Communications in Analysis and Geometry

Volume 30 (2022)

Number 5

Macroscopic stability and simplicial norms of hypersurfaces

Pages: 949 – 959



Hannah Alpert (Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, U.S.A.)


We introduce a $Z$-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.

Received 26 December 2017

Accepted 8 November 2019

Published 17 March 2023