In dimension three the curvature operator has three eigenvalues \(\lambda \ge \mu \ge \nu \) — twice the sectional curvatures, normalised so that the scalar curvature is the trace \(\operatorname {scal}= \lambda +\mu +\nu \) and the Ricci eigenvalues are \(\tfrac 12(\mu +\nu ), \tfrac 12(\lambda +\nu ), \tfrac 12(\lambda +\mu )\) — and the reaction term \(\operatorname {Rm}^2 + \operatorname {Rm}^\# \) of the evolution equation 66 is diagonal in the same frame. The associated ODE is

Along its solutions on \([0,T]\):

1. the ordering \(\lambda \ge \mu \ge \nu \) is preserved (le_preserved_lm, le_preserved_mn);

2. positive Ricci curvature, \(\mu + \nu {\gt} 0\), is preserved (ricci_pos_preserved);

3. \(\lambda \le C(\mu +\nu )\) is preserved for every \(C \ge 1/2\) (bound_preserved);

4. given (1)–(3) at time \(0\), for \(0 \le \delta \) with \(\delta (2C+1) \le 1\) the ratio \((\lambda -\nu )/(\mu +\nu )^{1-\delta }\) is nonincreasing (pinching_antitone).

Item (4) is the pinching estimate of Hamilton’s paper (Theorem 10.1) for the ODE: the traceless part of the curvature is dominated by a smaller power of the scalar curvature, so wherever the curvature blows up it becomes constant-sectional to leading order.

In dimension three positive Ricci curvature is preserved by the flow, and the eigenvalues pinch together relative to their size: the traceless part of \(\operatorname {Ric}\) is dominated by \(\operatorname {scal}^{1-\delta }\) for some \(\delta {\gt} 0\).

This is Lemma 89 transported from the ODE to the PDE by the tensor maximum principle 78: the sets \(\{ \mu +\nu \ge \epsilon \, \operatorname {scal}\} \), \(\{ \lambda \le C(\mu +\nu )\} \) and \(\{ \lambda -\nu \le C'(\mu +\nu )^{1-\delta }\} \) are closed, convex, and ODE-invariant. Convexity: the largest eigenvalue is convex and the smallest concave in the curvature operator, so \(\lambda - \nu \) and \(\lambda \) are convex, \(\mu + \nu = \operatorname {scal}- \lambda \) is concave, and \(t \mapsto t^{1-\delta }\) is concave increasing; each set is a sublevel set of a convex function against a concave one.

What is missing is more than a transfer. In order: the evolution equation 66 itself; the maximum principle on the bundle of curvature operators with the metric evolving on the fibres (Uhlenbeck’s trick), where 78 is proved for a fixed fibre; the convexity of the eigenvalue-defined sets, which needs the variational characterisation of eigenvalues; and Nagumo’s condition for these sets from the ODE invariance of 89, which is stated for eigenvalues and must be lifted to the operator ODE by equivariance.

Special to dimension three, where the Weyl tensor vanishes and \(\operatorname {Rm}\) is determined by \(\operatorname {Ric}\). The higher-dimensional analogues (Hamilton 1986 in dimension four, then Böhm–Wilking) are substantially harder and are not on this road.

\(|\nabla \operatorname {scal}| \le \eta \, \operatorname {scal}^{3/2}\) up to lower-order terms, so the scalar curvature is comparable at every pair of points as the flow shrinks.