In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4（-1）（H5（-1）） with constant mean curvature H satisfying H2 6634 （H2 114785 respectively） must be compact. Specially, we verify that any complete and stable hypersurface in the hyperbolic space H4（-1） （resp. H5（-1）） with constant mean curvature H satisfying H2 6643 （resp. H2 114785 ） must be compact. It shows that there is no manifold satisfying the conditions of some theorems in [7, 9].
It is shown that any solution to the semilinear problem{u（x,0=）u0（x）〈1,x∈[-1,1] u（±1,t）=0,t∈（0,T）, ut=uxx＋δ（1-u）^-p（x,t）∈（-1,1） ×（0,T）either touches 1 in finite time or converges smoothly to a steady state as t -~ ~e. Some extensions of this result to higher dimensions are also discussed.
