A topological gap theorem for the $\pi_2$-systole of positive scalar curvature 3-manifolds

TL;DR

This paper establishes a new topological gap theorem for the $oldsymbol{\pi_2 ext{-systole}}$ of positive scalar curvature 3-manifolds, improving the upper bound for nontrivial cases using inverse mean curvature flow.

Contribution

It proves a sharper upper bound for the $oldsymbol{\pi_2 ext{-systole}}$ in certain 3-manifolds, extending previous results with novel topological methods.

Findings

The $oldsymbol{\pi_2 ext{-systole}}$ is at most approximately $5.44oldsymbol{\pi}$ for non-quotient manifolds.

The result applies to manifolds with scalar curvature ≥ 1 and nonvanishing second homotopy group.

The proof uses Huisken and Ilmanen's weak inverse mean curvature flow.

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.