Rigidity of area-minimizing 2-spheres in 𝑛-manifolds with positive scalar curvature

We prove that the least area of noncontractible immersed spheres is no greater than 4 π 4π in any oriented manifold with dimension n + 2 ≤ 7 n+2≤ 7 which satisfies R ≥ 2 R≥ 2 and admits a continuous map to S 2 × T n S^2× T^n with nonzero degree. We also prove a rigidity result for the equality case. This can be viewed as a generalization of the result in [Comm. Anal. Geom. 18 (2010), pp. 821–830] to higher dimensions.