论文标题
带有连续指标的正质量定理的可移动奇异性
Removable singularity of positive mass theorem with continuous metrics
论文作者
论文摘要
在本文中，我们认为，与$ c^0 $ c^0 $ metric $ g $和$ g $的渐近平坦的riemannnian歧管$（m^n，g）$相对于封闭的界限子集$σ$和标量曲率$ r_g \ ge ge 0 $在$ m \ m \ m \ m \ setMinusσ$上平稳。对于给定的$ n \ le p \ le \ infty $，如果$ g \ in c^0 \ cap w^{1，p} $和hausdorff量级$ \ mathcal {h}^{n- \ frac {p} {p} $ \ MATHCAL {h}^{n-1}（σ）= 0 $当$ p = \ infty $时，我们证明每一端的ADM质量是无负的。此外，如果某个末端的ADM质量为零，那么我们证明$（m^n，g）$是对欧几里得空间的等值线，通过显示歧管在RCD方面具有非负RICCI曲率。这将[Lee-Lefloch2015]的结果从自旋到非自旋扩展，也可以改善[Shi-Tam2018]和[Lee2013]的结果。此外，对于$ p = \ infty $，这证实了李[Lee2013]的猜想。
In this paper, we consider asymptotically flat Riemannnian manifolds $(M^n,g)$ with $C^0$ metric $g$ and $g$ is smooth away from a closed bounded subset $Σ$ and the scalar curvature $R_g\ge 0$ on $M\setminus Σ$. For given $n\le p\le \infty$, if $g\in C^0\cap W^{1,p}$ and the Hausdorff measure $\mathcal{H}^{n-\frac{p}{p-1}}(Σ)<\infty$ when $n\le p<\infty$ or $\mathcal{H}^{n-1}(Σ)=0$ when $p=\infty$, then we prove that the ADM mass of each end is nonnegative. Furthermore, if the ADM mass of some end is zero, then we prove that $(M^n,g)$ is isometric to the Euclidean space by showing the manifold has nonnegative Ricci curvature in RCD sense. This extends the result of [Lee-LeFloch2015] from spin to non-spin, also improves the result of [Shi-Tam2018] and [Lee2013]. Moreover, for $p=\infty$, this confirms a conjecture of Lee [Lee2013].
