Non-degeneracy of Poincaré-Einstein four-manifolds satisfying a chiral curvature inequality

Abstract

A Poincaré-Einstein metric $g$ is called non-degenerate if there are no non-zero infinitesimal Einstein deformations of $g$, in Bianchi gauge, that lie in $L^2$. We prove that a 4-dimensional Poincaré-Einstein metric is non-degenerate if it satisfies a certain chiral curvature inequality. Write $R_{+}$ for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if $R_{+}$ is negative definite then $g$ is non-degenerate. This is a chiral generalisation of a result due to Biquard and Lee, that a Poincaré-Einstein metric of negative sectional curvature is non-degenerate

arXiv comment18 pages