# dg.differential geometry – Curvature estimate in Hamilton’s Ricci flow paper for traceless $\operatorname{Rm}$ on $4$-dimensional manifold Answer

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## dg.differential geometry – Curvature estimate in Hamilton’s Ricci flow paper for traceless $\operatorname{Rm}$ on $4$-dimensional manifold

In dimension $$4$$it is known that the curvature operator $$\operatorname{Rm} : \Lambda^2(M) \to \Lambda^2(M)$$ admits a block decomposition of the form $$\operatorname{Rm} = \begin{pmatrix} A & B \\ B^T & C\end{pmatrix}$$

Choosing convenient orthonormal basis, the matrices above can be written in diagonal form (ie $$A_{ij} = a_i \delta_{ij}$$, $$B_{ij} = b_i \delta_{ij}$$, $$C_{ij} = c_i \delta_{ij}$$).

In his “Four-manifolds with positive curvature operator” paper, Hamilton proved the following estimates:

If we choose successively positive constants
$$G$$ large enough $$H$$ large enough $$\delta$$ small enough $$J$$ large enough $$\varepsilon$$ small enough
$$K$$ large enough $$\theta$$ small enough, and $$L$$ large enough, with each depending on those chosen before, then the closed convex subset $$X$$ of $$\{M_{\alpha \beta } \geq 0 \}$$ defined by the inequalities

• $$(b_2 + b_3)^2 \leq G a_1 c_1$$
• $$a_3 \leq H a_1$$ other $$c_3 \leq H c_1$$
• $$(b_2 + b_3)^{2 + \delta} \leq J a_1 c_1 (a – 2b + c)^{\delta}$$
• $$(b_2 + b_3)^{2 + \varepsilon} \leq K a_1 c_1$$
• $$a_3 \leq a_1 + L a_1^{1 – \theta}$$ other $$c_3 \leq c_1 + L c_1^{1 – \theta}$$

is preserved under the Ricci flow for a suitable ODE.

For further context, one can see page $$165$$ of this bookwhere it is then claimed without any proof that an immediate consequence of the above estimates is the following estimate on the traceless Riemannian curvature operator:

$$\| \mathrm{Rm}} \| \leq \varepsilon R + C_{\varepsilon}$$

where $$\varepsilon$$ can be arbitrarily small and $$C_{\varepsilon} < \infty$$ is a constant. How does this estimate follow from the previous ones? I’ve tried to express $$\| \mathrm{Rm}} \|$$ in terms of the $$a_i$$‘s $$b_i$$‘sand $$c_i$$‘s in order to use the estimates proved for them, but that led me nowhere. How can one conclude this estimate from the previous ones? I can’t fill this hole in the paper. I’d appreciate any help!

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