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Lp harmonic 1-form on submanifold with weighted Poincaré inequality

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Abstract

We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is δ-stable or has sufficiently small total curvature, we establish two vanishing theorems for Lp harmonic 1-forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.

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References

  1. D. M. J. Calderbank, P. Gauduchon, M. Herzlich: Refined Kato inequalities and conformal weights in Riemannian geometry. J. Funct. Anal. 173 (2000), 214–255.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Carron: L 2-cohomologie et inégalités de Sobolev. Math. Ann. 314 (1999), 613–639. (In French.)

    Article  MathSciNet  MATH  Google Scholar 

  3. M. P. Cavalcante, H. Mirandola, F. Vitório: L 2 harmonic 1-forms on submanifolds with finite total curvature. J. Geom. Anal. 24 (2014), 205–222.

    Article  MathSciNet  MATH  Google Scholar 

  4. X. Chao, Y. Lv: L2 harmonic 1-forms on submanifolds with weighted Poincaré inequality. J. Korean Math. Soc. 53 (2016), 583-595.

    Article  MathSciNet  MATH  Google Scholar 

  5. N. T. Dung, K. Seo: Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature. Ann. Global Anal. Geom. 41 (2012), 447–460.

    Article  MathSciNet  MATH  Google Scholar 

  6. N. T. Dung, K. Seo: Vanishing theorems for L 2 harmonic 1-forms on complete submanifolds in a Riemannian manifold. J. Math. Anal. Appl. 423 (2015), 1594–1609.

    Article  MathSciNet  MATH  Google Scholar 

  7. H.-P. Fu, Z.-Q. Li: L 2 harmonic 1-forms on complete submanifolds in Euclidean space. Kodai Math. J. 32 (2009), 432–441.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. E. Greene, H. Wu: Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent. Math. 27 (1974), 265–298.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. E. Greene, H. Wu: Harmonic forms on noncompact Riemannian and K¨ahler manifolds. Mich. Math. J. 28 (1981), 63–81.

    Article  MATH  Google Scholar 

  10. D. Hoffman, J. Spruck: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27 (1974), 715–727.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Kawai:Operator Δ − aK on surfaces. Hokkaido Math. J. 17 (1988), 147–150.

    Article  MathSciNet  MATH  Google Scholar 

  12. K.-H. Lam: Results on a weighted Poincaré inequality of complete manifolds. Trans. Am. Math. Soc. 362 (2010), 5043–5062.

    Article  MATH  Google Scholar 

  13. P. Li: Geometric Analysis. Cambridge Studies in Advanced Mathematics 134, Cambridge University Press, Cambridge, 2012.

    Google Scholar 

  14. P. Li, R. Schoen: L p and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153 (1984), 279–301.

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Li, J. Wang: Complete manifolds with positive spectrum. J. Differ. Geom. 58 (2001), 501–534.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Miyaoka: L 2 harmonic 1-forms on a complete stable minimal hypersurface. Geometry and Global Analysis (T. Kotake et al., eds.). Int. Research Inst., Sendai 1993, Tŏhoku Univ., Mathematical Institute}, 1993, pp. 289–293.

    Google Scholar 

  17. B. Palmer: Stability of minimal hypersurfaces. Comment. Math. Helv. 66 (1991), 185–188.

    Article  MathSciNet  MATH  Google Scholar 

  18. N. D. Sang, N. T. Thanh: Stable minimal hypersurfaces with weighted Poincaré inequality in a Riemannian manifold. Commum. Korean. Math. Soc. 29 (2014), 123–130.

    Article  MATH  Google Scholar 

  19. K. Seo: L 2 harmonic 1-forms on minimal submanifolds in hyperbolic space. J. Math. Anal. Appl. 371 (2010), 546–551.

    Article  MathSciNet  MATH  Google Scholar 

  20. K. Seo: Rigidity of minimal submanifolds in hyperbolic space. Arch. Math. 94 (2010), 173–181.

    Article  MathSciNet  MATH  Google Scholar 

  21. K. Seo: L p harmonic 1-forms and first eigenvalue of a stable minimal hypersurface. Pac. J. Math. 268 (2014), 205–229.

    Article  MATH  Google Scholar 

  22. K. Shiohama, H. Xu: The topological sphere theorem for complete submanifolds. Compos. Math. 107 (1997), 221–232.

    Article  MathSciNet  MATH  Google Scholar 

  23. L.-F. Tam, D. Zhou: Stability properties for the higher dimensional catenoid in Rn+1. Proc. Am. Math. Soc. 137 (2009), 3451–3461.

    Article  MATH  Google Scholar 

  24. M. Vieira: Vanishing theorems for L 2 harmonic forms on complete Riemannian manifolds. Geom. Dedicata 184 (2016), 175–191.

    Article  MathSciNet  MATH  Google Scholar 

  25. S.-T. Yau: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25 (1976), 659–670; erratum ibid. 31 (1982), 607.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Yun: Total scalar curvature and L 2 harmonic 1-forms on a minimal hypersurface in Euclidean space. Geom. Dedicata. 89 (2002), 135–141.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Southeast University, 2 Sipailou, Xuanwu, Nanjing, Jiangsu, 211189, P. R. China

    Xiaoli Chao

  2. School of Mathematics, Wuhan University, Meiyuan 2nd Road, Wuhan, Hubei, 430072, P. R. China

    Yusha Lv

Authors
  1. Xiaoli Chao
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  2. Yusha Lv
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Correspondence to Xiaoli Chao.

Additional information

The research has been supported by NSFC and JSNSF.

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Chao, X., Lv, Y. Lp harmonic 1-form on submanifold with weighted Poincaré inequality. Czech Math J 68, 195–217 (2018). https://doi.org/10.21136/CMJ.2018.0415-16

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  • DOI: https://doi.org/10.21136/CMJ.2018.0415-16

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