Eigenvalue Estimates for Complete Submanifolds in the Hyperbolic Spaces

Abstract:

      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 $\frac{(n-1)^2(1-|H|^2)}{4}$.