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 $R_{g}\geq 6$ 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 $4\leq n\leq 7$, 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 $n\geq 5$. Our results imply that the $\epsilon$-gap length extremality of the standard $\mathbb{S}^3$ is stable under the Riemannian product with $\mathbb{R}^m$, $1\leq m\leq 4$ (see $D_{3}$. Question in Gromov's paper \cite{Gromov2017}, p.153).