Scalar Curvature Splittings II: Removal of Singularities

TL;DR

This paper extends previous work on scalar curvature by demonstrating that certain minimal geometries with singularities can be regularized through surgery, advancing understanding of scalar curvature in geometric analysis.

Contribution

It introduces surgery techniques to remove singularities in minimal geometries with positive scalar curvature, building on conformal deformation methods from Part I.

Findings

Singular sets can be eliminated via surgery in minimal geometries.

Conformal deformations preserve key geometric properties.

Results facilitate smoother models in scalar curvature studies.

Abstract

In Part I of this paper we have seen that any singular compact area minimizer in a positive scalar curvature manifold admits a conformal deformation to some minimal factor geometry that shares many properties with the minimizer, like the Ahlfors regularity, the validity of Poincare inequalities and the presence of tangent cones with positive scalar curvature. In this Part II we show that these geometries admit surgery style arguments eliminating the singular sets.