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



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.


p-Harmonic l-form Submanifolds 

Mathematics Subject Classification

Primary 53C21 53C25 



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).


  1. Cao, H., Shen, Y., Zhu, S.: The structure of stable minimal hypersurfaces in \(R^{n+1}\). Math. Res. Lett. 4, 637–644 (1997)MathSciNetCrossRefMATHGoogle Scholar
  2. Cavalcante, M.P., Mirandola, H., Vitório, F.: \(L^2\) harmonic 1-forms on submanifolds with finite total curvature. J. Geom. Anal. 24, 205–222 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. Chang, L.C., Guo, C.L., Anna, C.J.: Sung, \(p\)-harmonic 1-forms on complete manifolds. Arch. Math. 94, 183–192 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. Dung, N.T., Seo, K.: \(p\)-harmonic functions and connectedness at infinity of complete submanifolds in a Riemannian manifold. Annali di Matematica (2016). doi: 10.1007/s10231-016-0625-0 MATHGoogle Scholar
  5. Dung, N.T.: \(p\)-harmonic \(l\)-forms on Riemannian manifolds with a Weighted Poincaré inequality. Nonlinear Anal. 150, 138–150 (2017)MathSciNetCrossRefMATHGoogle Scholar
  6. Fu, H.P., Xu, H.W.: Total curvauture and \(L^2\) harmonic \(1\)-forms on complete submanifolds in space forms. Geom. Dedicata 144, 129–140 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. Han, Y.B., Pan, H.: \(L^p\) p-harmonic 1-forms on submanifolds in a Hadamard manifold. J. Geometry Phys. 107, 79–91 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. Han, Y.B., Zhang, Q.Y., Liang, M.H.: \(L^p\) p-harmonic 1-forms on locally conformally flat Riemannian manifolds. Kodai Math. J. 40, 518–536 (2017)Google Scholar
  9. Han, Y.B., Zhang, W.: \(L^p\) p-harmonic 1-forms on complete noncompact submanifolds in spheres (under review)Google Scholar
  10. Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27, 715–727 (1974)MathSciNetCrossRefMATHGoogle Scholar
  11. Lin, H.Z.: On the structure of submanifolds in Euclidean space with flat normal bundle. Results Math. 68, 313–329 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. Lin, H.Z.: Vanishing theorems for \(L^2\)-harmoinc forms on complete submanifolds in Euclidean space. J. Math. Anal. Appl. 425(2), 774–787 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. Lin, H.Z.: \(L^2\) harmonic forms on submanifolds in a Hadamard manifold. Nonlinear Anal. 125, 310–322 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. Li, P.: On the Sobolev constant and the \(p\)-spectrum of a compact Riemannian manifold. Ann. Sci. Éc. Norm. Sup. 13(4), 451–468 (1980)MathSciNetCrossRefMATHGoogle Scholar
  15. Li, P., Tam, L.F.: Harmonic functions and the structure of complete manifolds. J. Differ. Geom. 35, 359–383 (1992)MathSciNetCrossRefMATHGoogle Scholar
  16. Li, P., Wang, J.P.: Minimal hypersurfaces with finite index. Math. Res. Lett. 9, 95–103 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. Michael, J.H., Simon, L.M.: Sobolev and mean value inequalities on generalized submanifolds of \(R^n\). Commun. Pure Appl. Math. 24, 361–379 (1973)MathSciNetCrossRefMATHGoogle Scholar
  18. Schoen, R., Yau, S.T.: Harmonic maps and the topology of the stable hypersurfaces and manifolds with non-negatie Ricci curvature. Commun. Math. Helv. 51, 33–341 (1976)Google Scholar
  19. Wu, H.H.: The Bochner technique in differential geometry. Math. Rep. 3(2), 289–538 (1988)MathSciNetGoogle Scholar
  20. Zhang, X.: A note on \(p\)-harmoinic 1-forms on complete manifolds. Can. Math. Bull. 44, 376–384 (2001)CrossRefMATHGoogle Scholar
  21. Zhu, P., Fang, S.W.: A gap theorem on submanifolds with finite total curvature in spheres. J. Math. Anal. Appl. 413, 195–201 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. Zhu, P., Fang, S.W.: Finiteness of non-parabolic ends on submanifolds in shpere. Ann. Glob. Anal. Geom. 46, 187–196 (2014)CrossRefMATHGoogle Scholar
  23. Zhu, P.: On reduced \(L^2\) cohomology of hypersurfaces in spheres with finite total curvature. Ann. Braz. Acad. Sci. 88(4), 2053–2065 (2016)MathSciNetCrossRefMATHGoogle Scholar

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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