A complete proof of Hamilton's conjecture

In this paper, we give the full proof of a conjecture of R.Hamilton that for $(M^3,g)$ being a complete Riemannian 3-manifold with bounded curvature and with the Ricci pinching condition $Rc\ge \text{ep}Rg$, where $R>0$ is the positive scalar curvature and $\text{ep}>0$ is a uniform constant, $M^3$ is compact. One of the key ingredients to exclude the local collapse in singularities of the Ricci flow is the use of pinching-decaying estimate. The other important part of our argument is to role out the Type III singularity complete noncompact Ricci flow with positive Ricci pinching condition. We get this goal by obtaining an Ricci expander based on the monotonicity formula of weighted reduced volume.