Revisions to harmonic extension of a curve by different parametrization

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edited Oct 23, 2015 at 4:43

Paul

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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).

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:

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 is $h:\mathbb{D} \rightarrow \mathbb{C}$ 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 reallyreal question is how many (really different)? For $\Gamma$ the unit circle, the answer is one, you can prove it using Fourier series. butBut for instance, for the image of $e^{i\theta}+\frac{1}{2} e^{i2\theta}$, theI don't know how to proceed.

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).

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:

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 is $h:\mathbb{D} \rightarrow \mathbb{C}$ 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 really question is how many (really different)? For $\Gamma$ the unit circle, the answer is one, you can prove it using Fourier series. but for the image of $e^{i\theta}+\frac{1}{2} e^{i2\theta}$, the don't know how to proceed.

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).

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:

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.

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edited Oct 22, 2015 at 12:53

Paul

914
8
17

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).

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:

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 is $h:\mathbb{D} \rightarrow \mathbb{C}$ 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 really question is how many (really different)? For $\Gamma$ the unit circle, the answer is one, you can prove it using Fourier series. but for the image of $e^{i\theta}+\frac{1}{2} e^{i2\theta}$, the don't know how to proceed.

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).

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:

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.

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).

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:

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 is $h:\mathbb{D} \rightarrow \mathbb{C}$ 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 really question is how many (really different)? For $\Gamma$ the unit circle, the answer is one, you can prove it using Fourier series. but for the image of $e^{i\theta}+\frac{1}{2} e^{i2\theta}$, the don't know how to proceed.

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occurred Oct 18, 2015 at 13:44

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edited Oct 17, 2015 at 13:07

Paul

914
8
17

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).

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

Last but not least, in fact it is the point a realI 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$$\vert h_x \wedge h_y \vert(0)$ (the Jacobian in the planar case) with respect to the parametrization of $\gamma$. More precisely inif 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 enoughtenough to insure thisthe uniqueness.... it is just a normalization for the following question:

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 areis 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.

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).

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

Last but not least, in fact it is the point a real want to understand, even in the planar case. it is a kind a Scharwz theorem, I would like to maximize the the norm of $\vert h_x \wedge h_y \vert$ (the Jacobian in the planar case). More precisely in 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 enought to insure this uniqueness.... it is just a normalization for the following question:

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 are the $\theta \mapsto e^{i\theta}$, up to rotation, but I can't 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.

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).

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:

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.

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edited Oct 17, 2015 at 12:09

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edited Oct 16, 2015 at 14:54

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edited Oct 16, 2015 at 14:35

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asked Oct 16, 2015 at 13:25

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