Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition
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Mathematische Annalen
Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition Joel Fine1 · Kirill Krasnov2 · Michael Singer3 Received: 22 October 2019 / Revised: 3 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 curvature. 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 pure connection action S. The key step in the proof is that when R+ < 0, the Hessian of S is strictly positive modulo gauge.
Contents 1 Introduction . . . . . . . . . . . . . . . . . 1.1 Statement of the main result . . . . . . . 1.2 Outline of the proof . . . . . . . . . . . 1.3 Remark . . . . . . . . . . . . . . . . . 2 From Plebanski to the pure connection action 2.1 Some foundational 4-dimensional facts . 2.2 The Plebanski action . . . . . . . . . . 2.3 The pure connection action . . . . . . . 2.4 The gauge group . . . . . . . . . . . . . 3 The Hessian of the pure connection action . 3.1 The Hessian before gauge fixing . . . . 3.2 The Hessian after gauge fixing . . . . .
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Communicated by F. C. Marques. JF was supported by ERC consolidator Grant 646649 “SymplecticEinstein” and EoS Grant 30950721.
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Joel Fine [email protected]
1
Université libre de Bruxelles, Bruxelles, Belgium
2
University of Nottingham, Nottingham, UK
3
University College London, London, UK
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J. Fine et al. 3.3 Gauge fixing . . . . . . . 3.4 The proof of Theorem 3.1 4 The proof of local rigidity . . References . . . . . . . . . . . .
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1 Introduction 1.1
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