Abstract
We study a complete noncompact submanifold \(M^n\) in a sphere \(\mathbb {S}^{n+p}\). We prove that the dimension of the space of \(L^2\) harmonic \(1\)-forms on \(M\) is finite and there are finitely many non-parabolic ends on \(M\) if the total curvature of \(M\) is finite and \(n\ge 3\). This result is an improvement of Fu–Xu theorem on submanifolds in spheres and a generalized version of Cavalcante, Mirandola and Vitorio’s result on submanifolds in Hadamard manifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Cibotaru, D., Zhu, P.: Refined Kato inequalities for harmonic fields on Kähler manifolds. Pac. J. Math. 256, 51–66 (2012)
Fu, H.P., Xu, H.W.: Total curvature and \(L^2\) harmonic 1-forms on complete submanifolds in space forms. Geom. Dedic. 144, 129–140 (2010)
Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27, 715–727 (1974)
Leung, P.F.: An estimate on the Ricci curvature of a submanifold and some applications. Proc. Am. Math. Soc. 114, 1051–1061 (1992)
Li, P.: Lecture Notes on Geometric Analysis. Lecture Notes Series 6 Research Institute of Mathematics and Global Analysis Reseach Center. Seoul National University, Seoul (1993)
Li, P.: On the Sobolev constant and the \(p\)-spectrum of a compact Riemannian manifold. Ann. Sci. Éc. Norm. Super. 13, 451–468 (1980)
Li, P., Tam, L.F.: Harmonic functions and the structure of complete manifolds. J. Diff. Geom. 35, 359–383 (1992)
Li, P., Wang, J.P.: Minimal hypersurfaces with finite index. Math. Res. Lett. 9, 95–103 (2002)
Shiohama, K., Xu, H.W.: The topological sphere theorem for complete submanifolds. Compos. Math. 107, 221–232 (1997)
Wang, X.D.: On the \(L^2\)-cohomology of a convex cocompact hyperbolic manifold. Duke Math. J. 115, 311–327 (2002)
Zhu, P.: \(L^2\)-harmonic forms and finiteness of ends. An. Acad. Bras. Ciênc. 85, 457–471 (2013)
Zhu, P., Fang, S.W.: A gap theorem on submanifolds with finite total curvature in spheres. J. Math. Anal. Appl. 413, 195–201 (2014)
Acknowledgments
Both authors would like to thank professors Hongyu Wang and Detang Zhou for useful suggestions. P. Zhu was partially supported by NSFC Grants 11101352, 11371309, Fund of Jiangsu University of Technology Grants KYY13005, KYY 13031 and Qing Lan Project. S. Fang was partially supported by the University Science Research Project of Jiangsu Province 13KJB110029 and the Fund of Yangzhou University 2013CXJ001.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, P., Fang, S. Finiteness of non-parabolic ends on submanifolds in spheres. Ann Glob Anal Geom 46, 187–196 (2014). https://doi.org/10.1007/s10455-014-9418-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-014-9418-0