Fundamental tone of complete weakly stable constant mean curvature hypersurfaces in hyperbolic space

In this paper, we give an upper bound for the fundamental tone of stable constant mean curvature hypersurfaces in hyperbolic space. Let M be an n-dimensional complete non-compact constant mean curvature hypersurface with finite L^2-norm of the traceless second fundamental form. If M is weakly stable, then λ_1(M) is bounded above by n^2+O(n^{2+s}) for arbitrary s>0.