Essential minimal volume of Einstein 4-manifolds

2021/03/09 by Song, Antoine
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2103.05659

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, ess-Minvol(M), which is the limit, as δ>0 goes to 0, of the infimum of the volume of the δ-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^-1e(M) ≤ ess-Minvol(M) ≤ 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.

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