Scalar curvature rigidity for products of spheres and tori

TL;DR

This paper establishes scalar curvature rigidity results for products of spheres and tori, extending known inequalities and characterizing the geometry of manifolds under certain curvature and mapping conditions.

Contribution

It proves new scalar curvature rigidity theorems for manifolds mapping to products of spheres and tori, extending Gromov's torical inequality with sharp width bounds.

Findings

Rigidity results for manifolds mapping to $S^{n-m}\times\mathbb{T}^m$

Extension of Gromov's torical inequality with sharp bounds

Method combining weighted slicing and spectral Dirac operator

Abstract

We prove Llarull-type rigidity for $S^{n-m}\times\mathbb{T}^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times\mathbb{T}^m$ whose spherical projection is area non-increasing, and there exists $\psi\in C^\infty(M)$ with $-\Delta_M\psi-\frac{1}{2}|D_M\psi|^2+\frac{1}{2}\big(R_M-(n-m)(n-m-1)\big)\ge0$, then $(M,g)$ is isometrically covered by $S^{n-m}\times\mathbb{R}^m$. For bands, we extend Gromov's torical inequality and obtain sharp width bounds: $\text{dist}(\partial_-M,\partial_+M)\le 2\pi\sqrt{n/((n+1)\sigma)}$ when $R_M\ge (n-m)(n-m-1)+\sigma$. The method combines stable weighted slicing with a spectral Dirac operator argument.