On Gromov's rigidity theorem for polytopes with acute angles

TL;DR

This paper provides a detailed proof of Gromov's conjecture on scalar curvature extremality for convex polytopes with acute angles, using Dirac operator techniques and smoothing constructions.

Contribution

It offers the first complete proof of Gromov's theorem in the case of acute dihedral angles, clarifying the smoothing method involved.

Findings

Confirmed Gromov's scalar curvature extremality conjecture for acute-angled polytopes.

Developed a detailed smoothing construction essential for the proof.

Validated the use of Dirac operator techniques in this geometric context.

Abstract

In his ``Four Lectures", Gromov conjectured a scalar curvature extremality property of convex polytopes. Moreover, Gromov outlined a proof of the conjecture in the special case when the dihedral angles are acute. Gromov's argument relies on Dirac operator techniques together with a smoothing construction. In this paper, we give the details of such a smoothing construction, thereby providing a detailed proof of Gromov's theorem.