A Note on Scalar curvature comparison rigidity for compact domains

TL;DR

This paper generalizes scalar curvature rigidity results from convex polytopes to broader convex domains in Euclidean space, using harmonic spinors and boundary conditions, extending Shi-Tam's work.

Contribution

It introduces a new approach to scalar curvature comparison rigidity for convex domains, broadening Gromov's conjecture to more general shapes.

Findings

Proves scalar curvature and mean curvature comparison rigidity for convex domains.

Extends Gromov's conjecture to arbitrary convex Riemannian polytope type domains.

Provides a parallel to Shi-Tam's results on scalar curvature rigidity.

Abstract

We prove a generalization of Gromov's conjecture on scalar curvature rigidity of convex polytopes to arbitrary convex Riemannian polytope type domains via harmonic spinors on convex domians with boundary condition constructed by Brendle. In particular, we prove a rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain in Euclidean space, which is a parallel of Shi-Tam's results.