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...

Thanks in advance.

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.

  • 2
    $\begingroup$ 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. $\endgroup$ 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.

It sounds as though you are not aware of the work of Douglas and Rado on minimal surfaces using harmonic parametrization, which definitely addresses your questions in the space case. I'd recommend that you consult a source such as the discussion of this topic in H. Blaine Lawson's Lectures on minimal submanifolds for the details.

I think that the answer to (2) is also 'no' in any realistic sense, but that may depend on your ideal of 'realistic'.

  • $\begingroup$ 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 $9/8$. $\endgroup$
    – Paul
    Oct 18 '15 at 16:15

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