Type problem and the first eigenvalue

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.