Doubling a cube with positive scalar curvature

TL;DR

This paper clarifies a step in Gromov's 2013 work on scalar curvature limits and proves that a cube cannot have a positive scalar curvature metric with certain convexity and angle conditions, using a contradiction with the Geroch conjecture.

Contribution

It provides a detailed proof that a cube cannot admit a positive scalar curvature metric with mean convex faces and acute dihedral angles, extending Gromov's scalar curvature convergence results.

Findings

A cube cannot have a positive scalar curvature metric with mean convex faces and acute dihedral angles.

The proof constructs a positive scalar curvature metric on the torus, contradicting the Geroch conjecture.

Clarifies a previously sketched step in Gromov's 2013 scalar curvature work.

Abstract

In a 2013 paper, Gromov proves that if smooth Riemannian metrics $g_i$ converge to a smooth Riemannian metric $g$ uniformly, and $g_i$ have scalar curvature uniformly bounded below, then $g$ shares the same scalar curvature lower bound. In some places in the paper, the proofs are only sketched. In this paper we explain one of those sketched steps in detail. Specifically, we prove that a cube (the product $[0,1]^n$) cannot have a Riemannian metric with positive scalar curvature, such that the faces are mean convex, and such that the dihedral angles along the edges are all acute. The proof is accomplished by taking such a metric, and producing from it a metric on the $n$-torus with positive scalar curvature, contradicting the Geroch conjecture.