riemannian manifolds. Manuscripta Mathematica, 2014, 143(3-4): 483-489], which is covered in the case where f is a constant.
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    • References(8)
  • [1]
    Eells J, Lemaire L. Some properties of exponentially harmonic maps[J]. Proc Banach Center Pub, 1992, 27: 129-136.
    [2]
    Hong Jianqiao, Yang Yihu. Some results on exponentially harmonic maps[J]. Chinese Ann Math, 1993,6: 686-69.
    [3]
    Hong M C. Liouville theorems for exponentially harmonic functions on Riemannian manifolds[J]. Manuscripta Mathematica, 1992, 77(1): 41-46.
    [4]
    Wu J, Ruan Q, Yang Y H. Gradient estimate for exponentially harmonic functions on complete Riemannian manifolds[J]. Manuscripta Mathematica, 2014, 143(3-4): 483-489.
    [5]
    Kotschwar B, Ni L. Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula[J]. Annales scientifiques de lcole Normale Supérieure,2009, 42(1): 1-36.
    [6]
    Li P, Yau S T. On the parabolic kernel of the Schrdinger operator[J]. Acta Mathematica, 1986, 156(1): 153-201.
    [7]
    Wei G, Wylie W. Comparison geometry for the Bakry-Emery Ricci tensor[J]. Journal of Differential Geometry, 2009, 83(2): 337-405.
    [8]
    Li P. Lecture notes on geometric analysis[R]. Seoul: Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, 1993.
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    [1]
    Eells J, Lemaire L. Some properties of exponentially harmonic maps[J]. Proc Banach Center Pub, 1992, 27: 129-136.
    [2]
    Hong Jianqiao, Yang Yihu. Some results on exponentially harmonic maps[J]. Chinese Ann Math, 1993,6: 686-69.
    [3]
    Hong M C. Liouville theorems for exponentially harmonic functions on Riemannian manifolds[J]. Manuscripta Mathematica, 1992, 77(1): 41-46.
    [4]
    Wu J, Ruan Q, Yang Y H. Gradient estimate for exponentially harmonic functions on complete Riemannian manifolds[J]. Manuscripta Mathematica, 2014, 143(3-4): 483-489.
    [5]
    Kotschwar B, Ni L. Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula[J]. Annales scientifiques de lcole Normale Supérieure,2009, 42(1): 1-36.
    [6]
    Li P, Yau S T. On the parabolic kernel of the Schrdinger operator[J]. Acta Mathematica, 1986, 156(1): 153-201.
    [7]
    Wei G, Wylie W. Comparison geometry for the Bakry-Emery Ricci tensor[J]. Journal of Differential Geometry, 2009, 83(2): 337-405.
    [8]
    Li P. Lecture notes on geometric analysis[R]. Seoul: Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, 1993.
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