π2-systolic inequalities for 3-manifolds with positive scalar curvature

We discuss on a new systolic inequality for 3-manifolds with positive scalar curvature. It was proved by Bray, Brendle and Neves that if a closed 3-manifold has scalar curvature at least 1 and has nonzero second homotopy group, then its spherical 2-systole is bounded from above by 8π. Moreover, in the rigidity case the manifold is isometrically covered by a round cylinder. Recently the following gap theorem is proved by the speaker: if the manifold is topologically not a quotient of the cylinder, then the 2-systole is bounded by an improved universal constant which is approximately 5.44π. We will introduce this result along with related backgrounds and ideas. The proof of the new inequality uses Huisken and Ilmanen's weak formulation of inverse mean curvature flow, which is another main component of the talk.