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In this paper, we study the relationship between the type problem and theasymptotic 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 volumegrowth $\nu_*:=\liminf_{r\rightarrow +\infty} \frac{\log |B_r|}{\log r}$ isgiven as follows \[ {\Lambda_*} \lesssim \begin{cases} \nu_*^2,\ \ \&\nu_*\gg1,\\ \nu_*\log\frac{1}{\nu_*},&1<\nu_*\ll1. \end{cases} \] Theexponent $2$ for $\nu_*\gg1$ turns out to be the best possible.

Type problem and the first eigenvalue