  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 $Rc\geq \ep R g$, where $R>0$ is the positive
scalar curvature and $\ep>0$ 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.
