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Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition

Fine, J; Krasnov, K; Singer, M; (2020) Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition. Mathematische Annalen 10.1007/s00208-020-02097-z. (In press).

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Abstract

Let (M,g) be a compact oriented Einstein 4-manifold. 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 locally rigid: any other Einstein metric near to g is isometric to it. This is a chiral generalisation of Koiso's Theorem, which proves local rigidity of Einstein metrics with negative sectional curvatures. Our hypotheses are roughly one half of Koiso's. Our proof uses a new variational description of Einstein 4-manifolds, as critical points of the so-called poure connection action S. The key step in the proof is that when R_{+} < 0, the Hessian of S is strictly positive modulo gauge.

Type: Article
Title: Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition
DOI: 10.1007/s00208-020-02097-z
Publisher version: https://doi.org/10.1007/s00208-020-02097-z
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher's terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10112563
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