Singularity removal rigidity theorems for minimal hypersurfaces in manifolds with nonnegative scalar curvature

Abstract

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.