  In this paper, we study the relationship between the type problem and the
asymptotic behavior of the first eigenvalues $\lambda_1(B_r)$ of ``balls''
$B_r:=\{\rho<r\}$ on a complete Riemannian manfold $M$ as $r\rightarrow
+\infty$, where $\rho$ is a Lipschitz continuous exhaustion function with
$|\nabla\rho|\leq1$ a.e. on $M$. We show that $M$ is hyperbolic whenever \[
\Lambda_*:= \liminf_{r\rightarrow +\infty} \{ r^2 \lambda_1(B_r)\}
>18.624\cdots. \] Moreover, an upper bound of $\Lambda_*$ in terms of volume
growth $\nu_*:=\liminf_{r\rightarrow +\infty} \frac{\log |B_r|}{\log r}$ is
given as follows \[ {\Lambda_*} \lesssim \begin{cases} \nu_*^2,\ \ \
&\nu_*\gg1,\\ \nu_*\log\frac{1}{\nu_*},&1<\nu_*\ll1. \end{cases} \] The
exponent $2$ for $\nu_*\gg1$ turns out to be the best possible.
