# Which plane curves can be harmonically parametrized?

In this question, a “(closed oriented plane) curve$$\Gamma$$ will mean a continuous map $$f \colon \mathbb{U} \to \mathbb{C}$$ where $$\mathbb{U} := \{z\in\mathbb{C} : |z|=1\}$$ is the unit circle, modulo right-composition by orientation-preserving homeomorphisms $$\varphi\colon\mathbb{U}\to\mathbb{U}$$. Any such $$f$$ (i.e. any of the $$f\circ\varphi$$ defining the curve) will be called a “parametrization” of the curve $$\Gamma$$.

I'm willing to add any reasonable regularity conditions on $$\Gamma$$ if they help in answering the question, e.g., piecewise $$C^1$$ or even $$C^\infty$$.

Say that a curve $$\Gamma$$ is harmonically parametrizable iff it admits a parametrization $$f \colon \mathbb{U} \to \mathbb{C}$$ (see above) such that the Fourier coefficients $$(c_k(f))_{k\in\mathbb{Z}}$$ of $$f$$ are zero for all $$k<0$$: this then allows us to see $$f$$ as the restriction to $$\mathbb{U} = \partial\Delta$$ of a continuous function $$F$$ on the closed unit disk $$\overline{\Delta}$$ that is holomorphic on the open unit disk $$\Delta$$ (namely, $$F$$ is the Poisson integral of $$f$$, or equivalently $$F(z) = \sum_{k=0}^{+\infty} c_k z^k$$; see also this question).

More generally, if $$k_0\in\mathbb{Z}$$ say that $$\Gamma$$ is $$k_0$$-harmonically parametrizable iff it admits a parametrization $$f \colon \mathbb{U} \to \mathbb{C}$$ that $$c_k(f)=0$$ for all $$k. (So “harmonically parametrizable” means “$$0$$-harmonically parametrizable”, and the larger $$k_0$$ the stronger the condition.)

For example, if $$\Gamma$$ is a positively-oriented, sufficiently regular (I think “rectifiable” suffices), Jordan curve, then the Riemann conformal mapping theorem guarantees that we can find $$F \colon \Delta \to V$$ bijective and conformal, where $$V$$ is the bounded component of the complement of $$\Gamma$$, extending continuously to $$\partial\Delta = \mathbb{U}$$ (assuming $$\Gamma$$ is sufficiently regular), which shows that $$\Gamma$$ is harmonically parametrizable.

So, two related questions:

• Can we find a necessary and sufficient condition on a curve $$\Gamma$$ (again, assumed sufficiently regular if this is useful) for it to be harmonically parametrizable?

• What about $$k_0$$-harmonically parametrizable for $$k_0<0$$? Can we find, at least, a reasonable sufficient condition analogous to the “Jordan curve” condition for being harmonically parametrizable?

(For $$k_0>0$$, being $$k_0$$-harmonically parametrizable seems to say something about the moments of $$\Gamma$$ for the harmonic measure vanishing, and this is probably less interesting.)

PS: This question seems related to a kind of converse to the one I'm asking (I want to know if there exists a parametrization change which becomes harmonic, that other question is about what happens upon such a change).

First of all, this is a purely topological problem. Let $$\gamma:U\to C$$ be a curve. Your question is when this curve can be reparameterized so that the new parameterization is by boundary values of an analytic function in the unit disk.
It is necessary that $$\gamma$$ extends to a topologically holomorphic map of the unit disk to $$C$$ (topologically holomorphic map, a. k. a. polymersion, is a map which is topologically equivalent to $$z\mapsto z^n$$ near every point.
This condition is also sufficient. Indeed, once we have a topologically holomorphic map, we can pull back the complex analytic structure to the disk, and then use the Uniformization theorem. So for every topologically holomorphic map $$f$$ there is a homeomorphism $$\phi$$ of the disk such that $$f\circ\phi$$ is holomorphic.