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

Abstract:

We prove that the least area of noncontractible immersed spheres is no greater than $4\pi$ in any oriented manifold with dimension $n+2\leq 7$ which satisfies $R\geq 2$ and admits a continuous map to $\mathbf S^2\times T^n$ 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.