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Non-degeneracy of Poincaré–Einstein Four-Manifolds Satisfying a Chiral Curvature Inequality

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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 \({{\,\textrm{Rm}\,}}_+\) for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if \({{\,\textrm{Rm}\,}}_+\) 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.

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Acknowledgements

I would like to thank Rafe Mazzeo and Michael Singer for several helpful conversations on this topic. This research was supported by the ERC consolidator grant “SymplecticEinstein” 646649 and the Excellence of Science Grant 4000725.

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  1. Département de mathématique, Université libre de Bruxelles, Brussels, Belgium

    Joel Fine

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Fine, J. Non-degeneracy of Poincaré–Einstein Four-Manifolds Satisfying a Chiral Curvature Inequality. J Geom Anal 33, 249 (2023). https://doi.org/10.1007/s12220-023-01306-w

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