Advertisement

Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 195–217 | Cite as

L p harmonic 1-form on submanifold with weighted Poincaré inequality

  • Xiaoli Chao
  • Yusha Lv
Article

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 L p harmonic 1-forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.

Keywords

weighted Poincaré inequality δ-stability Lpharmonic 1-form property (Pϱ

MSC 2010

53C42 53C50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    G. Carron: L 2-cohomologie et inégalités de Sobolev. Math. Ann. 314 (1999), 613–639. (In French.)MathSciNetCrossRefMATHGoogle Scholar
  3. [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.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    X. Chao, Y. Lv: L2 harmonic 1-forms on submanifolds with weighted Poincaré inequality. J. Korean Math. Soc. 53 (2016), 583-595.MathSciNetCrossRefMATHGoogle Scholar
  5. [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.MathSciNetCrossRefMATHGoogle Scholar
  6. [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.MathSciNetCrossRefMATHGoogle Scholar
  7. [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.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    R. E. Greene, H. Wu: Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent. Math. 27 (1974), 265–298.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    R. E. Greene, H. Wu: Harmonic forms on noncompact Riemannian and K¨ahler manifolds. Mich. Math. J. 28 (1981), 63–81.CrossRefMATHGoogle Scholar
  10. [10]
    D. Hoffman, J. Spruck: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27 (1974), 715–727.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    S. Kawai:Operator Δ − aK on surfaces. Hokkaido Math. J. 17 (1988), 147–150.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    K.-H. Lam: Results on a weighted Poincaré inequality of complete manifolds. Trans. Am. Math. Soc. 362 (2010), 5043–5062.CrossRefMATHGoogle Scholar
  13. [13]
    P. Li: Geometric Analysis. Cambridge Studies in Advanced Mathematics 134, Cambridge University Press, Cambridge, 2012.Google Scholar
  14. [14]
    P. Li, R. Schoen: L p and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153 (1984), 279–301.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    P. Li, J. Wang: Complete manifolds with positive spectrum. J. Differ. Geom. 58 (2001), 501–534.MathSciNetCrossRefMATHGoogle Scholar
  16. [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. [17]
    B. Palmer: Stability of minimal hypersurfaces. Comment. Math. Helv. 66 (1991), 185–188.MathSciNetCrossRefMATHGoogle Scholar
  18. [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.CrossRefMATHGoogle Scholar
  19. [19]
    K. Seo: L 2 harmonic 1-forms on minimal submanifolds in hyperbolic space. J. Math. Anal. Appl. 371 (2010), 546–551.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    K. Seo: Rigidity of minimal submanifolds in hyperbolic space. Arch. Math. 94 (2010), 173–181.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    K. Seo: L p harmonic 1-forms and first eigenvalue of a stable minimal hypersurface. Pac. J. Math. 268 (2014), 205–229.CrossRefMATHGoogle Scholar
  22. [22]
    K. Shiohama, H. Xu: The topological sphere theorem for complete submanifolds. Compos. Math. 107 (1997), 221–232.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    L.-F. Tam, D. Zhou: Stability properties for the higher dimensional catenoid in Rn+1. Proc. Am. Math. Soc. 137 (2009), 3451–3461.CrossRefMATHGoogle Scholar
  24. [24]
    M. Vieira: Vanishing theorems for L 2 harmonic forms on complete Riemannian manifolds. Geom. Dedicata 184 (2016), 175–191.MathSciNetCrossRefMATHGoogle Scholar
  25. [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.MathSciNetCrossRefMATHGoogle Scholar
  26. [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.MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.School of Mathematics, Southeast UniversityJiangsuP. R. China
  2. 2.School of Mathematics, Wuhan UniversityHubeiP. R. China

Personalised recommendations