Macroscopic stability and simplicial norms of hypersurfaces

TL;DR

This paper develops a new inequality relating the macroscopic stability of hypersurfaces to their homology classes' simplicial norms, revealing limitations of scalar curvature bounds on hypersurface areas.

Contribution

It introduces a $bZ$--coefficient version of Guth's macroscopic stability inequality and applies it to relate hypersurface areas to Gromov simplicial norms in manifolds with scalar curvature bounds.

Findings

Lower bounds on hypersurface areas in terms of simplicial norms

Counterexamples showing positive scalar curvature does not bound hypersurface areas

Extension of Guth's inequality to $bZ$--coefficients

Abstract

We introduce a $\mathbb{Z}$--coefficient version of Guth's macroscopic stability inequality for almost-minimizing hypersurfaces. In manifolds with a lower bound on macroscopic scalar curvature, we use the inequality to prove a lower bound on areas of hypersurfaces in terms of the Gromov simplicial norm of their homology classes. We give examples to show that a very positive lower bound on macroscopic scalar curvature does not necessarily imply an upper bound on the areas of minimizing hypersurfaces.