Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition

TL;DR

This paper proves that Einstein 4-manifolds with a negative definite self-dual curvature component are locally rigid, meaning nearby Einstein metrics are isometric, extending Koiso's theorem to a chiral curvature condition.

Contribution

It introduces a new variational approach using the pure connection action and establishes local rigidity under a negative definiteness condition on R-plus.

Findings

Proves local rigidity for Einstein 4-manifolds with negative R-plus.

Develops a variational framework via the pure connection action.

Shows Hessian positivity modulo gauge when R-plus is negative definite.

Abstract

Let (M,g) be a compact oriented Einstein 4-manifold. Write R-plus for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if R-plus is negative definite then g is locally rigid: any other Einstein metric near to g is isometric to it. This is a chiral generalisation of Koiso's Theorem, which proves local rigidity of Einstein metrics with negative sectional curvatures. Our hypotheses are roughly one half of Koiso's. Our proof uses a new variational description of Einstein 4-manifolds, as critical points of the so-called poure connection action S. The key step in the proof is that when R-plus is negative definite, the Hessian of S is strictly positive modulo gauge.