Does there exist a holomorphic map from a neighborhood of $\mathbb C$ to $S^3 \subseteq \mathbb C^2$?



Holomorphic map from a neighborhood in $\mathbb C$ to S^3
Suppose such a map exists and is non-constant. Let $f,g$ be its components. By composing the map with a holomorphic function on the right and with an element of $U(2)$ on the left, we can assume that in a neighborhood of 0 we have $f(z)=z$ (i.e., we can take $f$ as a local coordinate) and $g(z)=1+az+\cdots$ with $a\neq 0$. Now let $b$ be a complex number such that $Re(ab)\neq 0$ and set $h(t)=bt, t\in\mathbb{R}$. Then on the curve given by $t\mapsto h(t)$ the equation $|f|^2+|g|^2=1$ becomes $$b^2t^2+1+2Re(ab)t+ o(t)=1,$$ which should hold for all $t$ sufficiently close to 0. This is impossible.


upd: this argument generalizes to maps $\mathbb{C}\to S^{2n-1},n>2$ and also implies the following "generalized maximum principle": for a holomorphic map $f:D\to\mathbb{C}^n,D\subset \mathbb{C}$ connected and contained in the closure of its interior, the maximum of $|f|$ can't be attained in an interior point, unless $f$ is constant. To see this one can notice that the moduli of the components of $f$ are subharmonic, so their sum, $|f|^2$, satisfies the maximum principle.


