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

Abstract

We prove that the least area of the non-contractible immersed spheres is no more than $4\pi$ in any oriented compact manifold with dimension $n+2\le 7$ which satisfies $R\ge 2$ and admits a 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 [2] to higher dimensions.

11 pages, fix a gap in section 3 in previous version

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