Let $D$ denote the unit complex 1-dimensional disc, together with the hyperbolic metric $h_D=\frac{4dz\wedge d\bar{z}}{(1-|z|^2)^2}$of curvature $-1$. By Nash's embedding theorem, we can always embed the disc $D$ real-analytically and isometrically into real Euclidean space ${\mathbb{R}}^n$ for some large $n$. (I think $n=5=2\dim_{\mathbb{R}}D+1$ is sufficient.)

I'm wondering whether we have a complex-analytic embeddeding of the disc $D$ into complex Euclidean space. Consider complex Euclidean space ${\mathbb{C}}^n$ with the standard Euclidean metric $h_{\mathbb{C}^n}=dz_1\wedge d\bar{z_1}+\cdots+dz_n\wedge d\bar{z_n}$. Does there exist a holomorphic map $f:D\to {\mathbb{C}}^n$ for some large $n$ that is at the same time an isometry, i.e. $f^*h_{\mathbb{C}^n}=h_D$?

If we write the map $f$ in terms of coordinates $f=(f_1,\ldots, f_n)$, this question has the an equivalent formulation solely in terms of holomorphic functions. This question is asking whether there are holomorphic functions $f_i:D\to {\mathbb{C}}$ on the disc, such that $\frac{4}{(1-|z|^2)^2}=f_1(z)\overline{f_1(z)}+\cdots + f_n(z)\overline{f_n(z)}$ for all $z\in D$.