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 $r\to +\mathrm{\infty }$, where $\rho$ is a Lipschitz continuous exhaustion function with $\mathrm{\mid }\mathrm{\nabla }\rho \mathrm{\mid }\le 1$ a.e. on $M$. We obtain several sharp results. First, if for all $r>r_0$ $$r^2{\lambda }_1(B_r)\ge \gamma >0,$$ we obtain a sharp estimate of the volume growth: $\mathrm{\mid }{B}_r\mathrm{\mid }\ge c{r}^{\mu (\gamma )}\mathrm{.}$ 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 hyperbolic 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 $\mathrm{\Delta }{\rho }^2\ge 2\mu (\gamma )>0$. In particular, when ${\inf }_M\mathrm{\Delta }{\rho }^2>4$, $M$ is hyperbolic and we get a sharp Hardy type inequality. Related results for finite volume case are also studied.