Quantitative marked length spectrum rigidity for surfaces

We consider a closed negatively curved surface $(M,g)$ with marked length spectrum sufficiently close (multiplicatively) to that of a hyperbolic metric $g_0$ on $M$. We show there is a smooth diffeomorphism $F:M\to M$ with derivative bounds close to 1, depending on the ratio of the two marked length spectrum functions. This is a two-dimensional version of our main result in [But25b].