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

Abstract

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.

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

References in corpus (4)

• Einstein metrics and smooth structures
• Entropies, volumes, and Einstein metrics
• Stable Cohomotopy Seiberg-Witten Invariants of Connected Sums of Four-Manifolds with Positive First Betti Number
• Homological invariance for asymptotic invariants and systolic inequalities

Cited by in corpus (1)

• Entropies, volumes, and Einstein metrics