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

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: \begin{enumerate} \item They have positive minimal volume entropy. \item They satisfy a strict version of the Gromov–Hitchin–Thorpe inequality, with a minimal volume entropy term. \item They nevertheless admit infinitely many distinct smooth structures for which no compatible Einstein metric exists. \end{enumerate}

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Published 1 January 2009