There are two ways, how one can define a conformal structure on an Riemann surface. Either in terms of a complex structure, or in terms of a Riemannian metric.
I understand, that the Möbius transformations form the set of all holomorphic automorphisms of the Riemann sphere $\hat{\mathbb{C}}$. What I don't understand is, why they can be understood as the (complete) set of conformal automorphisms in the sense of Riemannian geometry. More concretely: I don't see, that they act transitively on the set of conformally equivalent metrics:
Jost defines a conformal Riemannian metric on a Riemann surface in local coordinates as \begin{equation} \lambda(z)^2 dz d\bar{z} \end{equation} with $\lambda$ being a positive valued, smooth (in the sense of real analysis, I guess) function.
Now consider the Riemann sphere and choose (for simplicity) the metric with $\lambda(z) = 1$ on the Riemann sphere. If we now caluclate the pullback of this metric along a Möbius transformation \begin{equation} Z(z) = \frac{az + b}{cz + d} \end{equation} with $ad - bc = 1$, then the pullback metric is given by: \begin{equation} \left(\frac{1 + z \bar{z}}{|az + b|^2 + |cz + d|^2}\right) dz d\bar{z} \end{equation} But $\lambda(z)$ was just required to be smooth and positive (i.e. real) valued and of course one should find such functions on the Riemann sphere, which cannot be written as: \begin{equation} \frac{1 + z \bar{z}}{|az + b|^2 + |cz + d|^2} \end{equation}
Hence, there are conformal metrics, which cannot be reached by a Möbius transformation from the metric with $\lambda(z) = 1$. I.e.: There are different classes of metrics, which are related by a conformal transformation (in the sense of Riemannian geometry), but not by a Möbius transformation. But now the uniformization theorem tells us, that there is only one equivalence class of conformal structures on the Riemann sphere.
What am I overlooking?