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.

中文翻译：

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满足手征曲率条件的爱因斯坦四流形的局部刚性

令 (M,g) 是一个紧致的爱因斯坦四流形。将 g 的曲率算子作用于自对偶 2-形式的部分写出 R-plus。我们证明，如果 R-plus 是负定的，那么 g 是局部刚性的：靠近 g 的任何其他爱因斯坦度量都与它等距。这是 Koiso 定理的手征推广，它证明了具有负截面曲率的爱因斯坦度量的局部刚性。我们的假设大约是 Koiso 的一半。我们的证明使用了爱因斯坦 4-流形的新变分描述，作为所谓的浇连接动作 S 的临界点。证明的关键步骤是当 R-plus 是负定时，S 的 Hessian 是严格正模测量。