MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
up vote 4 down vote accepted

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 $\delta<1/4$.

share|cite|improve this answer
Thanks for your suggestions and pointing out the paper by Grove and Karcher! I am wondering if there is more elementary argument in the first step $l=0$, for examples, like the Jacobi vector fields comparison? – BewSMA Jul 1 '12 at 3:44
I do not see "more elementary" construction. – Anton Petrunin Jul 1 '12 at 15:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.