dg.differential geometry – Optimal lower bound on the volume of balls under a Sobolev inequality

Let $M$ be a complete non-compact $n$-dimensional ($n geq 3$)
Riemannian manifold with volume element
$dv$ such that, for every smooth compactly supported function $f : M to mathbb {R}$,
$$
bigg ( int_M |f|^{frac {2n}{n-2}} dvbigg)^{frac {n-2}{n}}
, leq , C int_M |nabla f|^2 dv
$$

where $C >0$ is the optimal constant of this Sobolev inequality in the Euclidean case
$M = mathbb{R}^n$. Is it true that
$$
mathrm {Vol} (B(x,r) ) , geq , mathrm{V}(r)
$$

where $B(x,r)$, $x in M$, is a ball of radius $r >0$ in $M$ and
$mathrm{V}(r)$ is the volume of a ball of radius $r$ in $mathbb{R}^n$?