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YCor
YCor
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Quasiconformal Mapsmaps in Arbitrary Dimensionsarbitrary dimensions

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Daniele Tampieri
Daniele Tampieri
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I am aware that a quasiconformal map satifies the formula $\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}$ where $$ \frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z} $$ where $sup\{\mu(z):z \in \text{Domain}\{f\}\}<1$$\sup\{\mu(z):z \in \text{Domain}\{f\}\}<1$ imposes a bound on the eccentricity of the ellipses in the image of $f$. For a multicomplex function $F(z_1, z_2, \dots, z_n)$, one could impose the quasiconformality condition for each variable $z_k$. I have read computer science articles about three-dimensional quasiconformal mappings, but no explicit formula relating the conjugate partial derivative to the partial derivative is given.

Question: Does a generalization of the formula $\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}$ apply to quasiconformal mappings in $\mathbb{R}^{2n+1}$?

I am aware that a quasiconformal map satifies the formula $\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}$ where $sup\{\mu(z):z \in \text{Domain}\{f\}\}<1$ imposes a bound on the eccentricity of the ellipses in the image of $f$. For a multicomplex function $F(z_1, z_2, \dots, z_n)$, one could impose the quasiconformality condition for each variable $z_k$. I have read computer science articles about three-dimensional quasiconformal mappings, but no explicit formula relating the conjugate partial derivative to the partial derivative is given.

Question: Does a generalization of the formula $\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}$ apply to quasiconformal mappings in $\mathbb{R}^{2n+1}$?

I am aware that a quasiconformal map satifies the formula $$ \frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z} $$ where $\sup\{\mu(z):z \in \text{Domain}\{f\}\}<1$ imposes a bound on the eccentricity of the ellipses in the image of $f$. For a multicomplex function $F(z_1, z_2, \dots, z_n)$, one could impose the quasiconformality condition for each variable $z_k$. I have read computer science articles about three-dimensional quasiconformal mappings, but no explicit formula relating the conjugate partial derivative to the partial derivative is given.

Question: Does a generalization of the formula $\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}$ apply to quasiconformal mappings in $\mathbb{R}^{2n+1}$?

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Talmsmen
Talmsmen
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Quasiconformal Maps in Arbitrary Dimensions

I am aware that a quasiconformal map satifies the formula $\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}$ where $sup\{\mu(z):z \in \text{Domain}\{f\}\}<1$ imposes a bound on the eccentricity of the ellipses in the image of $f$. For a multicomplex function $F(z_1, z_2, \dots, z_n)$, one could impose the quasiconformality condition for each variable $z_k$. I have read computer science articles about three-dimensional quasiconformal mappings, but no explicit formula relating the conjugate partial derivative to the partial derivative is given.

Question: Does a generalization of the formula $\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}$ apply to quasiconformal mappings in $\mathbb{R}^{2n+1}$?