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Vanishing theorems on complete manifolds with weighted Poincaré inequality and applications

Published online by Cambridge University Press:  11 January 2016

Hai-Ping Fu  and
Deng-Yun Yang
Hai-Ping Fu
Affiliation:
Department of Mathematics Nanjing University, Nanjing 210093, People’s Republic of China and Department of Mathematics Nanchang University, Nanchang 330031, People’s Republic of China, [email protected]
Deng-Yun Yang
Affiliation:
College of Mathematics and Information Science Jiangxi Normal University, Nanchang 330022, People’s Republic of China, [email protected]
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Abstract

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Two vanishing theorems for harmonic map and L2 harmonic 1-form on complete noncompact manifolds are proved under certain geometric assumptions, which generalize results of [13], [15], [18], [19], and [20]. As applications, we improve some main results in [2], [4], [6], [9], [12], [20], [22], [24], and [25].

Keywords

58E2053C42
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 206 , June 2012 , pp. 25 - 37
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

References

[1] Akutagawa, K., Yamabe metrics of positive scalar curvature and conformally flat manifolds, Differential Geom. Appl. 4 (1994), 239–258.Google Scholar
[2] Cao, H. D., Shen, Y., and Zhu, S., The structure of stable minimal hypersurfaces in R, Math. Res. Lett. 4 (1997), 637–644.CrossRefGoogle Scholar
[3] Carron, G., L2 harmonic forms on noncompact manifolds, preprint, arXiv:0704.3194 [math.DG] Google Scholar
[4] Cheng, X., Cheung, L. F., and Zhou, D. T., The structure of weakly stable constant mean curvature hypersurfaces, Tohoku Math. J. (2) 60 (2008), 101–121.Google Scholar
[5] Cheng, X. and Zhou, D. T., Manifolds with weighted Poincaré inequality and uniqueness of minimal hypersurfaces, Comm. Anal. Geom. 17 (2009), 139–154.Google Scholar
[6] do, M. P., Carmo, Q. L. Wang, and Xia, C. Y., Complete submanifolds with bounded mean curvature in a Hadamard manifold, J. Geom. Phys. 60 (2010), 142–154.Google Scholar
[7] Eells, J. and Sampson, J. H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160.Google Scholar
[8] Fu, H. P., The structure of δ-stable minimal hypersurface in Rn+1 , Hokkaido Math. J. 40 (2011), 103–110.Google Scholar
[9] Fu, H. P. and Li, Z. Q., On stable constant mean curvature hypersurfaces, Tohoku Math. J. (2) 62 (2010), 383–392.Google Scholar
[10] Fu, H. P. and Li, Z. Q., The structure of complete manifolds with weighted Poincaré inequality and minimal hypersurfaces, Internat. J. Math. 21 (2010), 1421–1428.Google Scholar
[11] Fu, H. P. and Xu, H. W., Weakly stable constant mean curvature hypersurfaces, Appl. Math. J. Chinese Univ. Ser. B 24 (2009), 119–126.Google Scholar
[12] Fu, H. P. and Xu, H. W., Vanishing results on complete manifolds with Poincaré inequality and applications, preprint, 2009.Google Scholar
[13] Lam, K. H., Results on weighted Poincaré inequality of complete manifolds, Trans. Amer. Math. Soc. 362 (2010), 5043–5062.Google Scholar
[14] Li, P. and Tam, L. F., Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), 359–383.Google Scholar
[15] Li, P. and Wang, J. P., Complete manifolds with positive spectrum, J. Differential Geom. 58 (2001), 501–534.CrossRefGoogle Scholar
[16] Li, and Wang, J. P., Minimal hypersurfaces with finite index, Math. Res. Lett. 9 (2002), 95–103.Google Scholar
[17] Li, and Wang, J. P., Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. ´ Ec. Norm. Supér. (4) 39 (2006), 921–982.Google Scholar
[18] Pigola, S., Rigoli, M., and Setti, A. G., Vanishing theorems on Riemannian manifolds, and geometric applications, J. Funct. Anal. 229 (2005), 424–461.Google Scholar
[19] Pigola, S., Rigoli, M., and Setti, A. G., Vanishing and Finiteness Results in Geometric Analysis: A Generalization of the Bochner Technique, Progr. Math. 266, Birkhäuser Basel, 2008.Google Scholar
[20] Schoen, R. and Yau, S. T., Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature, Comment. Math. Helv. 51 (1976), 333–341.CrossRefGoogle Scholar
[21] Schoen, R. and Yau, S. T., “Lectures on differential geometry” in Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, 1994.Google Scholar
[22] Seo, K., L2 harmonic 1-forms on minimal submanifolds in hyperbolic space, J. Math. Anal. Appl. 371 (2010), 546–551.Google Scholar
[23] Shiohama, K. and Xu, H. W., The topological sphere theorem for complete submanifolds, Compos. Math. 107 (1997), 221–232.CrossRefGoogle Scholar
[24] Wang, Q. L., Complete submanifolds in manifolds of partially non-negative curvature, Ann. Global Anal. Geom. 37 (2010), 113–124.Google Scholar
[25] Wang, Q. L. and Xia, C. Y., Complete submanifolds of manifolds of negative curvature, Ann. Global Anal. Geom. 39 (2011), 83–97.Google Scholar