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http://dx.doi.org/10.4134/JKMS.j150190

L2 HARMONIC 1-FORMS ON SUBMANIFOLDS WITH WEIGHTED POINCARÉ INEQUALITY  

Chao, Xiaoli (Department of Mathematics Southeast University)
Lv, Yusha (Department of Mathematics Southeast University)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.3, 2016 , pp. 583-595 More about this Journal
Abstract
In the present note, we deal with $L^2$ harmonic 1-forms on complete submanifolds with weighted $Poincar{\acute{e}}$ inequality. By supposing submanifold is stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^2$ harmonic 1-forms, which are some extension of the results of Kim and Yun, Sang and Thanh, Cavalcante Mirandola and $Vit{\acute{o}}rio$.
Keywords
weighted $poincar{\acute{e}}$ inequality; stable hypersurface; property ($\mathcal{P}_p$); $L^2$ harmonic 1-form;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 D. M. Calderbank, P. Gauduchon, and M. Herzlich, Refined Kato inequalities and conformal weights in Riemannian geometry, J. Funct. Anal. 173 (2000), no. 1, 214-255.   DOI
2 G. Carron, $L^2$-Cohomologie et inegalites de Sobolev, Math. Ann. 314 (1999), no. 4, 613-639.   DOI
3 M. P. Cavalcante, H. Mirandola, and F. Vitorio, $L^2$ harmonic 1-form on submanifolds with finite total curvature, J. Geom. Anal. 24 (2014), no. 1, 205-222.   DOI
4 X. Cheng, $L^2$ harmonic forms and stability of hypersurfaces with constant mean curvature, Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 2, 225-239.   DOI
5 N. T. Dung and K. Seo, Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature, Ann. Global Anal. Geom. 41 (2012), no. 4, 447-460.   DOI
6 N. T. Dung and K. Seo, Vanishing theorems for $L^2$ harmonic 1-forms on complete submanifolds in a Riemannian manifold, J. Math. Anal. Appl. 423 (2015), no. 2, 1594-1609.   DOI
7 N. T. Dung and C. J. Sung, Manifolds with a weighted Poincare inequality, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1783-1794.   DOI
8 H. P. Fu and Z. Q. Li, $L^2$ harmonic 1-forms on complete submanifolds in Euclidean space, Kodai Math. J. 32 (2009), no. 3, 432-441.   DOI
9 D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715-727.
10 J. J. Kim and G. Yun, On the structure of complete hypersurfaces in a Riemannian manifold of nonnegative curvature and $L^2$ harmonic forms, Arch. Math. (Basel) 100 (2013), no. 4, 369-380.   DOI
11 K. H. Lam, Results on a weighted Poincare inequality of complete manifolds, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5043-5062.   DOI
12 P. F. Leung, An estimate on the Ricci curvature of a submanifold and some applications, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1051-1061.   DOI
13 P. Li, Geometric Analysis, Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press, Cambridge, 2012.
14 B. Palmer, Stability of minimal hypersurfaces, Comment. Math. Helv. 66 (1991), no. 2, 185-188.   DOI
15 P. Li and J. Wang, Complete manifolds with positive spectrum, J. Differential Geom. 58 (2001), no. 3, 501-534.   DOI
16 V. Matheus, Vanishing theorems for $L^2$ harmonic forms on complete Riemannian manifolds, arXiv: 1407.0236v1.
17 R. Miyaoka, $L^2$ harmonic 1-forms on a complete stable minimal hypersurface, Geometry and global analysis (Sendai, 1993), 289-293, Tohoku Univ., Sendai, 1993.
18 N. N. Sang and N. T. Thanh, Stability minimal hypersurfaces with weighted Poincare inequality in a Riemannian manifold, Commun. Korean. Math. Soc. 29 (2014), no. 1, 123-130.   DOI
19 K. Seo, Rigidity of minimal submanifolds in hyperbolic space, Arch. Math. (Basel) 94 (2010), no. 2, 173-181.   DOI
20 K. Seo, $L^2$ harmonic 1-forms on minimal submanifolds in hyperbolic space, J. Math. Anal. Appl. 371 (2010), no. 2, 546-551.   DOI
21 K. Shiohama and H. Xu, The topological sphere theorem for complete submanifolds, Compos. Math. 107 (1997), no. 2, 221-232.   DOI
22 S. Tanno, $L^2$ harmonic forms and stability of minimal hypersurfaces, J. Math. Soc. Japan. 48 (1996), no. 4, 761-768.   DOI
23 S. T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659-670.   DOI
24 G. Yun, Total scalar curvature and $L^2$ harmonic 1-forms on a minimal hypersurface in Euclidean space, Geom. Dedicata. 89 (2002), 135-141.