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

TL;DR

This paper explores the relationship between Einstein metrics, volume entropy, and smooth structures on four-manifolds, showing that certain geometric properties are preserved under specific topological modifications and constructing examples with unique smooth structures.

Contribution

It introduces a novel connection between Einstein metrics and volume entropy, and constructs infinite families of four-manifolds with distinct smooth structures lacking Einstein metrics.

Findings

Minimal volume entropy is invariant under connected sum with nonessential manifolds.

Constructed four-manifolds with positive volume entropy satisfying a strict Gromov-Hitchin-Thorpe inequality.

Existence of infinitely many smooth structures without compatible Einstein metrics.

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.