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Abstract

Building upon dimension reduction techniques in the study of positive scalar curvature (PSC) geometry, we prove an effective version of the positive mass theorem (PMT) for asymptotically flat (AF) manifolds of dimension $n\leq 8$ with arbitrary ends (Theorem \ref{thm: 8dim Schoen conj}). Furthermore, we prove two "free of singularity type rigidity theorems" for minimal hypersurfaces with isolated singularities (Theorem \ref{prop: rigidity for minimal surface} and Theorem \ref{thm: georch free of singularity}). Our approach bypasses the need for N. Smale's regularity theorem for minimal hypersurfaces in generic $8$-dimensional compact manifolds, providing a direct derivation of the PMT for such AF manifolds (Theorem \ref{thm: pmt8dim}). Motivated by these developments, we further establish PMT for singular spaces (Theorems \ref{thm:pmt with singularity4}). These results assume only that the scalar curvature is non-negative in a strong spectral sense, a condition naturally aligned with the stability of minimal hypersurfaces in PSC ambient manifolds.