On the nonexistence of Einstein metric on 4-manifolds

TL;DR

This paper demonstrates that certain 4-manifolds, constructed via connected sums or surgeries on manifolds with nontrivial Seiberg-Witten invariants, cannot admit Einstein metrics, using Seiberg-Witten invariants and gluing formulae.

Contribution

It establishes new nonexistence results for Einstein metrics on a broad class of 4-manifolds using Seiberg-Witten theory and surgical techniques.

Findings

Certain 4-manifolds do not admit Einstein metrics.

Nonexistence results apply after multiple connected sums or surgeries.

Seiberg-Witten invariants are key to proving these results.

Abstract

By using the gluing formulae of the Seiberg-Witten invariant, we show the nonexistence of Einstein metric on manifolds obtained from a 4-manifold with nontrivial Seiberg-Witten invariant by performing sufficiently many connected sums or appropriate surgeries along circles or homologically trivial 2-spheres with closed oriented 4-manifolds with negative definite intersection form.