Smooth Structures and Einstein Metrics on $CP^2#5,6,7\bar{CP^2}$

TL;DR

This paper demonstrates the existence of multiple smooth structures on certain 4-manifolds that admit Einstein metrics with both positive and negative scalar curvature, and shows some structures do not admit Einstein metrics.

Contribution

It constructs new examples of 4-manifolds with distinct smooth structures that support Einstein metrics of different scalar curvatures, expanding understanding of Einstein metrics in topology.

Findings

Existence of Einstein metrics with positive scalar curvature on certain smooth structures.

Existence of Einstein metrics with negative scalar curvature on other smooth structures.

Infinitely many non-diffeomorphic structures without Einstein metrics.

Abstract

We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$ admits a smooth structure which has an Einstein metric of scalar curvature $s > 0$, a smooth structure which has an Einstein metric with $s < 0$ and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We show that there are infinitely many manifolds homeomorphic non-diffeomorphic to $CP^2#5\bar{CP^2}$ which do not admit an Einstein metric. We also exhibit new higher dimensional examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature. The main ingredients are recent constructions of exotic symplectic or complex manifolds with small topological numbers.