Stability of Einstein 4-manifolds satisfying a chiral curvature condition

Abstract

Let $(M,g)$ be a compact oriented Einstein four-manifold with Einstein constant $E$ and let ${\hat{R}}^{+}$ denote the action of the Riemann curvature tensor on self-dual two-forms. We show that if ${\hat{R}}^{+}<0$, then $g$ is strictly linearly stable for the Einstein-Hilbert functional, thus giving a chiral criterion for stability. Our result is stronger than the one by Fine-Krasnov-Singer, who conclude local rigidity from ${\hat{R}}^{+}<0$ by proving stability for a different action functional. Our proof proceeds by showing that $(M,g)$ admits a natural spin${}^h$ structure carrying a non-zero parallel spin${}^h$-spinor. We then apply a lower bound on the Lichnerowicz Laplacian on traceless symmetric two-tensors in the presence of such a spinor.

8 pages. Comments welcome!