Positive Scalar Curvature and Minimal Hypersurface Singularities

TL;DR

This paper extends minimal hypersurface techniques to all dimensions for positive scalar curvature problems, proving the positive mass theorem without spin assumptions and analyzing the structure of manifolds with positive scalar curvature.

Contribution

It develops methods to handle singularities in minimal slicings, extending regularity theory and proving key geometric results in all dimensions.

Findings

Positive mass theorem proved in all dimensions without spin assumption

Singular sets in slices have Hausdorff codimension at least three

Methods successfully extend to low-dimensional slices (1 or 2 dimensions)

Abstract

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of \cite{sy1} to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional slices. It is shown that the singular set in any slice is a closed set with Hausdorff codimension at least three. In particular for arguments which involve slicing down to dimension $1$ or $2$ the method is successful. The arguments can be viewed as an extension of the minimal hypersurface regularity theory to this setting of minimal slicings.