Gap Theorem on Riemannian manifolds using Ricci flow

Abstract

In this work, we consider complete non-compact manifolds with non-negative complex sectional curvature and small average curvature decay. By developing the Ricci flow existence theory, we show that complete non-compact manifolds with non-negative complex sectional curvature and sufficiently small average curvature decay are necessarily flat. We also prove a gap Theorem in the Euclidean volume case. In the compact case, we use the Ricci flow to generalise the celebrated Gromov-Ruh Theorem in this direction.Comment: 29 page