On an invariant curvature cone along 4-dimensional Ricci flow

In this paper, we study 4-dimensional complete noncompact manifolds (M,g) satisfying Rm(g) $\in {\mathfrak{C}}_{\eta ,\mu }$ via Ricci flow. Under the additional assumption of maximal volume growth, we prove topological and geometric gap theorems. We also study 4-dimensional complete manifolds satisfying a lower bound with respect to ${\mathfrak{C}}_{\eta ,\mu }$ and obtain regularity results for Gromov-Hausdorff limits of complete volume non-collapsed manifolds satisfying such curvature lower bounds.