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 $r\to +\mathrm{\infty }$, where $\rho$ is a Lipschitz continuous exhaustion function with $\mathrm{\mid }\mathrm{\nabla }\rho \mathrm{\mid }\le 1$ a.e. on $M$. We show that $M$ is hyperbolic whenever $${\mathrm{\Lambda }}_{\ast }:=\underset{r\to +\mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\{r^2{\lambda }_1(B_r)\}>18.624\cdots \mathrm{.}$$ Moreover, an upper bound of ${\mathrm{\Lambda }}_{\ast }$ in terms of volume growth ${\nu }_{\ast }:={\mathrm{lim\hspace{0.17em}inf}}_{r\to +\mathrm{\infty }}\frac{\log \mathrm{\mid }{B}_r\mathrm{\mid }}{\log r}$ is given as follows $${\mathrm{\Lambda }}_{\ast }\lesssim \{\begin{array}{ll}{\nu }_{\ast }^2,& {\nu }_{\ast }\gg 1,\\ {\nu }_{\ast }\log \frac{1}{{\nu }_{\ast }},& 1<{\nu }_{\ast }\ll 1.\end{array}$$ The exponent $2$ for ${\nu }_{\ast }\gg 1$ turns out to be the best possible.