# harmonic extension of a curve by different parametrization

Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ (i.e $\Delta h=0$ and $h=\gamma$ on $\partial\mathbb{D}$).

If I change the parametrization of $\gamma$ into $\gamma\circ \phi$, where $\phi$ is a diffeomorphism of $S^1$ :

1)is that true that the new harmonic extension can be write $h\circ \psi$ where $\psi$ is a diffeomorphism of $\mathbb{D}$. (it seems to be true in the planar convex case thanks to Rado-Kneser-Choquet theorem).

2) is there any chance to know how to "compute" $\psi$ with respect to $\phi$?

Last but not least, in fact it is the point I reallly want to understand, even in the planar case. it is a kind a Scharwz theorem: I would like to maximize the the norm of $\vert h_x \wedge h_y \vert(0)$ (the Jacobian in the planar case) with respect to the parametrization of $\gamma$. More precisely if the area bounded by the curve is equal to $\pi$ in the planar case, else we consider the area of the unique minimal surface when the curve is not planar but nice enough to insure the uniqueness.... it is just a normalization for the following question:

3) is there (always) a parametrization of the curve such that $\vert h_x \wedge h_y \vert(0) \geq 1$ (I am sure but I get no proof). is there even an optimal one? is there is many? For the circle, the only one is the $\theta \mapsto e^{i\theta}$, up to rotation, but I can't say anything for another curve.

I am open to any suggestion, I have read quickly the book of Peter Duren, Harmonic maps into the plane, but i seems to be not enough...

Edit (10/22/15):

My problem is indeed link to minimal surfaces, but not to the Palteau problem directly, but to the exterior Plateau problem. A simple by product is the following question:

let $\Gamma$ a planar curve, is there $h:\mathbb{D} \rightarrow \mathbb{C}$ such that $\frac{dh}{dz}=1$ and $h_{\vert \partial \mathbb{D}}$ is a monotone parametrisation of $\Gamma$. The answer is probably (clearly?) yes. But the real question is how many (really different)? For $\Gamma$ the unit circle, the answer is one, you can prove it using Fourier series. But for instance, for the image of $e^{i\theta}+\frac{1}{2} e^{i2\theta}$, I don't know how to proceed.

• I would suggest that you change the title of your question into something more descriptive. This will help catch the attention of the relevant people. – André Henriques Oct 16 '15 at 14:30

The answer to (1) in the space case is a definite 'no'. The actual image 'surface' $h(\Delta)$ in $\mathbb{R}^3$ can actually change when you reparametrize with $\phi:S^1\to S^1$, in which case, there can't be a reparametrization of the disk to match.
• Thanks for the Answer. The first point was indeed "easy". In fact I know quite well the solution of Douglas-Rado (through Struwe's book). lik in lawson, he minimise the energy among conformal parametrisation. But since my goal is 3)(1) and 2) are a try to make more explicit), it can't be adapted here. If we consider $h:z\mapsto z+ z^2/4$ , $\vert h(\mathbb{D})\vert=9/8$, but $\vert h'(0)\vert=1$ and any conformal reparametrization(using mobius group) will be below $9/8$. – Paul Oct 18 '15 at 16:15