Llarull type theorems on complete manifolds with positive scalar curvature

In this paper, without assuming that manifolds are spin, we prove that if a compact orientable, and connected Riemannian manifold $(M^n,g)$ with scalar curvature $R_g\ge 6$ admits a non-zero degree and $1$-Lipschitz map to $({\mathbb{S}}^3\times {\mathbb{T}}^{n-3},g_{{\mathbb{S}}^3}+g_{{\mathbb{T}}^{n-3}})$, for $4\le n\le 7$, then $(M^n,g)$ is locally isometric to ${\mathbb{S}}^3\times {\mathbb{T}}^{n-3}$. Similar results are established for noncompact cases as $({\mathbb{S}}^3\times {\mathbb{R}}^{n-3},g_{{\mathbb{S}}^3}+g_{{\mathbb{R}}^{n-3}})$ being model spaces (see Theorem \ref{noncompactrigidity1}, Theorem \ref{noncompactrigidity2}, Theorem \ref{noncompactrigidity3}, Theorem \ref{noncompactrigidity4}). We observe that the results differ significantly when $n=4$ compared to $n\ge 5$. Our results imply that the $\epsilon$-gap length extremality of the standard ${\mathbb{S}}^3$ is stable under the Riemannian product with ${\mathbb{R}}^m$, $1\le m\le 4$ (see $D_3$. Question in Gromov’s paper \cite{Gromov2017}, p.153).