Positive mass theorems on singular spaces and some applications

TL;DR

This paper extends positive mass theorems to singular spaces and minimal hypersurfaces with isolated singularities, using dimension reduction and spectral scalar curvature conditions, bypassing traditional regularity requirements.

Contribution

It introduces new positive mass theorems for asymptotically flat manifolds with arbitrary ends and singularities, employing novel rigidity theorems and spectral scalar curvature assumptions.

Findings

Effective positive mass theorem for AF manifolds of dimension ≤8

Rigidity theorems for minimal hypersurfaces with isolated singularities

Positive mass theorem established for singular spaces under spectral scalar curvature conditions

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.