Which plane curves can be harmonically parametrized?

Question

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 $k<k_0$. (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).

Answer 1

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.

Second, this topological problem is solved in the paper (under some smoothness conditions on the curve):

C. Curley and D. Wolitzer, Btranched immersions of surfaces, Michigan Math. J. 33 (1986) 131-144.

Unfortunately the answer is somewhat complicated and I do not reproduce it here.