I want to prove that there exists an $\mathbb{R}$-action on the unit disk $D$ in complex plane which preserves $i$ and $-i$, where $D=\{z\in \mathbb{C}~|~|z|\leq 1\}$.
This arises when we define the boundary map of Heegaard Floer homology and not a homework.
I know that one way to prove this is to find a conformal map from the strip $S$ to the disk $D$, where $S=\{z\in \mathbb{C}~|~0<\operatorname{Re} z<1\}$ and use $\mathbb{R}$-action on $S$.
However, I don't know any specific conformal map from $S$ to $D$.
Any ideas?