Scalar curvature rigidity for products of spheres and tori

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 $ψ\in C^\infty(M)$ with $-Δ_Mψ-\frac{1}{2}|D_Mψ|^2+\frac{1}{2}\big(R_M-(n-m)(n-m-1)\big)\ge0$, 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: $\text{dist}(\partial_-M,\partial_+M)\le 2π\sqrt{n/((n+1)σ)}$ when $R_M\ge (n-m)(n-m-1)+σ$. The method combines stable weighted slicing with a spectral Dirac operator argument.