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We prove that the least area of the non-contractible immersed spheres is nomore than $4\pi$ in any oriented compact manifold with dimension $n+2\leq 7$which satisfies $R\geq 2$ and admits a map to $\mathbf S^2\times T^n$ withnonzero degree. We also prove a rigidity result for the equality case. This canbe viewed as a generalization of the result in [2] to higher dimensions.

Rigidity of Area-Minimizing $2$-Spheres in $n$-Manifolds with Positive Scalar Curvature