p-Harmonic l-forms on Complete Noncompact Submanifolds in Sphere with Flat Normal Bundle

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Abstract

In this paper, we investigate a complete noncompact submanifold \(M^m\) in a sphere \(S^{m+t}\) with flat normal bundle. We prove that the dimension of the space of \(L^p\) p-harmonic l-forms (when \(m\ge 4\), \(2\le l\le m-2\) and when \(m=3\), \(l=2\)) on M is finite if the total curvature of M is finite and \(m\ge 3\). We also obtain that there are no nontrivial \(L^p\) p-harmonic l-forms on M if the total curvature is bounded from above by a constant depending only on mpl.

Keywords

p-Harmonic l-form Submanifolds 

Mathematics Subject Classification

Primary 53C21 53C25 

Notes

Acknowledgements

The author would like to thank the referee whose valuable suggestions make this paper more perfect. This work was written while the author visited Department of Mathematics of the University of Oklahoma in USA. He would like to express his sincere thanks to Professor Shihshu Walter Wei for his help, hospitality and support. This work was supported by the National Natural Science Foundation of China (Grant No.11201400), China Scholarship Council (201508410400), Nanhu Scholars Program for Young Scholars of XYNU and the Universities Young Teachers Program of Henan Province (2016GGJS-096).

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Copyright information

© Sociedade Brasileira de Matemática 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXinyang Normal UniversityXinyangPeople’s Republic of China

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