I have a question concerning lifts/descents of conformal metrics on Riemann surfaces after filling in a puncture. This is a soft question. I'll illustrate my confusion with an example.
Suppose $f:\mathbb{D}^*\to\mathbb{D}^*$ is a nonconstant holomorphic map on the punctured disk $\mathbb{D}^* = \mathbb{D}\setminus\{0\}$. By uniformization theorem there is a holomorphic covering map $\pi: \mathbb{D}\to\mathbb{D}^*$. One can choose some holomorphic lift of $f$, which we will denote $F:\mathbb{D}\to\mathbb{D}$, so that the following diagram commutes: $\require{AMScd}$ \begin{CD} \mathbb{D} @>{F}>> \mathbb{D}\\ @V{\pi}VV @VV{\pi}V\\ \mathbb{D}^* @>{f}>> \mathbb{D}^* \end{CD} By Riemann's theorem on removable singularities, $f$ admits a holomorphic extension $\hat{f}:\mathbb{D}\to\mathbb{D}$. Let us suppose that $f$ actually preserved the puncture, so $\hat{f}(0)=0$. Let $ds = \gamma(w)|dw|$ be the usual metric on the hyperbolic disk, and equip this metric on the filled disks in the bottom of our diagram. Does it make sense to pullback this filled metric to the covering space? In other words, put $d\tilde{s} = \pi^*(ds) = \gamma(\pi(z))|\pi'(z)||dz|$. Are there any hidden issues with us now looking at the diagram \begin{CD} (\mathbb{D},d\tilde{s}) @>{F}>> (\mathbb{D},d\tilde{s})\\ @V{\pi}VV @VV{\pi}V\\ (\mathbb{D},ds) @>{\hat{f}}>> (\mathbb{D},ds) \end{CD} My concern that by messing with the metrics here some of the standard properties one typically expects to hold now break. For instance, $(\mathbb{D},d\tilde{s})$ seems like it should not be conformally isomorphic to the standard disk anymore, since those points on the boundary which corresponded to $\pi^{-1}(0)$ are now a finite distance away from the center (I think?). So here are my questions:
- Is $F$ still holomorphic with respect $(\mathbb{D},d\tilde{s})$?
- If no, is it still smooth at least?
- The deck transformations for $\pi$ were originally isometries of the disk with respect to the standard hyperbolic metric. Are they isometries of $(\mathbb{D},d\tilde{s})$?
- Most importantly, does something like $||dF_z||_{d\tilde{s}} = ||d\hat{f}_{\pi(z)}||_{ds}$ hold? In other words, given a curve $\gamma$ in $(\mathbb{D},d\tilde{s})$ and its projection $\pi(\gamma)$ in $(\mathbb{D},ds)$, if $\hat{f}$ contracts the length of $\pi(\gamma)$ (as measured by $ds$) by a factor $\alpha\in(0,1)$, does $F$ contract the length of $\gamma$ (as measured by $d\tilde{s}$) by a factor of $\alpha$ as well?
I think (4) and (2) are at least true, but I'm not confident and worry that I've overlooked something (or maybe there is a flaw in the setup I've described which makes this all nonsense).