Gap Theorem on Riemannian manifolds using Ricci flow

Abstract: 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.