The size of isoperimetric surfaces in 3-manifolds and a rigidity result for the upper hemisphere

We characterize the standard ${\mathbb{S}}^3$ as the closed Ricci-positive 3-manifold with scalar curvature at least 6 having isoperimetric surfaces of largest area: $4\pi$. As a corollary we answer in the affirmative an interesting special case of a conjecture of Min-Oo’s on the scalar curvature rigidity of the upper hemisphere…