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

par Fine, Joel ;Krasnov, Kirill;Singer, Michael
Référence Mathematische Annalen, 55
Publication Publié, 2020-10-14

Article révisé par les pairs

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Résumé :

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.