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

TL;DR

This paper establishes an upper bound on the least area of non-contractible immersed 2-spheres in certain high-dimensional manifolds with positive scalar curvature, and proves a rigidity result when this bound is achieved.

Contribution

It generalizes previous results to higher dimensions, providing bounds and rigidity for minimal spheres in manifolds with scalar curvature at least 2.

Findings

Least area of non-contractible spheres is at most 4π

Rigidity occurs when the least area equals 4π

Results extend to manifolds of dimension up to 7

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 $n+2\leq 7$ which satisfies $R\geq 2$ 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.