Scalar Curvature Splittings I: Minimal Factors

TL;DR

This paper introduces minimal splitting factors with positive scalar curvature that resemble area minimizing hypersurfaces, providing new tools for analyzing scalar curvature constraints through splitting procedures.

Contribution

It defines canonical minimal splitting factors with positive scalar curvature, enhancing control over splitting procedures in scalar curvature studies.

Findings

Minimal splitting factors have positive scalar curvature.

They satisfy Poincare, Sobolev, and isoperimetric inequalities.

Singular points admit tangent cones with positive scalar curvature.

Abstract

Scalar curvature constraints can be studied by means of splitting procedures. The success of this strategy depends on the control we can get on its splitting factors. We introduce canonical so-called minimal splitting factors. They have positive scalar curvature while other properties strongly resemble those of area minimizing hypersurfaces. This includes the presence of Poincare, Sobolev and isoperimetric inequalities and the fact that singular points admit tangent cones but now with positive scalar curvature.