Entropies, volumes, and Einstein metrics
微分几何
2013-01-29 v2 几何拓扑
摘要
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.
引用
@article{arxiv.math/0410215,
title = {Entropies, volumes, and Einstein metrics},
author = {D. Kotschick},
journal= {arXiv preprint arXiv:math/0410215},
year = {2013}
}
备注
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