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arXiv · 2212.00526

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

Joel Fine
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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

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Joel Fine. 2022-12-01. Non-degeneracy of Poincar\'e-Einstein four-manifolds satisfying a chiral curvature inequality. https://arxiv.org/abs/2212.00526

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