# Extremal metric for image of a curve family

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.

• 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. – Lasse Rempe-Gillen Feb 6 at 1:30