  For a closed smooth manifold $M$, we define a variant of Gromov's minimal
volume called the essential minimal volume, as the infimum of the volume of
$(M_\infty,g_\infty)$, where $(M_\infty,g_\infty)$ lies in a natural closure of
the set of Riemannian manifolds $(M,h)$ with bounded sectional curvature
$|\sec|\leq 1$. In all dimensions, we prove the existence of metrics achieving
the essential minimal volume, and show that they generalize hyperbolic metrics.
We then estimate the essential minimal volume of closed Einstein $4$-manifolds
and complex surfaces with nonnegative Kodaira dimension in terms of their Euler
characteristic.
