This may seem like a crappy question, so I'll be upfront about my motives. I'm reading some sections in Hamilton's first Ricci flow paper, "Three-manifolds with Positive Ricci Curvature." In particular, I'm interested in section 10, "Pinching the eigenvalues." I'm supposed to present this section of the paper in isolation, but I would like to understand how it is used in the larger context of the paper for my own understanding.
My question: Can anyone who happens to know, in as much or as little detail as they like, give a high level summary of how the result of this section is used in the rest of the paper? Any amount of information or perspective would be appreciated.
The main theorem of the paper is that any 3-manifold admitting a metric with everywhere positive Ricci curvature also admits a metric with constant sectional curvature. (A topological corollary is that the manifold must be a quotient of a sphere.)
The idea of the proof is to take the given metric, evolve it by Ricci flow and show that it converges to a constant-curvature metric. Thus there are two main tasks: show that the curvature tensor approaches ("pinches towards") constant-curvature, and show that the flow actually exists and converges after rescaling.
It is the former point that section 10 is concerned with: showing that the curvature tensor approaches constant curvature, i.e. that the eigenvalues approach each other. The exact result that is proved is
$$\frac1{R^2} \left[ (\lambda - \mu)^2 + (\lambda - \nu)^2 + (\mu-\nu)^2 \right] \le C R^{-\delta}$$
where $\lambda,\mu,\nu$ are the curvature eigenvalues, $R$ is the scalar curvature $\lambda + \mu + \nu$ and $C,\delta>0$ are constants. It should be clear that the LHS is a dimensionless measurement of the deviation from constant curvature; so this estimate says that the deviation from constant curvature gets small when the scalar curvature gets large.
Elsewhere in the paper Hamilton proves that as the flow approaches its final time, the scalar curvature blows up to infinity somewhere on the manifold. He also proves bounds on the gradient of the scalar curvature, which then implies that in fact the scalar curvature blows up everywhere on the manifold. Combining this with the eigenvalue pinching estimate we see that in fact the deviation measurement must converge to zero as the flow approaches the final time.
The rest of the paper is then concerned with showing that after rescaling to constant volume, the flow actually converges to a final metric - once we have this, the pinching estimate tells us this metric must have constant curvature.