Singularity removal rigidity theorems for minimal hypersurfaces in manifolds with nonnegative scalar curvature
Peking University（北京大学）；Hangzhou Normal University（杭州师范大学）
Shihang He, Yuguang Shi, Haobin Yu
备注：
5 figures. This paper is the first part of the work arXiv:2502.18000v2. For improved readability, we have split the original text into two parts. All comments are welcome!
我们证明了两个\"奇点去除刚性定理\"，用于具有孤立奇点的极小超曲面在非负标量曲率流形中的情况（定理\ref{thm: rigidity for minimal surface}和\ref{thm: georch free of singularity}）。特别地，我们观察到一个新现象，即极值标量曲率条件迫使光滑性，这揭示了极小超曲面奇点在标量曲率几何中的一种积极影响。作为应用，我们得到了一个直接证明正质量定理（PMT）的证明，用于具有任意端点的渐近扁$8$-流形（定理\ref{thm: pmt8dim}），而无需使用N. Smale的通用正则性定理。关键成分是一个新的关于AF流形的谱版本的PMT，其证明依赖于具有$\mathbf{S}^1$对称性的渐近局部扁（ALF）流形的PMT证明。
英文摘要：
We prove two "Singularity removal rigidity theorems" for minimal hypersurfaces with isolated singularities in manifolds of nonnegative scalar curvature (Theorems \ref{thm: rigidity for minimal surface} and \ref{thm: georch free of singularity}). In particular, we observe a new phenomenon that the extremal scalar curvature condition forces smoothness, which reveals a kind of positive effect of minimal hypersurface singularities in scalar curvature geometry.
As an application, we obtain a direct proof of the positive mass theorem (PMT) for asymptotically flat $8$-manifolds with arbitrary ends (Theorem \ref{thm: pmt8dim}), without using N. Smale's generic regularity theorem. A key ingredient is a new spectral version of PMT for AF manifolds with arbitrary ends, whose proof relies on PMT for asymptotically locally flat (ALF) manifolds with $\mathbf{S}^1$-symmetry.
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