Type problem, the first eigenvalue and Hardy inequalities

In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues \lambda_{1}(B_{r}) of “balls” B_{r}:=\{\rho<r\} on a complete Riemannian manifold M as r\rightarrow +\infty , where \rho is a Lipschitz continuous exhaustion function with |\nabla\rho|\leq1 a.e. on M . We obtain several sharp results. First, if r^{2} \lambda_{1}(B_{r})\ge \gamma>0 for all r>r_{0} , we obtain a sharp estimate of the volume growth: |B_{r}|\ge cr^{\mu(\gamma)}. Moreover, when \gamma>j_{0}^{2}\approx 5.784 , where j_{0} denotes the first positive zero of the Bessel function J_{0} , then M is non-parabolic and we have a Hardy-type inequality. In the case where r_{0}=0 , a sharp Hardy-type inequality holds. These spectral conditions are satisfied if one assumes that \Delta\rho^{2}\geq2\mu(\gamma)>0 . In particular, when \inf_{M}\Delta\rho^{2}>4 , M is non-parabolic and we get a sharp Hardy-type inequality. Related results for finite volume case are also studied.