An essential relation between Einstein metrics, volume entropy, and exotic smooth structures

Details An essential relation between Einstein metrics, volume entropy, and exotic smooth structures An essential relation between Einstein metrics, volume entropy, and exotic smooth structures

2008-07-08

arxiv.org/abs/0807.1187v2

Mathematics

Differential Geometry

12 pages, v2; two references added, to appear in Math. Research Letters

Scientific paper

We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positive minimal volume entropy. 2) They satisfy a strict version of the Gromov-Hitchin-Thorpe inequality, with a minimal volume entropy term. 3) They nevertheless admit infinitely many distinct smooth structures for which no compatible Einstein metric exists.

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