On an invariant curvature cone along 4-dimensional Ricci flow

TL;DR

This paper investigates 4-dimensional Ricci flow on non-compact manifolds with curvature constraints, deriving topological, geometric, and regularity results under volume growth and lower bound conditions.

Contribution

It introduces new gap theorems and regularity results for 4D Ricci flow manifolds constrained by an invariant curvature cone.

Findings

Established topological and geometric gap theorems under maximal volume growth.

Proved regularity of Gromov-Hausdorff limits for volume non-collapsed manifolds with curvature bounds.

Analyzed curvature operator behavior within the invariant cone during Ricci flow.

Abstract

In this paper, we study 4-dimensional complete non-compact manifold with its curvature operator in $\mathfrak{C}_{\eta,\mu}$ via Ricci flow. We obtain topological and geometric gap theorems assuming such manifold has maximal volume growth. We also study 4-dimensional complete manifold with lower bound of $\mathfrak{C}_{\eta,\mu}$ and obtain regularity results for Gromov-Hausdorff limit of complete volume non-collapsed manifolds with lower bound of $\mathfrak{C}_{\eta,\mu}$.