Sharpening a gap theorem: nonnegative Ricci and small curvature concentration

We sharpen a gap theorem of Chan & Lee for nonnegative Ricci curvature manifolds that have positive asymptotic volume ratio and small enough scale-invariant integral curvature (so-called "curvature concentration"), by showing that the curvature concentration need only depend linearly on the asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci flow solution with faster than $1/t$ curvature decay, which allows us to shift the limiting contradiction argument to time infinity and thus obtain an explicit bound on the size of the gap.