I have been reading John H. Hubbard's book *Teichmüller Theory vol. 1* and I am a little bit concerned with his definition of Quasiconformal Surface.

Definition:A Quasiconformal surface $S$ is a topological surface with a Riemann-surface structure; two Riemann surface structures on $S$ define the same quasiconformal structure if the identity map between them is quasiconformal.If $S_1,$ $S_2$, are two quasiconformal surfaces, a map $f:S_1 \to S_2$ is quasiconformal if it is a quasiconformal homeomorphism for one, hence all, analytic structures on each $S_1$ and $S_2$. This implies that all quasiconformal maps are isomorphisms.

I do understand the definition and one it's advantage is that it is easy to prove that if two compact quasiconformal surfaces $S_1$ and $S_2$ are homeomorphic, then they are isomorphic as quasiconformal surfaces.

However, I am use to define structure like that in a similar way as we define manifolds. Hence, if I had to give a definition of Quasiconformal Surface, I would say that it is a topological surface $S$ together with a maximal atlas that contains all the charts for which the transition maps are quasiconformal maps.

My question is are those 2 definitions equivalent (can I define a specific Riemann surface structure from a maximal atlas which is unique up to quasiconformal maps?), my first idea to solve this problem was to use the measurable Riemann mapping theorem which allow us to find local quasiconformal maps that will satisfy a Beltrami equation. The problem is that I am not sure if this argument will work for any surface, we might have to do the process on infinitely many charts (I want to consider surfaces with puncture and boundaries as well.)