Collapsing and group growth as obstructions to Einstein metrics on some smooth 4-manifolds

TL;DR

This paper investigates how collapsing and fundamental group growth hinder the existence of Einstein metrics on various smooth 4-manifolds, including infrasolvmanifolds, elliptic surfaces, and geometrizable manifolds.

Contribution

It identifies specific obstructions related to collapsing and fundamental group growth that prevent Einstein metrics on certain classes of 4-manifolds.

Findings

Collapsing and fundamental group growth obstruct Einstein metrics.

Infrasolvmanifolds with non-virtually nilpotent fundamental groups lack Einstein metrics.

Certain elliptic and geometrizable 4-manifolds cannot admit Einstein metrics due to these obstructions.

Abstract

We show that a combination of collapsing and excessive growth from the fundamental group impedes the existence of Einstein metrics on several families of smooth four-manifolds. These include infrasolvmanifolds whose fundamental group is not virtually nilpotent, most elliptic surfaces of zero Euler characteristic, geometrizable manifolds with hyperbolic factor geometries in their geometric decomposition, and higher graph four-manifolds without purely negatively curved pieces.