Scalar curvature rigidity for products of spheres and tori
Tsz-Kiu Aaron Chow
TL;DR
This work proves Llarull-type rigidity for maps from closed spin manifolds to sphere-torus products (S^{n-m}×T^m) in dimensions 3≤n≤7, under a scalar-curvature lower bound adjusted by the spherical factor and a degree-nonzero map with area-nonincreasing projection. The authors introduce stable weighted slicing to encode torus directions and combine it with a spectral Llarull argument via twisted Dirac operators to obtain rigidity: (M,g) is isometrically covered by (S^{n-m}×R^m, g_{S^{n-m}}+g_{R^m}) and the auxiliary function ψ is constant. They extend the theory to incomplete settings by establishing sharp torical-band width bounds dist(∂_-M,∂_+M) ≤ 2π√( n / ((n+1)σ) ) under R_M ≥ (n-m)(n-m-1)+σ, and show a refined bound d(∂_-M,∂_+M) ≤ 2π/√σ when a σ-perturbed inequality holds. The approach blends stable weighted slicing with spectral Dirac-operator methods and μ-bubble techniques to connect curvature, topology, and macroscopic geometry in mixed sphere-torus geometries.
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.