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Let $U\subset \mathbb{C}$ be a domain and $\Gamma$ some family of curves in $U$ with $\textrm{mod}(\Gamma)<\infty$ and such that $\rho$ is an extremal metric for the modulus. Suppose we are given a $C^\infty$ ($C^1$ should probably suffice) diffeomorphism $f: U\to V\subset \mathbb{C}$. Can we say anything about an extremal metric for $f(\Gamma)$? Hopefully one can write it in terms of $f$ and $\rho$?

There are plenty of well-known ways to control $\textrm{mod}(f(\Gamma))$ in generic situations (say, $f$ is quasiconformal), but I am wondering if we can say something precise about an extremal metric. Thanks.

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    $\begingroup$ Upon first glance, I would not think that you can expect to be able to write an extremal metric explicitly (unless f is conformal). Consider a diffeomorphism f of the unit square to itself that preserves the y-coordinate, but not the x-coordinate. For a family of horizontal curves, the family and hence the metric stays the same, whereas for some families of vertical curves the modulus & metric might be dramatically different. $\endgroup$
    – Lasse Rempe
    Commented Feb 6, 2019 at 1:30

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