A topological gap theorem for the ${\pi }_2$-systole of positive scalar curvature 3-manifolds

Abstract

Let $M$ be a closed orientable 3-manifold with scalar curvature greater than or equal to 1. If $M$ has nonvanishing second homotopy group, then it is known that the ${\pi }_2$-systole of $M$ (i.e. the minimal achievable area of homotopically nontrivial spheres) is at most $8\pi$. We prove the following gap theorem: if $M$ is further not a quotient of $S^2\times S^1$, then the ${\pi }_2$-systole of $M$ is no greater than an improved constant $c\approx 5.44\pi$. This statement follows as a new topological application of Huisken and Ilmanen's weak inverse mean curvature flow.

v3 update: proof details added, presentation imporoved. 13 pages, 3 figures. Final version, accepted by Duke Math J