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p-harmonic 1-forms on complete manifolds

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Abstract

Let (M m, g) be a complete non-compact manifold with asymptotically non-negative Ricci curvature and finite first Betti number. We prove that any bounded set of p-harmonic 1-forms in L q(M), 0 < q < ∞, is relatively compact with respect to the uniform convergence topology.

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References

  1. Chen J.T., Sung C.J.: Dimension estimate of harmonic forms on complete manifolds. Pacific J. Math. 1, 91–109 (2007)

    Article  MathSciNet  Google Scholar 

  2. J. Eells and L. Lemaire, Selected Topics in Harmonic Maps, CBMS Regional Conf. Series. 50, Amer. Math. Soc, (1995).

  3. P. Hajłasz and P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sér. I. Math. 320 (1995).

  4. P. Li, On the Sobolev constant and the p-spectrum of a compact Riemannian manifold, Ann. Sci. École. Norm. Sup. 13 (1980).

  5. P. Li, Lecture notes on geometric analysis, Lecture Notes Series No. 6, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University, Korea (1993).

  6. Li P., Tam L.-F.: Green’s functions, harmonic functions, and volume comparisons. J. Diff. Geom. 41, 277–318 (1995)

    MATH  MathSciNet  Google Scholar 

  7. Moser J.: On Harnack’s theorem for elliptic equations. Comm. Pure. Appl. Math. 14, 577–591 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  8. Saloff-Coste L.: Uniformly elliptic operators on Riemannian manifolds. J. Diff. Geom. 36, 417–450 (1992)

    MATH  MathSciNet  Google Scholar 

  9. L. Saloff-Coste, Aspects of Sobolev-Type Inequalities, London Mathematical Society Lecture Note Series Vol 289, Cambridge University (2002).

  10. Sung C.J., Tam L.F., Wang J.: Bounded harmonic maps on a class of manifolds, Proc. Amer. Math. Soc. 123, 2241–2248 (1996)

    MathSciNet  Google Scholar 

  11. Tam L.-F.: A note on harmonic forms on complete manifolds. Proc. Amer. Math. Soc. 126, 3097–3108 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Wu H.: The Bochner technique in differential geometry. Math. Rep. 3, 289–538 (1988)

    MathSciNet  Google Scholar 

  13. Zhang X.: A note on p-harmonic 1-forms on complete manifolds. Canad. Math. Bull. 44, 376–384 (2001)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National Chung Cheng University, Chiayi, Taiwan, ROC

    Liang-Chu Chang

  2. Department of Mathematics, National Cheng Kung University, Tainan, Taiwan

    Cheng-Lin Guo

  3. Department of Mathematics, National Tsing-Hua University, Hsinchu, Taiwan

    Chiung-Jue Anna Sung

Authors
  1. Liang-Chu Chang
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  2. Cheng-Lin Guo
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  3. Chiung-Jue Anna Sung
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Correspondence to Chiung-Jue Anna Sung.

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The third author was partially supported by NSC.

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Chang, LC., Guo, CL. & Sung, CJ.A. p-harmonic 1-forms on complete manifolds. Arch. Math. 94, 183–192 (2010). https://doi.org/10.1007/s00013-009-0079-3

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  • DOI: https://doi.org/10.1007/s00013-009-0079-3

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