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

Fine, Joel; Krasnov, Kirill; Singer, Michael

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+

Citation

Fine, J., Krasnov, K., & Singer, M. (2020). Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition. Mathematische Annalen, 379, 569–588. https://doi.org/10.1007/s00208-020-02097-z

• Journal Article Type: Article
• Acceptance Date: Oct 6, 2020
• Online Publication Date: Oct 14, 2020
• Publication Date: Oct 14, 2020
• Deposit Date: Oct 16, 2020
• Publicly Available Date: Oct 15, 2021
• Journal: Mathematische Annalen
• Print ISSN: 0025-5831
• Electronic ISSN: 1432-1807
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 379
• Pages: 569–588
• DOI: https://doi.org/10.1007/s00208-020-02097-z
• Public URL: https://nottingham-repository.worktribe.com/output/4967702
• Publisher URL: https://link.springer.com/article/10.1007/s00208-020-02097-z