[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.b150986

RIGIDITY OF IMMERSED SUBMANIFOLDS IN A HYPERBOLIC SPACE

Nguyen, Thac Dung (Department of Mathematics, Mechanics and Informatics Hanoi University of Sciences (HUS-VNU))

Publication Information

Bulletin of the Korean Mathematical Society / v.53, no.6, 2016 , pp. 1795-1804 More about this Journal

Abstract

Let $M^n$ , $2{\leq}n{\leq}6$ be a complete noncompact hypersurface immersed in ${\mathbb{H}}^{n+1}$ . We show that there exist two certain positive constants 0 < ${\delta}{\leq}1$ , and ${\beta}$ depending only on ${\delta}$ and the first eigenvalue ${\lambda}_1(M)$ of Laplacian such that if M satisfies a ( ${\delta}$ -SC) condition and ${\lambda}_1(M)$ has a lower bound then $H^1(L^2(M))=0$ . Excepting these two conditions, there is no more additional condition on the curvature.

Keywords

immersed hypersurface; harmonic forms; the first eigenvalue; ${\delta}$ -stablity; stable hypersurface;

Citations & Related Records

Reference

1	H. D. Cao, Y. Shen, and S. Zhu, The structure of stable minimal hypersurfaces in ${\mathbb{R}}^{n+1}$ , Math. Res. Lett. 4 (1997), no. 5, 637-644. DOI
2	G. Carron, $L^2$ harmonic forms on non compact manifolds, Lectures given at CIMPA's summer school "Recent Topics in Geometric analysis", Arxiv:0704.3194v1.
3	G. Carron, Une suite exacte en $L^2$ -cohomologie, Duke Math. J. 95 (1998), no. 2, 343-372. DOI
4	L. F. Cheung and P. F. Leung, Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001), no. 3, 525-530. DOI
5	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
6	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
7	H. P. Fu and D. Y. Yang, Vanishing theorems on complete manifolds with weighted Poincare inequality and applications, Nagoya Math. J. 206 (2012), 25-37. DOI
8	D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715-727.
9	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. 100 (2013), no. 4, 369-380. DOI
10	P. F. Leung, An estimate on the Ricci curvature of a submanifold and some applications, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1051-1063. DOI
11	P. Li and L. F. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), no. 2, 359-383. DOI
12	P. Li and J. P. Wang, Stable minimal hypersurfaces in a nonnegatively curved manifold, J. Reine Angew. Math. 566 (2004), 215-230.
13	L. Ni, Gap theorems for minimal submanifolds in ${\mathbb{R}}^{n+1}$ , Comm. Anal. Geom. 9 (2001), no. 3, 641-656. DOI
14	K. Seo, $L^2$ harmonic 1-forms on minimal submanifolds in hyperbolic space, J. Math. Anal. Appl. 371 (2010), no. 2, 546-551. DOI