Non-degeneracy of Poincar\'e-Einstein four-manifolds satisfying a chiral curvature inequality

TL;DR

This paper proves that 4-dimensional Poincaré-Einstein metrics satisfying a specific chiral curvature inequality are non-degenerate, extending previous results related to negative sectional curvature.

Contribution

It introduces a new curvature inequality condition involving the self-dual part of the curvature operator to establish non-degeneracy of Poincaré-Einstein metrics.

Findings

If R_+ is negative definite, then the metric is non-degenerate.

The result generalizes previous work by Biquard and Lee.

Provides a new criterion for non-degeneracy based on chiral curvature.

Abstract

A Poincar\'e-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 Poincar\'e-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 Poincar\'e-Einstein metric of negative sectional curvature is non-degenerate