Concise expression for a specific holomorphic map $f:D\times D\longrightarrow \mathbb{B}^{2}$

Question

Let $\mathbb{B}^{2}\subset \mathbb{C}^{2}$ be the unit ball and $D=\{z\in\mathbb{C}:|z|<1\}$. $f=(f_{1},f_{2}):D\times D\longrightarrow \mathbb{B}^{2}$ is a holomorphic map satisfying $\{det df=0\}\subset{\{z_{1}z_{2}=0\}}$. We can assume that $f(0,0)=(0,0)$. So can we choose another appropriate coordinate system centered at $(0,0)$ so that $f$ has a concise expression under this coordinate system? For example, can we find new coordinates $z_{1}, z_{2}$ such that $f_{1}(z_{1},z_{2})=z_{1}^{m}$, $f_{1}(z_{1},z_{2})=z_{1}^{s}z_{2}^{t}$, where $m$, $s$ and $t$ are integers? This problem comes from our consideration on local expressions of high-dimensional complex hyperbolic metrics with singularities along a simple normal crossing divisor.

Answer 1

Unfortunately, it doesn't look like the condition $\det(df)\subset(z_1z_2=0)$ is strong enough. Consider, for example, the map $$f:(z_1,z_2)\to (z_1, z_2^3+z_1z_2).$$

Clearly, the differential of the map is vanishing along the smooth curve $3z_2^2+z_1=0$ (and so one can change the coordinates in the source to make this curve the axis $z_1=0$). However, the image of this curve $\det(df)=0$ has parameterisation $(-3t^2, -2t^3)$. I.e, the curve $f(\{\det(df)=0\})$ has a cusp at $(0,0)$. So it is rather clear, that there is no change of coordinates that would make the image smooth.