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

*(English)*Zbl 1183.53038Following a terminology introduced by M. Gromov in [J. Differ. Geom. 18, 1–147 (1983; Zbl 0515.53037)], it is said that a closed connected manifold \(V\) is essential if for some map \(f:V\rightarrow K\) into an aspherical space the induced top-dimensional homomorphism on homology does not vanish.

The first main result of the paper shows that the minimal volume entropy of a connected closed manifold \(M\) doesn’t change for the connected sum of \(M\) with a connected closed orientable manifold \(N\) which is not essential.

This result is used to extract new information about exotic smooth structures on 4-manifolds and existence of Einstein metrics.

The following main result of the paper gives a positive answer to an open question which stems from work of C. LeBrun in [Math. Res. Lett. 3, No. 2, 133–147 (1996; Zbl 0856.53035)]. The authors show that there exists an infinite family of 4-manifolds with the following properties:

The first main result of the paper shows that the minimal volume entropy of a connected closed manifold \(M\) doesn’t change for the connected sum of \(M\) with a connected closed orientable manifold \(N\) which is not essential.

This result is used to extract new information about exotic smooth structures on 4-manifolds and existence of Einstein metrics.

The following main result of the paper gives a positive answer to an open question which stems from work of C. LeBrun in [Math. Res. Lett. 3, No. 2, 133–147 (1996; Zbl 0856.53035)]. The authors show that there exists an infinite family of 4-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. This implies that their homotopy type satisfies all the restrictions known so far to the existence of an Einstein metric.
- 3)
- They nevertheless each admit infinitely many distinct smooth structures for which no compatible Einstein metric exists.

Reviewer: Cornelia-Livia Bejan (Iaşi)

##### MSC:

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

57R55 | Differentiable structures in differential topology |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

57R57 | Applications of global analysis to structures on manifolds |