0
$\begingroup$

We have that the pseudo-hyperbolic metric in the open unit disk $\mathbb D$ is defined by
$$ \rho(z,w) = |\phi_w(z)|, \qquad \phi_w(z) = \frac{w - z}{1 - \overline w z}$$ where $z,w \in \mathbb D.$ It is also know that $\phi_w(z)$ is an automorphism of the unit disk.

Then for $r$, $0 \lt r \lt 1$ and $\alpha \in \mathbb D$, the set $$ P(\alpha,r) = \{ z \in \mathbb D : \rho(z,\alpha) \lt r \}$$ is the pseudo-hyperbolic ball with center $\alpha$ and radius $r.$

I'm able to show that $P(\alpha,r)$ = $\phi_\alpha(r\mathbb D).$ Then it follows (at least intuitively) that since $\phi_\alpha$ maps circles to circles that $\phi_\alpha(r\mathbb D) $ is a Euclidean ball, which implies that $P(\alpha,r)$ is a Euclidean ball.

I have two problems that I've been struggling with:

(1): I'd like to have a few algebraic steps in order to show more rigorously that $\phi_\alpha(r\mathbb D) $ is indeed a Euclidean ball.

(2): More importantly, determining explicitly the center and radius of the Euclidean ball. I've found online (numerous places) that the Euclidean center $\beta$ and radius $R$ are given by $$ \beta = \frac{(1 - r^2)\alpha}{1 - r^2 |\alpha|^2} \qquad \text{and} \qquad R = \frac{r(1 - |\alpha|^2)}{1 - r^2 |\alpha|^2}$$

respectively. Every article says that the center and radius can be determined from a straightforward calculation but never actually shows the calculation. (I'm beginning to think it's a secret.) Rudin remarks in his book Function Theory of the Unit Ball that $z \in P(\alpha,r)$ if and only if $|\phi_\alpha(z)| \lt r.$ He then says that if we square this and use the definition of $\phi_\alpha(z)$, then a little manipulation will give the desired formulas for the center and radius. I've tried squaring $|\phi_\alpha(z)| \lt r$ and then working forward and backwards to come up with the algebraic steps to solve for the center and radius but have not been successful. Maybe there is also a more sophisticated way to find the center and radius using the fact that $\phi_\alpha(z)$ is an automorphism of the unit disk

I'm hoping that someone could help me out with problems (1) and (2). Any help would be greatly appreciated!

$\endgroup$
0
$\begingroup$

Let $w \in B(0,1)$, then we can write $w = |w|e^{i\theta}$ for some $\theta \in \mathbb{R}$. Consider the ball $B(0,r)$, then what line through the origin will $\phi_{|w|}(B(0,r))$ be symmetric about?

Once you figure this out, you should be able to identify the two intersection points of the circle and line, say $\alpha$ and $\beta$. Then, by the symmetry, the true euclidean center will be the midpoint between $\alpha$ and $\beta$ and its radius will follow. Note then, once you have the euclidean center and radius, the only difference between $|w|$ and $w$ is the rotation $e^{i\theta}$, which does not affect the radius and will completely determine the Euclidean ball with center: midpoint $m_{\alpha,\beta}$.

As for justification of circles to circles, note that: $P(w,r) = \phi_w(P(0,r))$. Using what you have above with this should be sufficient.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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