Eigenvalue Estimates for Complete Submanifolds in the Hyperbolic Spaces
Received:May 29, 2012  Revised:February 19, 2013
Key Word: finite $L^q$ norm curvature   first eigenvalue   hyperbolic space   stable hypersurface.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11261038), the Natural Science Foundation of Jiangxi Province (Grant Nos.2010GZS0149; 20132BAB201005) and Youth Science Foundation of Eduction Department of Jiangxi Province (Grant No.GJJ11044).
 Author Name Affiliation Haiping FU Department of Mathematics, Nanchang University, Jiangxi 330031, P. R. China Yongqian TAO Department of Mathematics, Nanchang University, Jiangxi 330031, P. R. China
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In this paper, we study upper bounds of the first eigenvalue of a complete noncompact submanifold in an $(n+p)$-dimensional hyperbolic space $\mathbb{H}^{n+p}$. In particular, we prove that the first eigenvalue of a complete submanifold in $\mathbb{H}^{n+p}$ with parallel mean curvature vector $H$ and finite $L^q(q\geq n)$ norm of traceless second fundamental form is not more than $\frac{(n-1)^2(1-|H|^2)}{4}$. We also prove that the first eigenvalue of a complete hypersurfaces which has finite index in $\mathbb{H}^{n+1}(n\leq 5)$ with constant mean curvature vector $H$ and finite $L^q(2(1-\sqrt{\frac{2}{n}})< q<2(1+\sqrt{\frac{2}{n}}))$ norm of traceless second fundamental form is not more than $\frac{(n-1)^2(1-|H|^2)}{4}$.