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I want to understand Prop 6 in the paper "Convergence of the Q-curvarture flow on $S^4$" by Simon Brendle. I understand that for every $p\in B^4$, where $B^4=\{x\in\mathbb{R}^5: |x|\leq 1\}$, $$\phi(x)=p+\frac{1-|p|^2}{1+2\langle p,x\rangle+|p|^2}(x+p) $$defines a conformal diffeomorphism from $S^4$ to $S^4$, where $S^4=\{x\in\mathbb{R}^5: |x|=1\}$ is the sphere (This fact was proved in the paper in P.11). My questions are:

(i) How is the formula derived? I want to understand how can we know the conformal map is related to some $p\in B^4$ and why the formula is in this particular form. Of course, if I already know the formula I can just check by definition it is a conformal map.

(ii) Is every conformal diffeomorphism from $S^4$ to $S^4$ in this form? I think this is related to Liouville's Theorem.

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    $\begingroup$ actually the set that you call $B^4$ is usually called $B^5$ $\endgroup$
    – Mircea
    Commented Jun 12, 2012 at 9:45

5 Answers 5

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Conformal diffeomorphisms of $S^n$ correspond to hyperbolic isometries of hyperbolic space $\mathbb H^{n+1}$ -- the idea is to think of $S^n$ as the visual sphere for hyperbolic space, all conformal diffeos extend uniquely to a hyperbolic isometry.

For (ii), no. Hyperbolic isometries have various forms. Your $\phi$ does not give you any elliptic or parabolic elements.

You can work out similar formula for a general conformal map. What you do is you write-out your hyperbolic isometry in your favourite model of hyperbolic geometry, and then restrict it to the sphere. Ratcliffe's book is maybe one of the better sources for these types of formula, but they're available in many places.

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Another approach to conformal maps is to use stereographic projection and work on $R^n$. There, you basically want to show that the only conformal maps are higher dimensional analogues of Mobius transformations. I'm not sure, but i think the survey article by Lee and Parker on the Yamabe problem in the Bulletin of the AMS goes through the details. It's also a really nice exercise to try to work this all out yourself.

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  • $\begingroup$ Like that one gets exactly all conformal maps of $S^n$ having a fixed point (the one corresponding to infinity), which also are all of those given by the above formula, as pointed out by Ryan Budney. Is there a more general formula for all the conformal diffeomorphisms? $\endgroup$
    – Mircea
    Commented Jun 12, 2012 at 14:34
  • $\begingroup$ @Mircea: just guessing: aren't the remaining ones conjugate to the ones in the OP by a rotation of S^n sending the point at infinity to a point in (the open subset corresponding to) R^n? Or are there any others? $\endgroup$
    – Qfwfq
    Commented Oct 3, 2017 at 8:28
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See my answer to a question about conformal diffeomorphims of a sphere.

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Here is a geometric illustration for the conformal map $\phi$ on $S^1$. On $S^n$ this is the same.

The conformal map $\phi$ on $S^1$

I hope this is enough to realize the formule given by Brendle in his paper.

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    $\begingroup$ Could you explain please? Your answer needs to be expanded substantially if you want it to be understood. Also note that you can embed images into posts. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 3, 2017 at 5:39
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When I first saw this equation, I think it was just the inversion of unit sphere with respect to some sphere with center $-p$ and radius $r$, then I tried to write it in a form that look likes inversion: $$\varphi=\frac{1-|p|^2}{|x+p|}\cdot\frac{x+p}{|x+p|}+p$$ So actually you fix point $p$ first, then draw a sphere $T$ with origin at $-p$ and radius $r=\sqrt{1-|p|^2}.$ Then above map is just the inversion of unit sphere with respect to this sphere $T$. This map actually looks similar to the conformal maps constructed by Yau "Lectures on Differential geometry" and Nadirashvili "A differential invariant related to the first steklov eigenvalue", but I think they built such sphere $T$ with origin outside of unit sphere. In this way I guess you can construct infinitely many different transformations.

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