Singular surfaces in $\mathbb{C}^2$ can be described by knots or links: $z_1^2-z_2^2=0$ is the Hopf link, $z_1^2-z_2^3=0$ is the trefoil, etc. I want to know how to determine the metric on the surface around these singularities.
If I proceed how I might usually, I can do this for the Hopf link. Starting with a metric on $\mathbb{C}^2$
$$ds^2=dz_1d\bar{z}_1^2-dz_2d\bar{z}_2^2$$
(This negative sign looks strange but iI need to be that for the coordinates I am going to use...I don't see a real problem with it?) I want to determine the metric around an embedded surface described by $z_1^2-z_2^2=\alpha^2$, and at $\alpha=0$ this surface will be singular. Using the coordinates $$z_1=\alpha \cosh\theta,\qquad z_2=\alpha\sinh\theta$$ The induced metric from $\mathbb{C}^2$ is simply the conical one, $$ds^2=d\alpha^2+\alpha^2d\theta^2,$$ with the singularity at $\alpha=0$.
But, if I try this with the trefoil, $z_1^2-z_2^3=\alpha^2$ I can't seem to find any coordinates that are amenable. If I repeat the same style, $$z_1=\alpha \cosh\theta,\qquad z_2=\alpha^{2/3} \cosh^{2/3}\theta$$ I can satisfy the polynomial equation but the metric becomes very complicated. If I try something like $$z_1=\alpha e^{i\theta},\qquad z_2=\alpha^{2/3}e^{2i\theta/3}$$ Then the metric might look better but the polynomial won't be satisfied with a real $\alpha$.
Can anyone give some advice?