Harmonic p-forms on Hadamard manifolds with finite total curvature

  • Peijun Wang
  • Xiaoli Chao
  • Yilong Wu
  • Yusha lv


In the present note, the geometric structures and topological properties of harmonic p-forms on a complete noncompact submanifold \(M^{n}(n\ge 4)\) immersed in Hadamard manifold \(N^{n+m}\) are discussed, where \(M^{n}\) and \(N^{n+m}\) are assumed to have flat normal bundle and pure curvature tensor, respectively. Firstly, under the assumption that \(M^{n}\) satisfies the \((\mathcal {P}_\rho )\) property (i.e., the weighted Poincaré inequality holds on \(M^{n}\)) and the \((p,n-p)\)-curvature of \(N^{n+m}\) is not less than a given negative constant, using Moser iteration, the space of all \(L^{2}\) harmonic p-forms on \(M^{n}\) is proven to have finite dimensions if \(M^{n}\) has finite total curvature. Furthermore, if the total curvature is small enough or \(M^{n}\) has at most Euclidean volume growth, two vanishing theorems are, respectively, established for harmonic p-forms. Note that the two vanishing theorems extend several previous results obtained by H. Z. Lin.


Harmonic p-form Hadamard manifold \(({\mathcal {P}}_\rho )\) property Finite total curvature Euclidean volume growth Vanishing theorem 

Mathematics Subject Classification

58A10 53C42 53C50 



The authors would like to express their sincere thanks to the Editor and anonymous reviewer for their valuable comments, which have helped significantly improve the presentation of this paper.


  1. 1.
    Calderbank, D.M.J., Gauduchon, P., Herzlich, M.: Refined Kato inequalities and conformal weights in Riemannian geometry. J. Funct. Anal. 173(1), 214–255 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carron, G.: Faber–Krahn isoperimetric inequalities and consequences. Actes de la Table Ronde de Géométrie Différentielle. 1, 205–232 (1992)zbMATHGoogle Scholar
  3. 3.
    Cavalcante, M.P., Mirandola, H., Vitório, F.: \(L^{2}\) harmonic \(1\)-forms on submanifolds with finite total curvature. J. Geom. Anal. 24(1), 205–222 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chao, X.L., Lv, Y.S.: \(L^{2}\) harmonic \(1\)-forms on submanifolds with weighted Poincaré inequality. J. Korean Math. Soc. 53(3), 583–595 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dung, N.T., Seo, K.: Vanishing theorems for \(L^{2}\) harmonic \(1\)-forms on complete submanifolds in a Riemannian manifold. J. Math. Anal. Appl. 423(2), 1594–1609 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dung, N.T., Le Hai, N.T., Thanh, N.T.: Eigenfunctions of the weighted Laplacian and a vanishing theorem on gradient steady Ricci soliton. J. Math. Anal. Appl. 416(2), 553–562 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27, 715–727 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lam, K.H.: Results on a weighted Poincaré inequality of complete manifolds. Trans. Am. Math. Soc. 362(10), 5043–5062 (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Li, P.: On the Sobolev constant and the \(p\)-spectrum of a compact Riemannian manifold. Ann. Sci. École Norm. Sup. (4) 13(4), 451–468 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, P., Wang, J.P.: Complete manifolds with positive spectrum. J. Differ. Geom. 58(3), 501–534 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li, H.Z.: \(L^{2}\) harmonic forms on a complete stable hypersurfaces with constant mean curvature. Kodai Math. J. 21(1), 1–9 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lin, H.Z.: On the structure of submanifolds in Euclidean space with flat normal bundle. Results Math. 68(3–4), 313–329 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lin, H.Z.: \(L^{2}\) harmonic forms on submanifolds in a Hadamard manifold. Nonlinear Anal. 125, 310–322 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lin, H.Z.: Vanishing theorems for \(L^{2}\) harmonic forms on complete submanifolds in Euclidean space. J. Math. Anal. Appl. 425(2), 774–787 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ma, L., Yang, Y.: \(L^{2}\) forms and Ricci flow with bounded curvature on complete non-compact manifolds. Geom. Dedicata 119, 151–158 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ma, L.: Some properties of non-compact complete Riemannian manifolds. Bull. Sci. Math. 130(4), 330–336 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Miyaoka, R.: \(L^{2}\) Harmonic \(1\)-Forms on a Complete Stable Minimal Hypersurface. Geometry and Global Analysis, pp. 289–293. Tohoku University, Sendai (1993)zbMATHGoogle Scholar
  18. 18.
    Nguyen, D.S., Nguyen, T.T.: Stable minimal hypersurfaces with weighted Poincaré inequality in a Riemannian manifold. Commun. Korean Math. Soc. 29(1), 123–130 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Palmer, B.: Stability of minimal hypersurfaces. Comment. Math. Helv. 66(2), 185–188 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tanno, S.: \(L^{2}\) harmonic forms and stability of minimal hypersurfaces. J. Math. Soc. Jpn. 48(4), 761–768 (1996)CrossRefzbMATHGoogle Scholar
  21. 21.
    Vieira, M.: Vanishing theorems for \(L^{2}\) harmonic forms on complete Riemannian manifolds. Geom. Dedicata 184, 175–191 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhu, P.: \(L^{2}\) harmonic forms and stable hypersurfaces in space forms. Arch. Math. (Basel) 97(3), 271–279 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsSoutheast UniversityNanjingPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China

Personalised recommendations