Essential minimal volume of Einstein 4-manifolds

The minimal volume of a closed manifold $M$ is the infimum of the volume of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We introduce a variant called the essential minimal volume, $\mathrm{e}\mathrm{s}\mathrm{s}-\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}(M)$, which is the limit, as $\delta >0$ goes to $0$, of the infimum of the volume of the $\delta$-thick part of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We show that, for some universal constant $C>0$, any closed Einstein 4-manifold $M$ with Euler characteristic $e(M)$ satisfies $$C^{-1}e(M)\le \mathrm{e}\mathrm{s}\mathrm{s}-\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}(M)\le Ce(M)\mathrm{.}$$ As a corollary, these inequalities are true for the essential minimal volume of closed complex surfaces of nonnegative Kodaira dimension. We conjecture that those linear bounds in fact hold for the minimal volume.