Holomorphic curve with unit norm

Question

Is there an open set $U \subset \mathbb{C}$ and a holomorphic function $\gamma: U \to \mathbb{C}^{2}$ such that $\forall z \in U: \|\gamma(z) \| =1$? if the answer is yes, can the method of unit speed reparametrization of real curves be generalized in the holomorphic setting?

Answer 1

If $U$ is connected, no, unless both components of $\gamma $ are constant. Take $U$ to be the unit disc for convenience. Let $\gamma =(f,g).$ Suppose $|f|^2 + |g|^2 \equiv 1.$ Write $$f(re^{it}) = \sum_{n=0}^{\infty}a_nr^ne^{int}, \,\,g(re^{it}) = \sum_{n=0}^{\infty}b_nr^ne^{int}.$$ We then get

$$2\pi = \int_0^{2\pi}(|f(re^{it})|^2 + |g(re^{it})|^2)\,dt = 2 \pi \sum_{n=0}^{\infty} (|a_n|^2+|b_n|^2)r^{2n}.$$

That's not possible unless $a_n,b_n = 0$ for $n>0.$

Answer 2

You're asking whether there are two (presumably nonconstant) analytic functions $f$, $g$ on (presumably connected) $U$ such that $f \overline{f} + g \overline{g} = 1$. Use the Cauchy-Riemann equations in the Wirtinger form $\dfrac{\partial f}{\partial z} = f'$, $\dfrac{\partial \overline{f}}{\partial z} = 0$, to obtain $\overline{f} f' + \overline{g} g' = 0$. But then (after getting rid of the cases $f' \equiv 0$ or $g \equiv 0$), $\dfrac{\overline{f}}{\overline{g}} = - \dfrac{{g'}}{{f'}}$ is both analytic and conjugate-analytic (away from the zeros of $g$ and $f'$), and therefore must be constant. But that would lead to $|f|$ being constant, which can't happen for a nonconstant analytic function.

Answer 3

As the others have already pointed out, the answer is no. In fact, much more is true. For example, if $\Omega$ is any strictly pseudoconvex domain in $\mathbb{C}^n$ (the unit ball is an example of this), then a holomorphic curve $\phi : \mathbb{C} \to \mathbb{C}^n$ can't have order of contact more than $2$ with the boundary $\partial\Omega$.

Generalizations of the above idea lead to the study of domains of finite type (at least in dimension 2; the higher dimensional case gets a little technical).