摘要
Motivated by Ni's optimal gap Theorem in Kähler geometry, in this work, we study the gap phenomenon of Riemannian manifolds with non-negative curvature and sub-critical scaling invariant curvature decay. Using Ricci flow smoothing methods, we show that complete non-compact manifolds with non-negative complex sectional curvature and sufficiently small average curvature decay in an integral sense are necessarily flat. The second main result concerns complete non-compact three-manifolds with non-negative Ricci curvature of quadratic decay. By combining Ricci flow smoothing with the methods in Kähler geometry, we show that if the curvature decays slightly faster even in an average sense, the manifold must be flat. This strengthens a result of Reiris. In the compact case, we generalize the celebrated Gromov-Ruh Theorem in this direction.