We show that non-compact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions. The main ingredient in the proof is to show, via a cohomogeneity-one approach, that non-compact homogeneous spaces admitting an ideal isomorphic to ${\mathfrak{sl}}_2(\mathbb{R})$ admit no homogeneous Einstein metrics of negative Ricci curvature.

中文翻译：

---

9 维和 10 维的阿列克谢夫斯基猜想

我们表明，非紧致齐次空间与 9 维或 10 维欧几里得空间不微分，不允许负 Ricci 曲率的齐次爱因斯坦度量，只有三个潜在的例外。证明的主要成分是通过一种同构方法证明非紧致同构空间允许理想同构${\mathfrak{SL}}_2(\mathbb{电阻})$ 不承认负 Ricci 曲率的齐次爱因斯坦度量。