AI 中文总结
本文建立了爱因斯坦度量及里奇曲率有下界度量的非孤立奇点可去性结果，证明特定奇异集外有界的度量可延拓，还可用于证明Schoen的标量曲率奇点猜想。
AI 中文摘要
本文建立了爱因斯坦度量及里奇曲率有下界的度量的可去奇点结果。设n≥2，在闭n维流形上，证明了里奇曲率在余维数大于3−1/(n−1)的奇异集外有下界的L∞黎曼度量可典范延拓为RCD空间。由此，利用新的爱因斯坦度量可去奇点定理，证明了4维时，任意在余维数大于3−1/3的奇异集外有L∞奇点的爱因斯坦度量可跨奇异集光滑延拓，可能需改变光滑结构。高维时，在非塌缩RCD空间的正则集上构造了C^{1,α}黎曼流形结构，该空间在余维数大于2的集合外是有界|Ric|的黎曼流形。所得结果可用于证明Schoen关于标量曲率奇点的猜想，适用于连续度量、L∞度量或与光滑背景度量充分接近的度量。
英文摘要
In this paper we establish removable singularities results for Einstein metrics and for metrics with Ricci curvature bounded below. Let $n\geq 2$. On a closed $n$-manifold, we show that an $L^\infty$-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension $> 3- \frac{1}{n-1}$ canonically extends to an $\mathrm{RCD}$ space. As a consequence, using a new removable singularity theorem for Einstein metrics, we prove that in dimension $4$ any Einstein metric with $L^\infty$ singularities of codimension $>3-\frac{1}{3}$ extends smoothly across the singular set, possibly after changing the smooth structure. In higher dimensions, we construct a $C^{1,α}$-Riemannian manifold structure on the regular set of a non-collapsed $\mathrm{RCD}$ space that is a Riemannian manifold with bounded $|\mathrm{Ric}|$ outside a set of codimension $>2$. Our results can be used to give a proof of Schoen's conjecture on scalar curvature singularities for metrics that are either continuous, or $L^\infty$ and sufficiently close to a smooth background metric.
Comments44 pages. Comments are welcome!
