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Harmonic p-forms on Hadamard manifolds with finite total curvature

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

Abstract

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.

Keywords

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

Mathematics Subject Classification

58A10 53C42 53C50 

Notes

Acknowledgements

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.

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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

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