# 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 (i.e $$A_{ij} = a_i \delta_{ij}$$, $$B_{ij} = b_i \delta_{ij}$$, $$C_{ij} = c_i \delta_{ij}$$).

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

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

is preserved under the Ricci flow for a suitable ODE.

For further context, one can see page $$165$$ of this book, where 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:

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

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 $$\| \mathring{\mathrm{Rm}} \|$$ in terms of the $$a_i$$'s, $$b_i$$'s and $$c_i$$'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!

• Could you clarify what book you are referring to? When I click on the link, I get an article that starts on page 167. Jun 24, 2022 at 2:12
• @DeaneYang I guess it was a mistake on my part to call it a book. The link is correct. The page I referred to is page $165$ of the PDF and page $329$ of the article (where the first estimates I cited were proved, which the authors then claim in the corollary $5.2.7$ (located at the end of page $168/332$) implies the estimate I'm having trouble with Jun 24, 2022 at 2:17

After a lot more time thinking about it, I think I've figured it out. Let $$\Gamma$$ be a constant such that $$\Gamma \|g \odot g \| = 1$$ (where $$\odot$$ denotes the Kulkarni Nomizu product, and the only reason I don't make $$\Gamma$$ explicit here is because it depends on the conventions for the definitions of $$g \odot g$$ and the inner product, but they all agree up to a factor). Clearly, we can control $$\| \mathrm{Rm} \|^2$$ once we control $$A, B, C$$, i.e
$$\| \mathrm{Rm} \|^2 \leq\beta( \|A \|^2 + 2\|B \|^2 + \|C \|^2)$$
where $$\beta$$ is some constant. For convenience, I'll commit the (harmless) abuse of denoting both the operators on $$\Lambda^{2}_{\pm}(M)$$ and its associated $$(0, 4)$$ tensors by the same letters. Recalling that $$R = \mathrm{tr}(A) = \mathrm{tr}(C)$$, by a very similar argument to the one seen in the end of Chapter 9 of Peter Topping's lectures on Ricci flow , we see that the estimates provided by Hamilton imply that for all $$\varepsilon > 0$$ (however small) there exists a constant $$C_{\varepsilon}$$ such that $$2 \max\left\{\left\|A - \frac{2}{3}\Gamma R (g \odot g)\right\| , \left\|C - \frac{2}{3} \Gamma R (g \odot g)\right\| \right\} \leq \varepsilon R+C_{\varepsilon}$$ The norm of $$B$$ is controlled by $$b_3^2$$ which is controlled by $$(b_3 + b_2)^{2 + \delta}$$ which in turn is controlled by $$K a_1 c_1$$. The latter is controlled by the estimate above. Putting all this together we get almost exactly the desired following inequality (it's worth noting this may not exactly be $$\mathring{\mathrm{Rm}}$$ and some undesirable $$\varepsilon^2 R^2$$ terms may appear, but since we'll just use the fact that $$\varepsilon$$ can be made arbitrarily small in the end, this will not matter).
$$\left\|\mathrm{Rm} - \frac{2}{3}\Gamma R (g \odot g)\right\| \leq \varepsilon R+C_{\varepsilon}$$