# Harmonic *p*-forms on Hadamard manifolds with finite total curvature

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## 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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