Length distortion of volume-preserving Lipschitz mappings

We show that every area-preserving Lipschitz map between metric surfaces distorts the length of almost every curve by at most a multiplicative factor, answering a question posed by the first author and Ntalampekos. We give two different proofs. The first is entirely intrinsic and extends to higher dimensions under additional assumptions. The second relies on non-smooth uniformization theory and yields the conclusion that the family of curves intersecting the purely 2-unrectifiable part of a metric surface in a set of positive length is exceptional.