paper

Entropies, volumes, and Einstein metrics

arXiv:math/0410215 · doi:10.1007/978-3-642-22842-1_2

Abstract

We survey the definitions and some important properties of several asymptotic invariants of smooth manifolds, and discuss some open questions related to them. We prove that the (non-)vanishing of the minimal volume is a differentiable property, which is not invariant under homeomorphisms. We also formulate an obstruction to the existence of Einstein metrics on four-manifolds involving the volume entropy. This generalizes both the Gromov--Hitchin--Thorpe inequality and Sambusetti's obstruction.

This is a substantial revision and expansion of the 2004 preprint, which I prepared in spring of 2010 and which has since been published. The version here is essentially the published one, minus the problems introduced by Springer production

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