Essential minimal volume of Einstein 4-manifolds

TL;DR

This paper introduces the concept of essential minimal volume for 4-manifolds, establishing linear bounds in terms of Euler characteristic for Einstein 4-manifolds and complex surfaces.

Contribution

It defines the essential minimal volume and proves universal linear bounds relating it to Euler characteristic for Einstein 4-manifolds.

Findings

Essential minimal volume is bounded linearly by Euler characteristic.

Bounds hold for Einstein 4-manifolds and nonnegative Kodaira dimension surfaces.

Conjecture that minimal volume also satisfies similar linear bounds.

Abstract

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, $\mathrm{ess-Minvol}(M)$, 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 $$C^{-1}e(M) \leq \mathrm{ess-Minvol}(M) \leq Ce(M).$$ 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.