标题： 正标量曲率和孤立的圆锥奇点 显示英文标题

标题： Positive scalar curvature and isolated conical singularity

摘要： 我们证明了一个关于孤立圆锥奇点的Geroch型结果。 即，我们证明在$ X \# T^n $上不存在具有孤立圆锥奇点的黎曼度量$g$，该度量在正则部分具有非负标量曲率，并且在某点为正。 特别地，这表明在环面（tori）上不存在具有孤立圆锥奇点且标量曲率为正的度量。 我们还证明了在$X \# T^n$上具有有限多个孤立圆锥奇点的标量平坦的黎曼度量$g$必须是平坦的，并且可以在奇点处光滑延拓。 我们不事先假设$X$上的圆锥奇点是一个流形点；即，圆锥奇点的横截面可能不是球面的。 显示英文摘要

摘要： We prove a Geroch type result for isolated conical singularity. Namely, we show that there is no Riemannian metric $g$ on $ X \# T^n $ with an isolated conical singularity which has nonnegative scalar curvature on the regular part, and is positive at some point. In particular, this implies that there is no metric on tori with an isolated conical singularity and positive scalar curvature. We also prove that a scalar flat Riemannian metric $g$ on $X \# T^n$ with finitely many isolated conical singularities must be flat, and extend smoothly across the singular points. We do not a priori assume that a conically singular point on $X$ is a manifold point; i.e., the cross section of the conical singularity may not be spherical.