estimate of metric tensors in terms of curvatures

Question

I would appreciate if someone knows how to get the following estimates:

Let $\rho_m$ is a sequence of real numbers approaching $\infty$. Consider a sequence of Riemannian metrics $g^{(m)}$ on $S^3$ such that they satisfy the following two assumptions:

1. For all sectional curvatures of $(S^3, g^{(m)})$ we have $$1-A\rho_m^{-\frac{1}{4}}\leq K(g^{(m)})(x)\leq 1+A\rho_m^{-\frac{1}{4}}$$ for any $x\in S^3$ and $A$ is a positive constant independent of $m$.
2. The covariant derivatives of Riemann curvature tensors have uniform bounds, i.e., $$\sup_{S^3}|\nabla^lRm(g^{(m)})|_{g^{(m)}}\leq B(l),$$ where $l=0,1,2,\ldots$, and $B(l)$ does not depend on $m$.

Want to show the following estimates: for a small positive constant $\delta>0$ we have \begin{equation} ||g^{(m)}-g_0||_{C^l(S^3,g_0)}\leq C(l)\rho_m^{-\delta }, \end{equation} where $g_0$ is the metric with constant sectional curvature one and $C(l)$, $l=0,1,2,\ldots$, are constants independent of $m$.

Any hint or reference will be really appreciated!

Answer 1

All your assumptions survive after arbitrary smooth reparametriztion of $\mathbb S^3$. Therefore maximum you can expect is that for any $n$ there is a reparamtrization of $\mathbb S^3$ such that \begin{equation} ||h^{(m)}-g_0||_{C^l(S^3,g_0)}\leq C(l)\rho_m^{-\delta }, \end{equation} holds for the pullback $h^{(m)}$ of $g^{(m)}$.

This is indeed true, the exponential maps give such reparametrizations between balls of radius $r<\pi$ for all larde $n$. It remains to glue two such balls in a neigborhood of the equator. To do this choose a nice partition of unity $m_1, m_2$ for these balls, rotate parametrizations so that they are almost identical near equator and send a point to the baricenter of its images with the masses $m_1$ and $m_2$ (see How to Conjugate $C^1$-close group actions by Grove and Karcher).

It seems that $\delta=1/4$ will do, but for sure it works for any $\delta<1/4$.