  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 for all $r>r_0$ \[ r^2 \lambda_1(B_r)\ge \gamma>0, \] 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 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
$\Delta\rho^2\geq2\mu(\gamma)>0$. In particular, when $\inf_M\Delta\rho^2>4$,
$M$ is hyperbolic and we get a sharp Hardy type inequality. Related results for
finite volume case are also studied.
