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

TL;DR

This paper establishes singularity removal rigidity theorems for minimal hypersurfaces in nonnegative scalar curvature manifolds, revealing that extremal scalar curvature enforces smoothness and providing a new proof of the positive mass theorem in 8 dimensions.

Contribution

It introduces new rigidity theorems for minimal hypersurfaces with isolated singularities and applies them to prove the positive mass theorem in higher dimensions without generic regularity assumptions.

Findings

Extremal scalar curvature conditions enforce smoothness of minimal hypersurfaces.

A new spectral version of the positive mass theorem is established for asymptotically flat manifolds.

The positive mass theorem is proved for 8-dimensional manifolds with arbitrary ends without using Smale's theorem.

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.