Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition
Abstract
Let (M,g) be a compact oriented Einstein 4-manifold. Write R-plus for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if R-plus 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-plus is negative definite, the Hessian of S is strictly positive modulo gauge.
Keywords
Cite
@article{arxiv.1910.09790,
title = {Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition},
author = {Joel Fine and Kirill Krasnov and Michael Singer},
journal= {arXiv preprint arXiv:1910.09790},
year = {2020}
}
Comments
16 pages. v2 small cosmetic adjustments, text agrees with published version, to appear in Mathematische Annalen