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 ) (Mⁿ,g) with scalar curvature R g ≥ 6 Rg≥ 6 admits a non-zero degree and 1 1 -Lipschitz map to ( S 3 × T n − 3 , g S 3 + g T n − 3 ) (S³× Tⁿ⁻³,g_{S³}+g_{Tⁿ⁻³}) , for 4 ≤ n ≤ 7 4≤ n≤ 7 , then ( M n , g ) (Mⁿ,g) is locally isometric to S 3 × T n − 3 S³× Tⁿ⁻³ . Similar results are established for non-compact cases as ( S 3 × R n − 3 , g S 3 + g R n − 3 ) (S³× Rⁿ⁻³,g_{S³}+g_{Rⁿ⁻³}) being model spaces. We observe that the results differ significantly when n = 4 n=4 compared to n ≥ 5 n≥ 5 . Our results imply that the ϵ ε -gap length extremality of the standard S 3 S^3 is stable under the Riemannian product with R m R^m , 1 ≤ m ≤ 4 1≤ m≤ 4 (see D 3 D₃ . Question in Gromov’s paper [Foundations of mathematics and physics one century after Hilbert, Spring, Cham, 2018, p. 153]).