A proof of Gromov's cube inequality on scalar curvature

TL;DR

This paper proves Gromov's cube inequality for scalar curvature in all dimensions using Dirac operator methods, extending previous results limited to dimensions up to 8 and providing a stronger version of the inequality.

Contribution

The authors establish Gromov's cube inequality in all dimensions with optimal constants, using Dirac operator techniques, surpassing the minimal surface approach.

Findings

Proved Gromov's cube inequality in all dimensions.

Derived a strengthened version of the inequality.

Provided a new proof method via Dirac operators.

Abstract

Gromov proved a cube inequality on the bound of distances between opposite faces of a cube equipped with a positive scalar curvature metric in dimension $\leq 8$ using minimal surface method. He conjectured that the cube inequality also holds in dimension $\geq 9$. In this paper, we prove Gromov's cube inequality in all dimensions with the optimal constant via Dirac operator method. In fact, our proof yields a strengthened version of Gromov's cube inequality, which does not seem to be accessible by minimal surface method.