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Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E∗(du^2+dv^2)$ and where $E>0$ is an harmonic function?

I know that this metric is a special kind of Liouville's metric, but is there a possible application?

$E=e^f$ and if $f=logE$, with $E>0$ and harmonic, we have $f_u$$_u$+$f_v$$_v$$+(f_u^2+f_v^2)=0$, that seems the problem of the artificial boundary of the KPZ equation with Cole-Hopf tranformation.

Thanks you in advance!

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  • $\begingroup$ I remember having read the words "coordonnées isothermes" in Einstein's "Théorie du champ unifié", so maybe an application to theoretical physics is possible. $\endgroup$
    – Sylvain JULIEN
    Commented May 21, 2016 at 9:34
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    $\begingroup$ Every surface admits local isothermal coordinates (a result going back to Gauss, at least for real analytic surfaces). These coordinates are very convenient for calculations and make some results more transparent. They are particularly useful when studying minimal surfaces, for example. $\endgroup$
    – José Figueroa-O'Farrill
    Commented May 21, 2016 at 10:22
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    $\begingroup$ Thanks for the comments!! I'm looking for something on isothermal coordinates, but with "focus" on $E$ harmonic. $\endgroup$
    – MathDG
    Commented May 21, 2016 at 10:37

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