Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2016; 53(6): 1795-1804

Online first article September 22, 2016      Printed November 30, 2016

https://doi.org/10.4134/BKMS.b150986

Copyright © The Korean Mathematical Society.

Rigidity of immersed submanifolds in a hyperbolic space

Nguyen Thac Dung

No. 334, Nguyen Trai Road

Abstract

Let $M^{n}, 2\leq n\leq6$ be a complete noncompact hypersurface immersed in $\HH^{n+1}$. We show that there exist two certain positive constants $0<\delta\leq1$, 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

MSC numbers: 53C42, 58C40