  In this work, we use the Ricci flow approach to study the gap phenomenon of
Riemannian manifolds with non-negative curvature and sub-critical scaling
invariant curvature decay. The first main result is a quantitative Ricci flow
existence theory without non-collapsing assumption. We use it to show that
complete non-compact manifolds with non-negative complex sectional curvature
and sufficiently small average curvature decay are necessarily flat. The second
main result concerns three-manifolds with non-negative Ricci curvature of
quadratic decay. By combining our newly established curvature estimate and
method in K\"ahler geometry, we show that if the curvature decays slightly
faster even in average sense, the manifold must be flat. This strengthens a
result of Reiris. In the compact case, we use the Ricci flow regularization to
generalize the celebrated Gromov-Ruh Theorem in this direction.
