On Gromov's dihedral extremality and rigidity conjectures

TL;DR

This paper introduces a new index theory for manifolds with polyhedral boundaries and proves Gromov's dihedral extremality and rigidity conjectures across all dimensions, advancing understanding of scalar and mean curvature comparisons.

Contribution

It develops a novel index theory for polyhedral manifolds and proves two longstanding conjectures by Gromov regarding dihedral extremality and rigidity in all dimensions.

Findings

Proved Gromov's dihedral extremality conjecture for scalar and mean curvature comparisons.

Established Gromov's dihedral rigidity conjecture for certain positively curved manifolds.

Developed a new index theory applicable to manifolds with polyhedral boundary.

Abstract

In this paper, we develop a new index theory for manifolds with polyhedral boundary. As an application, we prove Gromov's dihedral extremality conjecture regarding comparisons of scalar curvatures, mean curvatures and dihedral angles between two compact manifolds with polyhedral boundary in all dimensions. We also prove Gromov's dihedral rigidity conjecture for a class of positively curved manifolds with polyhedral boundary in all dimensions.