I am wondering if there exists a complete Riemannian manifold (noncompact) such that $$\lim\limits_{r\rightarrow \infty}\frac{vol(B(x,r))}{e^{r^3}}=\infty$$
where the denominator denotes the volume of a geodesic ball around $x$.
Does anyone know an example?
Best wishes
You can construct metrics with essentially any volume growth you like. Consider
$$ g = dr^2 + f^2(r) d\theta^2 $$ in polar coordinates - so long as $f$ is of the form $f(r) = r + o(r)$ as $r \to 0$, this defines an honest metric on $\mathbb R^2$. Such a metric has volume form $ f(r) dr \wedge d \theta,$ and balls $B(0,R) = \{ r< R \}$,
and thus $$ {\rm Vol}(B(0,R)) = 2 \pi \int_0^R f.$$
Thus (away from $r=0$) any smooth increasing function is ${\rm Vol\ }B(R)$ for some metric. These metrics are all complete, since any curve escaping to infinity has length at least $\int_{r_0}^\infty dr = \infty.$